# Wide-band Unambiguous Quantum Sensing via Geodesic Evolution

Ke Zeng  Key Laboratory of Atomic and Subatomic Structure and Quantum Control
(Ministry of Education), and School of Physics, South China Normal University,
Guangzhou 510006, China  Xiaohui Yu  Key Laboratory of Atomic and Subatomic
Structure and Quantum Control (Ministry of Education), and School of Physics,
South China Normal University, Guangzhou 510006, China  Martin B. Plenio
Institut für Theoretische Physik und IQST, Universität Ulm, Albert-Einstein-
Allee 11, 89081 Ulm, Germany  Zhen-Yu Wang<EMAIL_ADDRESS>Key
Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of
Education), and School of Physics, South China Normal University, Guangzhou
510006, China  Guangdong Provincial Key Laboratory of Quantum Engineering and
Quantum Materials, and Guangdong-Hong Kong Joint Laboratory of Quantum Matter,
South China Normal University, Guangzhou 510006, China

###### Abstract

We present a quantum sensing technique that utilizes a sequence of $\pi$
pulses to cyclically drive the qubit dynamics along a geodesic path of
adiabatic evolution. This approach effectively suppresses the effects of both
decoherence noise and control errors while simultaneously removing unwanted
resonance terms, such as higher harmonics and spurious responses commonly
encountered in dynamical decoupling control. As a result, our technique offers
robust, wide-band, unambiguous, and high-resolution quantum sensing
capabilities for signal detection and individual addressing of quantum
systems, including spins. To demonstrate its versatility, we showcase
successful applications of our method in both low-frequency and high-frequency
sensing scenarios. The significance of this quantum sensing technique extends
to the detection of complex signals and the control of intricate quantum
environments. By enhancing detection accuracy and enabling precise
manipulation of quantum systems, our method holds considerable promise for a
variety of practical applications.

#### Introduction. —

Accurate characterization of the qubit environment holds significant
importance across a range of applications, spanning from quantum information
processing to quantum sensing [1, 2, 3, 4]. A widely utilized technique for
achieving this is the implementation of dynamical decoupling (DD) pulse
sequences [5, 6]. These sequences serve to filter out environmental noise,
thereby extending the quantum coherence time, as well as to extract and
amplify signals of specific frequencies [3, 4, 7]. Consequently, qubits under
such sequences become highly sensitive quantum sensors, presenting diverse
applications. For instance, nitrogen-vacancy (NV) centers [8, 9, 10] subjected
to DD pulse sequences enable nanoscale nuclear magnetic resonance (NMR) [11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 7], spin label
detection [27, 28, 29], spin cluster imaging [30, 31, 32, 33, 34, 35, 36, 37,
38], and AC field sensing [39, 40, 41, 42, 43, 44, 45, 46]. Furthermore, they
can be employewd for controlling nearby single nuclear spins [47, 48, 49, 50,
51, 52] in the context of quantum information processing [53, 54], quantum
simulations [55], and quantum networks [56, 57, 58, 59].

However, DD pulse sequences used for quantum sensing, such as the commonly
employed Carr-Purcell-Meiboom-Gill (CPMG) [60, 61] and XY8 [62] sequences
encounter an issue known as spectral leakage and spurious resonance. These
complications make the interpretation of the sensor’s recorded signal
challenging and can result in ambiguous signal identification [63, 64]. DD
pulses introduce abrupt temporal state flipping of the qubit sensor, causing a
pronounced frequency modulation at a specific frequency [indicated by the gray
line in Fig. 1(d)]. This modulation leads to a strong resonance at the
flipping frequency, enabling frequency-selective sensing. However, it also
generates resonances at other frequencies, as evident in its Fourier transform
[represented by the gray squares in Fig. 1(d)]. Consequently, signals outside
the target sensing frequency band can contribute to the sensor’s response and
each individual frequency signal produces multiple signal peaks in the
measured spectrum. Complex signals with a wide frequency range often exhibit
significant signal overlap, posing challenges for analysis. Additionally,
quantum heterodyne methods employed to down-convert high-frequency signals for
sensing can exacerbate signal overlap [45, 44, 43]. Furthermore, the limited
power of control pulses introduces spurious signals at unexpected frequencies
[63]. These factors collectively present obstacles to reliably measuring
environmental signals, particularly when various background noise sources are
not fully characterized. With this goal in mind sequence timing has been
optimised [65, 66] to eliminate higher-harmonics and sequence randomization
has been explored to mitigates spurious signals [67, 68]. However, both
approaches can address only specific their aspects and can solve these only
partially.

In this Letter, we present a novel approach called cyclic geodesic driving,
which enables unambiguous sensing of signals across a broad frequency range on
qubit sensors. Our method involves the application of a sequence of
$\pi$-pulses to implement accelerated high-fidelity quantum adiabatic driving,
effectively inducing periodic evolution of the quantum sensor along the
geodesic path on the Bloch sphere. As a result, the resonance frequency of the
quantum sensor aligns with the frequency of the periodic evolution, while
simultaneously mitigating the impact of environmental background noise. By
employing this technique, each individual frequency signal generates a single,
distinct signal response within the wide frequency band. This eliminates
undesirable signal overlap, enabling precise characterization of complex
external environments and signals. For achieving arbitrary high frequency
resolution, our method can be combined with synchronized readout techniques
[39, 40]. Furthermore, our approach exhibits remarkable resilience against
control errors, as it is the counterpart of quantum adiabatic control.
Overall, our proposed method of cyclic geodesic driving offers a robust and
effective solution for wide-band, unambiguous signal sensing, enabling
accurate analysis of intricate quantum systems and external environments.

Figure 1:  Quantum sensing via geodesic jumping. (a) Upper panel: Repeated
application of a sequence of $N$ $\pi$ pulses realizes cyclic quantum
adiabatic evolution along the geodesic in (b) or (c). Lower panel: Combined
with synchronized readout techniques for arbitrary frequency resolution. (b)
$\text{GD}_{y}$, where the closed geodesic is sampled by $N$ $\pi$ pulses. The
red solid line illustrates the trajectory of the state evolution starting at
$|1\rangle$ for $N=12$. When $\omega_{{\rm scan}}$ matches the frequency
$\nu_{n}$ of the target, a resoance occurs. (c) As (b) but for $\text{GD}_{z}$
which uses a horizontal geodesic, e.g., for robust heterodyne sensing of high-
frequency signal. The resonant condition is accurately tuned by $T_{{\rm
scan}}$ and the frequency $\omega_{{\rm ctr}}$ of control field. (d) The
resulting modulation function $F(t)$ (red line for $N=20$) and its Fourier
components (red cycles) where $f_{k}=0$ for all $1<k<N-1$. The gray line and
squares are the corresponding ones for DD pulse sequences.

#### Adiabatic shortcut by jumping. —

Our sensing scheme utilizes a sequence of periodic $\pi$ pulses [Fig. 1 (a)]
which achieves cyclic quantum adiabatic evolution along a geodesic [69, 70,
71, 72] defined by the control Hamiltonian for the qubit sensor

$H_{c}(t)=\frac{E(t)}{2}(|+_{\phi}\rangle\langle+_{\phi}|-|-_{\phi}\rangle\langle-_{\phi}|),$
(1)

where the instantaneous eigenstates $|\pm_{\phi}\rangle$ are varied by a
parameter $\phi=\phi(t)$ starting with $\phi(0)=0$ at the initial time $t=0$.
For adiabatic evolution of the parameter $\phi(t)$, the evolution operator for
the quantum state driven by $H_{c}(t)$ reads [73, 69, 74]

$U_{c}(\phi)=e^{-i\varphi_{+}(t)}|+_{\phi}\rangle\langle+_{0}|+e^{-i\varphi_{-}(t)}|-_{\phi}\rangle\langle-_{0}|,$
(2)

which transfers the initial state $|\pm_{0}\rangle$ to the instantaneous
eigenstate $|\pm_{\phi}\rangle$ at a later time. We use the Born-Fock gauge
$\langle\pm_{\phi}|\frac{d}{d\phi}|\pm_{\phi}\rangle=0$ such that
$\varphi_{\pm}(t)=\pm\frac{1}{2}\int_{0}^{t}E(t^{\prime})dt^{\prime}$ are the
dynamic phases. To realize the adiabatic evolution $U_{c}(\phi)$ by $H_{c}(t)$
in finite times, we choose

$\displaystyle|+_{\phi}\rangle=$
$\displaystyle\cos(\frac{\phi}{2})|+_{0}\rangle+\sin(\frac{\phi}{2})|-_{0}\rangle,$
(3) $\displaystyle|-_{\phi}\rangle=$
$\displaystyle-\sin(\frac{\phi}{2})|+_{0}\rangle+\cos(\frac{\phi}{2})|-_{0}\rangle,$

which connect two orthonormal states $|\pm_{0}\rangle$ via a geodesic curve
[75]. In our scheme we apply a sequence of control $\pi$ pulses via Eq. (1)
with $\phi=\phi_{j}$ at the moments $T_{j}=T_{{\rm scan}}\frac{2j-1}{2N}$
($j=1,2,\ldots$), where $N$ is the pulse number in one periodic of evolution,
see Fig. 1(a). We use the linear relation $\phi_{j}=\omega_{{\rm scan}}T_{j}$,
where frequency $\omega_{{\rm scan}}$ can be negative or positive depending on
the change of $\phi_{j}$ in time. Each control $\pi$ pulse has a time duration
$t_{j}$ such that the pulse area
$\int_{T_{j}-t_{j}/2}^{T_{j}+t_{j}/2}E(t)dt=\pi$. Between the $\pi$ pulses,
there is no control, i.e., $E(t)=0$ if $\phi\notin\\{\phi_{j}\\}$. We remove
the dynamic phases at the final time of the evolution, we introduce a $\pi$
phase shift to the pulses in the second-half of the sequence. According to
Refs. [69, 70, 71], the sequence realizes $U_{c}(\phi)$ with unit-fidelity at
the middle of any successive path points
$\phi=\bar{\phi}_{j}\equiv(\phi_{j}+\phi_{j-1})/2$. For other values of
$\phi\notin\\{\bar{\phi}_{j}\\}$ the difference between $U_{c}(\phi)$ and the
actual evolution implemented by $H_{c}(t)$ is negligible when $N$ is
sufficiently large. In contrast to conventional shortcuts to adiabaticity [74]
to accerate adiabatic process, in our method the instantaneous eigenstates of
the control Hamiltonian $H_{c}(t)$ are the same as the evolution path
$|\pm_{\phi}\rangle$ in Eq. (2). This avoids the use of counter-diabatic
fields and retains the intrinsic robustness of traditional adiabatic process
[71].

Figure 2:  Quantum spectroscopy. (a) Population signal of XY8 sequence (blue
solid line) for spectroscopy of AC fields with the frequencies
$\\{\nu_{j}/2\pi\\}=\\{500,1500.05,2499.88\\}$ kHz by varying the pulse
interval $\tau$. The 1500.05 kHz and 2499.88 kHz AC fields distort the signal
centered at 500 kHz via the 3rd and 5th harmonics, respectively. Yellow dashed
line is the result when there is dephasing noise (with a control-free
decoherence time $T_{2}^{*}\approx 2$ $\mu$s) and control errors (with about
$20$% drift on the amplitude of control field). (b) As (a) but by using
$\text{GD}_{y}$. The 500 kHz signal dip is not distorted by the 1500.05 kHz
and 2499.88 kHz AC fields. (c) Power spectrum for the AC signal fields in (c)
by using the synchronized readout technique in Ref. [40, 39]. The resonances
due to higher harmonics make the signal unidentifiable even through the
expected spectral resolution is about 1 Hz. (d) As (c) but by using
$\text{GD}_{y}$. (e) Signal of XY8 sequence for the detection of a
${}^{1}\text{H}$ spin with its frequency indicated by a red arrow. The
spurious resonance (centered around the pink vertical line) produced by a
${}^{13}\text{C}$ spin in diamond distorts the ${}^{1}\text{H}$ spin signal.
Yellow dashed line is the result when there are $2\pi\times 2$ MHz detuning
error and 30% of amplitude drift in the control field. (f) As (e) but by using
$\text{GD}_{y}$, where the spurious signals disappear. See [76]for details of
simulation.

####  Figure 3:  (a) Quantum heterodyne spectroscopy of a signal field with
frequencies
$\\{\nu_{j}\\}=\omega_{q}+2\pi\times\\{-84,-68,-56,-50,-42,-2,72,90\\}$ kHz by
using the protocol in Refs. [43, 44] with the CPMG sequences. $\omega_{q}$
could be at the GHz range. The blue solid line (the line with yellow filling)
is the simulation with (without) dephasing noise that induces a control-free
decoherence time $T_{2}^{*}\approx 2$ $\mu$s. The pulse interval $\tau$ is
varied to measure the spectrum. Because for each AC signal frequency $\nu_{j}$
resonance dips occur whenever $1/(2\tau)=(\nu_{j}-\omega_{{\rm ctr}})/(2\pi
k)$, ($k=\pm 1,\pm 3,\pm 5,\ldots$), the dips at
$1/(2\tau)=(\nu_{j}-\omega_{q})/(2\pi)$ indicated by red vertical arrows are
obscured by other resonance dips (vertical lines). Different number
$N^{\prime}$ of $\pi$ pulses are used for different range of $\tau$ to insure
that the sequence times are smaller than 1 ms. (b) [(c)]: As (a) but by using
Protocol with fixed $\Delta_{{\rm scan}}=0$ ($\omega_{{\rm scan}}=2\pi\times
80$ kHz]). All the dips only appear at the right frequencies $\nu_{j}$. See
[76] for details of simulation.

#### Unambiguous wide-bandwidth robust sensing. —

To demonstrate the concept of quantum sensing through the aforementioned
geometric control, we examine a qubit coupled to its environment via the
interaction Hamiltonian

$H_{{\rm int}}(t)=\frac{1}{2}\sigma_{z}B(t),$ (4)

where the Pauli operator $\sigma_{z}=|1\rangle\langle 1|-|0\rangle\langle 0|$
and $B(t)$ could be a classical field or a quantum operator in a rotating
frame which includes possible dephasing noise. We use the control Hamiltonian
$H_{c}(t)=\Delta(t)\frac{\sigma_{z}}{2}+\Omega(t)\frac{\sigma_{x}}{2}$ and the
states $|+_{0}\rangle=|1\rangle$ and $|-_{0}\rangle=|0\rangle$ for the
geodesic in Eq. (3) for sensing. This geodesic driving around the $y$-axis
($\text{GD}_{y}$) is sketched in Fig. 1(b). In the rotating frame of
$H_{c}(t)$, $H_{{\rm int}}(t)$ becomes $\tilde{H}_{{\rm
int}}(t)\approx\frac{1}{2}U_{c}^{\dagger}\sigma_{z}U_{c}B(t)$. For the
evolution Eq. (2), we find the approximated transformation [76]

$U_{c}^{\dagger}\sigma_{z}U_{c}\approx\cos\phi\sigma_{z};~{}U_{c}^{\dagger}\sigma_{x}U_{c}\approx\sin\phi\sigma_{z};~{}U_{c}^{\dagger}\sigma_{y}U_{c}\approx
0,$ (5)

which have the nice property that they only depend on the geometric paramteter
$\phi(t)$. We obtain [76]

$\tilde{H}_{{\rm int}}(t)\approx F(t)\frac{\sigma_{z}}{2}B(t),$ (6)

where when $N$ is sufficiently large the modulation function
$F(t)\approx\cos(\omega_{{\rm scan}}t)$ has only one Fourier component over a
large frequency band, see Fig. 1(d). Conventional DD sequences [3, 4] also
induce modulation factors $F(t)$ to the $\sigma_{z}$ operator with
$F(t)\in\\{\pm 1\\}$ for ideal sequences [77, 3, 4]. However, those modulation
factors have multiple Fourier components that complicate the interpretation of
the sensor’s signal [65, 63, 64], see Fig. 1(d) for equally spaced DD
sequences [3, 4, 60, 61, 62, 80, 79, 78].

In Figs. 2(a)-(d) we simulate the measured spectrum of a classical AC field
with $B(t)=\sum_{j=1}^{3}b_{j}\cos(\nu_{j}t+\theta_{j})$, where $\nu_{j}$ are
the frequencies of different components. For the result of Fig. 2(a) obtained
by the widely-used robust XY8 sequence with an interpulse duration $\tau$
[62], all the frequencies ($\\{\nu_{j}/2\pi\\}=\\{500,1500.05,2499.88\\}$ kHz)
cause transitions of the sensor states at $1/(2\tau)\approx 500$ kHz via the
1st, 3rd, and the 5th harmonics. This problem of ambiguous spectral overlap is
not solved even when we improve the frequency resolution sufficiently high via
the synchronized readout technique (Qdyne) [39, 40], see Fig. 2(c), because
all the frequency components contribute to the readout signal in Qdyne.
However, using $\text{GD}_{y}$, only the frequency $\nu_{1}/2\pi=500$ kHz
contributes to the dip at the resonance $\omega_{{\rm scan}}=\nu_{1}$, because
we have the effective Hamiltonian $\tilde{H}_{{\rm
int}}\approx\frac{1}{4}b_{1}\cos\theta_{1}\sigma_{z}$ from Eq. (6) after
rotating wave approximation [76]. The phase dependence on the effective
Hamiltonian (and hence the signal) allows for arbitrary frequency resolution
with synchronized readout, see Figs. 2(d) and 1(a). The results also show that
our protocol is more resistant to control errors and dephasing noise. The
already strong robustness of $\text{GD}_{y}$ is enhanced further with
increasing number $N$ of pulses, see Fig. 4.

It is interesting that cyclic geodesic driving also fully solve the problems
of spurious response due to finite-width pulses [63], as detailed in [76]. In
Fig. 2(e) and (f), we simulate the quantum sensing of a single proton spin
(${}^{1}{\rm H}$) by an NV center. A ${}^{13}{\rm C}$ spin in diamond is also
coupled to the NV center as a noise source. For this case $B(t)$ is a quantum
operator [76, 48, 49]. For the result Fig. 2(e) obtained by XY8 sequences, the
spurious response from the ${}^{13}{\rm C}$ spin disturb the target proton
spin signal and lead to misidentification of ${}^{13}{\rm C}$ nuclei for
proton. On the contrary, $\text{GD}_{y}$ provides a clean signal dip in the
spectrum [Fig. 2(f)], because when $\omega_{{\rm scan}}$ matches the frequency
$\nu_{1}$ of the target ${}^{1}{\rm H}$, $\tilde{H}_{{\rm
int}}\approx\frac{1}{2}a_{1}^{x}\sigma_{z}I_{1}^{x}$ (where $I_{1,z}$ is the
${}^{1}{\rm H}$ spin operator and $a_{1}^{x}$ is the hyperfine strength [76])
and the effect of the ${}^{13}{\rm C}$ spin is removed.

Figure 4:  Control fidelity with respect to amplitude and detuning errors for
(a) XY8 sequence, (b) $\text{GD}_{y}$ with $N=20$, (c)$\text{GD}_{y}$ with
$N=80$, (d) CPMG sequence with $40$ pulses for heterodyne sensing, (e)
$\text{GD}_{z}$ with $N=10$, (f) $\text{GD}_{z}$ with $N=10$ but each pulse is
replaced by a Knill pulse. All the protocols have the same sequence time
length, and the ideal Rabi frequency of the control is $2\pi\times 50$ MHz.

#### Unambiguous heterodyne sensing. —

The idea can be generalized to other settings. Consider the sensing of a
multi-frequency signal field $\vec{B}(t)=(B_{x},B_{y},B_{z})$ with frequencies
$\nu_{j}$ much larger than the Rabi frequency of the control field. We use
$\text{GD}_{z}$ [see Fig. 1(c)] with
$|\pm_{0}\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$ for the geodesic in Eq. (3)
to sense the traverse part $B_{\perp}(t)\equiv
B_{x}+iB_{y}=\sum_{j}b_{j}e^{i\alpha_{j}}\cos(\nu_{j}t+\theta_{j})$. The
relevant Hamiltonian reads

$H=\frac{1}{2}(\omega_{q}+\delta_{t})\sigma_{z}+(\frac{1}{2}B_{\perp}(t)\sigma_{+}+{\rm
h.c.})+H_{{\rm ctr}},$

where $\omega_{q}$ is the frequency of the qubit and $\delta_{t}$ is an
unknown fluctuation. For NV qubits, $\omega_{q}$ (e.g.,
$\omega_{q}=D+\gamma_{e}B_{z}$ with $D=2\pi\times 2.87$ GHz and
$\gamma_{e}\approx 2\pi\times 2.8$ MHz/G) can be adjusted by changing the
magnetic field $B_{z}$. The control Hamiltonian $H_{{\rm
ctr}}=\Omega(t)\cos(\omega_{{\rm ctr}}t+\phi)\sigma_{x}$ has a controllable
detuning $\Delta_{{\rm scan}}=\omega_{{\rm ctr}}-\omega_{q}$. In the rotating
frame with respect to $\frac{1}{2}(\omega_{q}+\delta_{t})\sigma_{z}+H_{{\rm
ctr}}$, we obtain the effective interaction [76]

$\tilde{H}_{{\rm
int}}(t)\approx\frac{1}{4}\sum_{j}b_{j}e^{i(\alpha_{j}-\theta_{j})+i(\omega_{{\rm
ctr}}+\omega_{{\rm scan}}-\nu_{j})t}\sigma_{+}+{\rm h.c.}.$ (7)

Under the resonance condition for a signal frequency $\omega_{{\rm
ctr}}+\omega_{{\rm scan}}=\nu_{n}$ and when $b_{j}\ll|\nu_{n}-\nu_{j}|$ for
$j\neq n$, $\tilde{H}_{{\rm
int}}(t)\approx\frac{1}{4}b_{n}e^{i(\alpha_{n}-\theta_{n})}\sigma_{+}+{\rm
h.c.}$ picks up the signal only at the frequency $\nu_{n}$.

As exemplified in Fig. 3(a), the heterodyne sensing using the CPMG sequences
produce multiple dips at $1/(2\tau)=\pm(\nu_{j}-\omega_{q})/k$
$(k=1,3,5,\ldots)$, which implies ambiguity sensor responses especially when
$\\{\nu_{j}\\}$ happen to be close to the qubit frequency $\omega_{q}$. In
contrast, our geodesic driving gives clear signal dips for accurate
determination of all the signal frequencies, see Fig. 3(b). Our method is also
more resilient to dephasing noise (Fig. 3) and is more robust against control
errors (Fig. 4). The robustness can be further enhanced by combining composite
pulse techniques [see Fig. 4 (e) for the result where each pulse is replaced
by a Knill pulse [81, 78, 66]].

#### Conclusion. —

We propose a quantum sensing scheme based on a quantum adiabatic shortcut
along a geodesic path. This scheme offers the capability to resolve complex
broadband signals and addresses the issues of spectral overlap and spurious
signals that arise in existing DD-based quantum sensing methods. Notably, our
approach allows for arbitrary frequency resolution through the utilization of
synchronized readout techniques. Moreover, it exhibits robustness against
control errors and effectively suppresses unwanted decoherence noise. The
versatility of our method extends beyond signal detection; it can also be
employed for the detection and control of various quantum objects, including
single nuclear spins, spin clusters, and mechanical oscillators. Furthermore,
our scheme holds promise for applications aimed at mitigating crosstalk in
qubit arrays. In summary, our proposed quantum sensing scheme based on a
quantum adiabatic shortcut along a geodesic path provides a universal solution
for accurate signal detection, offering improved performance over existing
methods. Its potential applications encompass a wide range of quantum systems
and address key challenges in the field.

###### Acknowledgements.

This work was supported by National Natural Science Foundation of China (Grant
No. 12074131), the Natural Science Foundation of Guangdong Province (Grant No.
2021A1515012030), the ERC Synergy grant HyperQ (grant no. 856432) and the BMBF
Zukunftscluster QSense: Quantensenoren für die biomedizinische Diagnostik
(QMED) (grant no 03ZU1110FF).

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Supplemental Material:
Wide-band Unambiguous Quantum Sensing via Geodesic Evolution

##  I Spectral responses

###  I.1 Higher harmonic and spurious responses in DD sequences.

Here we provide a background on DD based quantum sensing. Consider the qubit-
enviroment Hamiltonian

$H_{{\rm int}}(t)=\frac{1}{2}\sigma_{z}B(t).$ (S1)

A sequence of DD $\pi$ pulses is applied to the qubit, where every $\pi$ pulse
aims to flip the qubit and to induce a transformation
$\sigma_{z}\rightarrow-\sigma_{z}$ in the interaction picture with respective
to the DD control. Let $U_{{\rm DD}}(t)$ be the evolution operator driven by
the DD control. In the rotating frame of the DD control, $H_{{\rm int}}(t)$
becomes

$\tilde{H}_{{\rm int}}(t)=\frac{1}{2}U_{{\rm
DD}}^{\dagger}(t)\sigma_{z}U_{{\rm DD}}(t)B(t).$ (S2)

For the idealized situation where the DD sequences use infinitely sharp
$\delta$-shape $\pi$ pulses (where the pulse duration $t_{p}$ equals 0),
$\tilde{H}_{{\rm int}}(t)=\frac{1}{2}F(t)\sigma_{z}B(t)$ with
$F(t)\in\\{+1,-1\\}$. In realistic experiments the $\pi$ pulses have a nonzero
duration $t_{p}$ because the control has a finite Rabi frequency $\Omega$. In
general,

$\tilde{H}_{{\rm
int}}(t)=\frac{1}{2}\sum_{\alpha=x,y,z}F_{\alpha}(t)\sigma_{\alpha}B(t).$ (S3)

When the $\pi$ pulses have a constant high $\Omega$, $t_{p}=\pi/\Omega$. For
the control Hamiltonian $H_{c}(t)=\Omega\frac{\sigma_{x}}{2}$, we have during
the first $\pi$ pulse $\tilde{H}_{{\rm
int}}(t)=\frac{1}{2}\left[\sigma_{z}\cos(\Omega t)+\sigma_{y}\sin(\Omega
t)\right]B(t)$. A plot of the $F_{\alpha}(t)$ for a XY8 sequence is shown in
Fig. S1, and the corresponding Fourier transforms of $F_{\alpha}(t)$ are shown
by the blue lines in Fig. S2. Under the ideal situation that there is no
control error and that the pulse duration $t_{p}=0$, the modulation factor

$F(t)=F_{z}(t)=\sum_{k=1}^{\infty}\frac{4\sin(k\frac{\pi}{2})}{k\pi}\cos(k\frac{\pi}{\tau}t),$
(S4)

has harmonics of the frequency $\pi/\tau$ for an interpulse spacing $\tau$ of
equally spaced DD sequences. For $t_{p}>0$, $F_{x}(t)$ and $F_{y}(t)$ can have
non-zero values during the $\pi$ pulses and their Fourier transforms have many
peaks, which cause spurious signals in quantum sensing.

Figure S1:  XY8 sequence and its induced modulation functions $F_{\alpha}(t)$.
The $\pi$ pulses of the XY8 sequence have a constant high $\Omega=2\pi\times
10$ MHz and a time interval, $\tau$, between pulses. The value of $\tau$
determines the frequency $1/(2\tau)=1$ MHz for sensing. Figure S2:  The power
spectrum of $F_{\alpha}(t)$ ($\alpha=x,y,z$) for the XY8 sequence in Fig. S1
(blue lines) and for the geodesic evolution with $N=16$ in Fig. S3 (orange
line). $f_{\alpha}(\omega)=\frac{1}{T_{{\rm seq}}}\int_{0}^{T_{{\rm
seq}}}F_{\alpha}(t)e^{i\omega t}dt$. We choose the same total sequence time
$T_{{\rm seq}}=16$ $\mu$s for both methods. Both methods have the strongest
spectral response for $f_{z}(\omega)$ at $\omega/(2\pi)=1/(2\tau)=\omega_{{\rm
scan}}/(2\pi)=1$ MHz. For the XY8 sequences, there are also resonances at
$k/(2\tau)$ ($k=3,5,7,\ldots$) due to higher harmonics. The nonzero pulse
width ($\pi/\Omega$) also results in complicated spurious peaks appeared in
$f_{x}(\omega)$ and $f_{y}(\omega)$. For the method of geodesic evolution,
only one resonance response exists in a wide range of spectrum. For
$f_{z}(\omega)$, we have unambiguous frequency response at
$\omega=\omega_{{\rm scan}}$, and only tiny higher harmonics appear at
$|\omega|\geq(N-1)|\omega_{{\rm scan}}|$, c.f. Fig. 1 in the main text. For
$f_{x}(\omega)$ and $f_{y}(\omega)$, weak resonances appear at $(N\pm
2)\omega_{{\rm scan}}$, $(N\pm 1)\omega_{{\rm scan}}$, and $N\omega_{{\rm
scan}}$. Figure S3:  The modulation functions $F_{\alpha}(t)$ induced by the
geodesic evolution using $\text{GD}_{y}$. The plots show a complete period of
$F_{\alpha}(t)$. The control field has a Rabi frequency $\Omega=2\pi\times 10$
MHz. The control parameters $N=20$ and $\omega_{{\rm scan}}/(2\pi)=1$ MHz.

###  I.2 Quantum sensing by geometric evolution

Here we provide details on the operator dynamics in the rotating frame of the
control
$H_{c}(t)=\frac{E(t)}{2}(|+_{\phi}\rangle\langle+_{\phi}|-|-_{\phi}\rangle\langle-_{\phi}|)$
[Eq. (1) in the main text] with $\phi=\phi(t)$ and

$\displaystyle|+_{\phi}\rangle=$
$\displaystyle\cos(\frac{\phi}{2})|+_{0}\rangle+\sin(\frac{\phi}{2})|-_{0}\rangle,$
(S5) $\displaystyle|-_{\phi}\rangle=$
$\displaystyle-\sin(\frac{\phi}{2})|+_{0}\rangle+\cos(\frac{\phi}{2})|-_{0}\rangle.$
(S6)

Without lost of generality, we assume that $\sigma_{z}=|1\rangle\langle
1|-|0\rangle\langle 0|$, $\sigma_{x}=|1\rangle\langle 0|+|0\rangle\langle 1|$,
and $\sigma_{x}=-i|1\rangle\langle 0|+i|0\rangle\langle 1|$ with
$|+_{0}\rangle=|1\rangle$ and $|-_{0}\rangle=|0\rangle$. In the rotating frame
of the evolution
$U_{c}(t)=\mathcal{T}e^{-i\int_{0}^{t}H_{c}(t^{\prime})dt^{\prime}}$, we write

$U_{c}^{\dagger}\sigma_{z}U_{c}=\sum_{\alpha=x,y,z}F_{\alpha}(t)\sigma_{\alpha}.$
(S7)

A plot of the $F_{\alpha}(t)$ for $\text{GD}_{y}$ is shown in Fig. S3. The
corresponding Fourier transforms are shown by the yellow lines in Fig. S2. One
can see that the geodesic evolution removes higher harmonics and spurious
responses over a wide frequency band. For the Fourier transform of $F_{z}(t)$,
we have unambiguous frequency response at $\omega=\omega_{{\rm scan}}$ given
that $\phi(T_{j})=\omega_{{\rm scan}}T_{j}$. Tiny higher harmonics only appear
at the frequency $\omega>(N-1)\omega_{{\rm scan}}$. For the Fourier transforms
of $F_{x}(t)$ and $F_{y}(t)$, weak resonances only appear at
$\omega>(N-2)\omega_{{\rm scan}}$. Therefore, one can arbitary remove unwanted
spectral respoances via a sufficently large $N$.

When the control $H_{c}(t)$ realizes the ideal adiabatic evolution
$U_{c}(\phi)=e^{-i\varphi_{+}(t)}|+_{\phi}\rangle\langle+_{0}|+e^{-i\varphi_{-}(t)}|-_{\phi}\rangle\langle-_{0}|$
[6, 7, 8], we have

$\displaystyle U_{c}^{\dagger}\sigma_{z}U_{c}$
$\displaystyle=\cos\phi\sigma_{z}-\sin\phi[e^{i\varphi(t)}|1\rangle\langle
0|+{\rm h.c.}],$ $\displaystyle U_{c}^{\dagger}\sigma_{x}U_{c}$
$\displaystyle=\sin\phi\sigma_{z}+\cos\phi[e^{i\varphi(t)}|1\rangle\langle
0|+{\rm h.c.}],$ (S8) $\displaystyle U_{c}^{\dagger}\sigma_{y}U_{c}$
$\displaystyle=ie^{i\varphi(t)}|1\rangle\langle 0|+{\rm h.c.}.$

where the difference of dynamic phases
$\varphi(t)=\varphi_{+}(t)-\varphi_{-}(t)=\int_{0}^{t}E(s)ds$ increases with
time and $e^{i\varphi(t)}$ is a fast oscillating factor with a zero mean.
Neglecting terms involving $e^{i\varphi(t)}$ in a way similar to the rotating
wave approximation, we obtain Eq. (5) in the main text, i.e.,

$\displaystyle U_{c}^{\dagger}\sigma_{z}U_{c}$
$\displaystyle\approx\cos\phi\sigma_{z},$ $\displaystyle
U_{c}^{\dagger}\sigma_{x}U_{c}$ $\displaystyle\approx\sin\phi\sigma_{z},$ (S9)
$\displaystyle U_{c}^{\dagger}\sigma_{y}U_{c}$ $\displaystyle\approx 0,$

and

$U_{c}^{\dagger}(\sigma_{z}+i\sigma_{x})U_{c}=\sigma_{z}e^{i\phi}.$ (S10)

The operator dynamics can also be analyzed as follows. The control $H_{c}(t)$
with $\phi=\phi_{j}$ is used to realize the $j$th $\pi$ pulse which has the
propagator
$U_{j}=\exp\left[-i\frac{\pi}{2}\left(\cos\phi_{j}\sigma_{z}+\sin\phi_{j}\sigma_{x}\right)\right]$.
After $n$ such $\pi$ pulses, we have

$U_{c}=U_{n}U_{n-1}\cdots U_{2}U_{1}.$ (S11)

This gives the transformation after $n$ $\pi$ pulses,
$U_{c}^{\dagger}\sigma_{z}U_{c}=\sigma_{z}\cos\Theta_{n}+\sigma_{x}\sin\Theta_{n}$,
where $\Theta_{n}=2\sum_{j=1}^{n}(-1)^{j+1}\phi_{j}$. Using the relation
$\phi_{j}=\omega_{{\rm scan}}T_{{\rm scan}}\frac{2j-1}{2N}$, we obtain
$\Theta_{n}=(-1)^{n+1}\frac{n}{N}\omega_{{\rm scan}}T_{{\rm scan}}$, where the
factor $(-1)^{n}$ is caused by the $\pi$ phase shifts of the dynamic phases.
Neglecting the fast oscillating term $\sigma_{x}\sin\Theta_{n}$ in a way
similar to the rotating wave approximation, we obtain after $n$ $\pi$ pulses
$U_{c}^{\dagger}\sigma_{z}U_{c}\approx\sigma_{z}\cos\phi_{n}$. Since
$\phi_{j}=\phi(T_{j})=\omega_{{\rm scan}}T_{j}$, we have
$U_{c}^{\dagger}\sigma_{z}U_{c}\approx\sigma_{z}\cos\phi(t)$.

###  I.3 Sensing of single nuclear spins by $\text{GD}_{y}$

Here we provide details of quantum sensing by geometric evolution for the case
of sensing of single nuclear spins by a single NV center. The Hamiltonian of a
NV electron spin and its surrounding nuclear spins reads

$H=-\sum_{j}\gamma_{j}\vec{I}_{j}\cdot(B_{z}\hat{z})+|1\rangle\langle
1|\sum_{j}\vec{A}_{j}\cdot\vec{I}_{j}+H_{c}(t),$ (S12)

where $\vec{I}_{j}$ is the spin-$\frac{1}{2}$ operator for the $j$th nuclear
spin, $\gamma_{j}$ is the gyromagnetic ratio, and
$\vec{A}_{j}=(A_{j}^{x},A_{j}^{y},A_{j}^{z})$ describes the hyperfine
interaction at the spin location. $B_{z}\hat{z}$ is a magnetic field with an
amplitude $B_{z}$ and a direction denoted by the unit vector $\hat{z}$.
$H_{c}(t)=\Delta(t)\frac{\sigma_{z}}{2}+\Omega(t)\frac{\sigma_{x}}{2}$ as in
the main text for $\text{GD}_{y}$. We has neglected the week nuclear-nuclear
coupling in the Hamiltonian because it is much smaller than the electron-
nuclear coupling.

The nuclear spins have the effective precession frequency
$\vec{\nu}_{j}\equiv\nu_{j}\hat{\nu}_{j}=\gamma_{j}B_{z}\hat{z}-\frac{1}{2}\vec{A}_{j}$,
where $\hat{\nu}_{j}$ is a unit vector. In the rotating frame with respect to
$-\sum_{j}\vec{\nu}_{j}\cdot\vec{I}_{j}$, the Hamiltonian becomes

$\tilde{H}(t)=H_{{\rm int}}(t)+H_{c}(t),$ (S13) $H_{{\rm
int}}(t)=\frac{\sigma_{z}}{2}B(t),$ (S14)

where

$B(t)=\sum_{j}\frac{1}{2}a_{j}^{x}(I_{j}^{+}e^{-i\nu_{j}t}+\text{h.c.})+\frac{1}{2}a_{j}^{z}I_{j}^{z},$
(S15)

$a_{j}^{z}=\vec{A}_{j}\cdot\hat{\nu}_{j}$, and
$a_{j}^{x}=|\vec{A}_{j}-a_{j}^{z}\hat{\nu}_{j}|$. Defining
$\hat{z}_{j}\equiv\hat{\nu}_{j}$,
$\hat{x}_{j}\equiv(\vec{A}_{j}-a_{j}^{z}\hat{\nu}_{j})/a_{j}^{x}$, and
$\hat{y}_{j}\equiv\hat{z}_{j}\times\hat{x}_{j}$, we write
$I_{j}^{\mu}=\vec{I}_{j}\cdot\hat{\mu}_{j}$ with $\mu\in\\{x,y,z\\}$ and
$I_{j}^{\pm}=I_{j}^{x}\pm iI_{j}^{y}$. In the rotating frame with respect to
$H_{c}(t)$, the Pauli operator $\sigma_{z}$ becomes

$U_{c}^{\dagger}\sigma_{z}U_{c}\equiv\sum_{\alpha=x,y,z}F_{\alpha}(t)\sigma_{\alpha},$
(S16)

where $F_{\alpha}(t)$ are oscillating factors. See Figs. S3 and S1 for
exemplified plots for $\text{GD}_{y}$ and XY8 sequences, respectively. For the
case of $\text{GD}_{y}$,

$U_{c}^{\dagger}\sigma_{z}U_{c}\approx F(t)\sigma_{z}$ (S17)

in the relevant frequency range. With a sufficiently large $N$,

$F(t)=F_{z}(t)\approx\cos(\omega_{{\rm scan}}t).$ (S18)

Therefore, when $\omega_{{\rm scan}}=\nu_{n}$ for the $n$th spin and assuming
that $\omega_{{\rm scan}}\gg a_{j}^{\alpha}$ and $a_{j}^{\alpha}$ are much
smaller than the differences between $\nu_{n}$ and other $\nu_{j}$, we obtain

$\tilde{H}_{{\rm int}}\approx\frac{1}{2}a_{n}^{x}\sigma_{z}I_{n}^{x},$ (S19)

where the qubit sensor couples to only one nucler spin.

###  I.4 Heterodyne sensing by $\text{GD}_{z}$

We consider a single-qubit sensor exposed to a high-frequency signal field
$\vec{B}(t)=(B_{x},B_{y},B_{z})$ and a dephasing noise $\delta_{t}$, with the
Hamiltonian

$\displaystyle H_{0}$
$\displaystyle=\left(\omega_{q}+\delta_{t}+B_{z}\right)\frac{\sigma_{z}}{2}+B_{x}\frac{\sigma_{x}}{2}+B_{y}\frac{\sigma_{y}}{2},$
(S20)
$\displaystyle=\left(\omega_{q}+\delta_{t}+B_{z}\right)\frac{\sigma_{z}}{2}+(\frac{1}{2}B_{\perp}\sigma_{+}+{\rm
h.c.}),$ (S21)

where $B_{\perp}\equiv
B_{x}-iB_{y}=\sum_{j}b_{j}e^{i\alpha_{j}}\cos(\nu_{j}t+\theta_{j})$ and
$\sigma_{+}=\frac{1}{2}(\sigma_{x}+i\sigma_{y})$. Adding a control $H_{{\rm
ctr}}=\Omega(t)\cos(\omega_{{\rm ctr}}t+\phi)\sigma_{x}$ with a detuning
$\Delta_{{\rm scan}}=\omega_{{\rm ctr}}-\omega_{q}$, the total Hamiltonian is
$H=H_{0}+H_{{\rm ctr}}$. In the rotating frame with respect to
$\frac{1}{2}\omega_{{\rm ctr}}\sigma_{z}$, the Hamiltonian becomes

$\displaystyle H^{\prime}=$ $\displaystyle\left(\delta_{t}+B_{z}-\Delta_{{\rm
scan}}\right)\frac{\sigma_{z}}{2}$
$\displaystyle+\left[\frac{1}{2}B_{\perp}(t)\sigma_{+}e^{i\omega_{{\rm
ctr}}t}+{\rm h.c.}\right]+\Omega(t)\frac{\sigma_{\phi}}{2},$ (S22)

where

$\sigma_{\phi}\equiv\sigma_{x}\cos\phi+\sigma_{y}\cos\phi.$ (S23)

The control part $H_{c}=\Omega(t)\frac{\sigma_{\phi}}{2}$ realizes a sequence
of $\pi$ pulses for accelerated quantum adiabatic evolution along the geodesic
and prolongs the coherence time of the qubit by suppressing the nosie in a way
similar to DD [7]. On the other hand, $B_{z}$ is a fast oscillating field at a
frequency much larger than the Rabi frequency $\Omega(t)$. Therefore, the
effect of the term $\left(\delta_{t}+B_{z}-\Delta_{{\rm
scan}}\right)\frac{\sigma_{z}}{2}$ can be neglected. Similar to Eq. (S10) in
Sec. I.2, in the rotating frame of
$U_{c}(t)=\mathcal{T}e^{-i\int_{0}^{t}H_{c}(t^{\prime})dt^{\prime}}$,
$U_{c}^{\dagger}\sigma_{x}U_{c}\approx\cos\phi\sigma_{x}$ and
$U_{c}^{\dagger}\sigma_{y}U_{c}\approx\sin\phi\sigma_{x}$, which implies

$U_{c}^{\dagger}\sigma_{+}U_{c}\approx\frac{1}{2}\sigma_{x}e^{i\phi}\approx\frac{1}{2}\sigma_{x}e^{i\omega_{{\rm
scan}}t}.$ (S24)

Therefore, in the rotating frame of $U_{c}(t)$, we obtain the effective
Hamiltonian

$\displaystyle\tilde{H}_{{\rm int}}(t)$ $\displaystyle\approx
U_{c}^{\dagger}\left[\frac{1}{2}B_{\perp}(t)\sigma_{+}e^{i\omega_{{\rm
ctr}}t}+{\rm h.c.}\right]U_{c}$ (S25)
$\displaystyle\approx\frac{1}{4}B_{\perp}(t)\sigma_{x}e^{i(\omega_{{\rm
ctr}}+\omega_{{\rm scan}})t}+{\rm h.c.},$
$\displaystyle\approx\frac{1}{8}\sum_{j}b_{j}e^{i(\alpha_{j}-\theta_{j})+i(\omega_{{\rm
ctr}}+\omega_{{\rm scan}}-\nu_{j})t}\sigma_{x}+{\rm h.c.},$
$\displaystyle=\sum_{j}\frac{b_{j}}{4}\cos\left[(\omega_{{\rm
ctr}}+\omega_{{\rm scan}}-\nu_{j})t+(\alpha_{j}-\theta_{j})\right]\sigma_{x}.$

When the frequency of control matches a signal frequency $\nu_{n}$, i.e.,

$\omega_{{\rm ctr}}+\omega_{{\rm scan}}=\nu_{n},$ (S26)

and the signal is weak $|b_{j}/(\nu_{j}-\nu_{n})|\ll 1$, the qubit sensor
extracts the signal at this frequency with the Hamiltonian

$\tilde{H}_{{\rm
int}}(t)\approx\frac{1}{2}b_{n}\cos(\alpha_{n}-\theta_{n})\frac{\sigma_{x}}{2},$
(S27)

by applying the rotating wave approximation.

For a state $\rho$ after the control of a duration $T$, the population
remaining at the initial state is
$P_{|\psi_{0}\rangle\langle\psi_{0}|}\equiv{\rm
Tr}(\rho(T)|\psi_{0}\rangle\langle\psi_{0}|)$. For the resonant case
$\omega_{{\rm ctr}}+\omega_{{\rm scan}}=\nu_{n}$,
$P_{|\psi_{0}\rangle\langle\psi_{0}|}=\cos^{2}\left[\frac{1}{4}b_{n}\cos(\theta_{n}-\alpha_{n})T\right]$,
while for the off-resonant case, $P_{|\psi_{0}\rangle\langle\psi_{0}|}\approx
1$.

##  II Simulation details

###  II.1 AC field sensing

The AC field used for Figs. 2 (a)-(b) of the main text is
$B(t)=\sum_{j}b_{j}\cos(\nu_{j}t+\theta_{j})$, where
$\\{\nu_{j}\\}=2\pi\times\\{500,1500.05,2499.875\\}$ kHz,
$\\{b_{j}\\}=2\pi\times\\{5,10,10\\}$ kHz, and $\theta_{j}=0$.

Without control errors and dephasing noise, the Hamiltonian for the simulation
reads $H=B(t)\frac{\sigma_{z}}{2}+H_{c}(t)$. For the XY8 sequences,
$H_{c}(t)=\frac{1}{2}\Omega(t)\sigma_{\mu}$ when applying $\mu$ ($\mu=x,y$)
pulses (see Fig. S1 for the pulse sequence). For $\text{GD}_{y}$,
$H_{c}(t)=\Delta(t)\frac{\sigma_{z}}{2}+\Omega(t)\frac{\sigma_{x}}{2}$. In the
simulations of both protocols, we choose $\Omega(t)=\Omega=2\pi\times 10$ MHz
for the Rabi frequency during the $\pi$ pulses when there is no control
errors. In $\text{GD}_{y}$, we use the detuning of the control
$\Delta=\Omega\cot\phi_{j}$ for the $j$th pulse according to the ideal value
of $\Omega$. For the sensing using XY8 sequences, we use the resonant
frequency $1/(2\tau)$ at the 1st harmonic to detect the signal, and we repeat
the XY8 unit 16 times implying the whole XY8 sequence has $N^{\prime}=128$
$\pi$ pulses and a total duration $T_{{\rm seq}}=128\tau$. To compare the
results for the same sequence duration $T_{{\rm seq}}$, we apply $M=64$ cycles
of the cyclic evolution when we use the $\text{GD}_{y}$ (with $N=25$). In both
protocols the qubit is initially prepared in the superposition state
$|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ and we measure the population
remaining at the initial state $|+\rangle$ as the signal.

For the results with dephasing noise and control errors, we add a noise term
$\delta B(t)$ to $B(t)$ and an amplitude flutuation to the Rabi frequency
$\Omega(t)$ in both protocols. $\delta B(t)$ is modeled by a stochastic
Ornstein-Uhlenbeck (OU) process [5] with a correlation time $\tau_{B}=4$ ms
and a standard deviation $\delta_{B}=2\pi\times 0.1$ MHz [2], which gives a
decoherence time $T_{2}^{*}\approx 2$ $\mu$s, i.e., a time when the phase
coherence decays to $1/e$ of its original value in the absence of quantum
control. In the modeling of the control fluctuation, we change the value of
$\Omega(t)$ to $[1+\eta(t)](1+\epsilon)\Omega(t)$, where the static bias
$\epsilon=20\%$ and the time-dependent fluctuation $\eta(t)$ is modeled by a
stochastic OU process with a correlation time of $1$ ms and a standard
deviation $2.4\times 10^{-3}$ [1]. The results are the average of 6500 Monte
Carlo simulations. The initial state is
$|\psi_{0}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$, and we measure the
population remaining at the initial state as the signal.

###  II.2 Arbitrary high-resolution spectroscopy

Combining the sensing sequences with the method of synchronized readout [3, 4]
can improve the spectral resolution. In the simulations for Figs. 2 (c) and
(d) of the main text, we use the same AC field
[$B(t)=\sum_{j}b_{j}\cos(\nu_{j}t+\theta_{j})$ with
$\\{\nu_{j}\\}=2\pi\times\\{500,1500.05,2499.875\\}$ kHz,
$\\{b_{j}\\}=2\pi\times\\{5,10,10\\}$ kHz, $\theta_{j}=0$] and sequence
parameters ($\Omega=2\pi\times 10$ MHz, $N^{\prime}=128$, $M=64$, and $N=25$)
as in Sec. II.1.

We repeatedly apply the same sequence (each has the duration $T_{{\rm
seq}}=128\tau$) for $10^{4}$ times to obtain the synchronized readout signals
[see the lower panel of Fig. 1(a) in the main text]. At the end of each
sequence we make a projection measurement and record the measured values.
There is a delay of $t_{r}$ for readout and initialization between the end of
a sequence and the starting time of its subsequent sequence. The readout
process at the end of the $n$th sequence yields at the moment $t$ a random
photon number $y_{n}(t)={\rm Pois}[C(1-r{\rm Bn}[p_{n}]]$, where Pois denotes
a Poisson process and Bn is a Bernoulli process that takes the value $1$ with
a probability of $p_{n}$ and the value 0 with probability $1-p_{n}$ [3]. We
initialize the qubit sensor at the state
$|\psi_{0}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ before each sequence. The
probability $p_{n}$ is the population signal of finding the qubit in the state
$|+_{y}\rangle=(|0\rangle+i|1\rangle)/\sqrt{2}$ at the end of the sequence. We
choose $r=0.35$, $t_{r}=2.32$ $\mu$s, and $C\approx 0.105n$ with $n=10$
according to the parameters in Ref. [3]. A discrete Fourier transform
$\tilde{y}$ of the photon numbers $\\{y_{n}(t)\\}$ results in the power
spectrum $|\tilde{y}|/\mathcal{N}$ shown in Figs. 2 (c) and (d). Here
$\mathcal{N}$ is a fixed scaling factor for all the protocols.

###  II.3 Sensing of single nuclear spins

In Figs. 2 (e) and (f) of the main text for the sensing of a proton spin, we
perform the simulations by using the Hamiltonian Eq. (S12). The control
$H_{c}(t)=\Delta(t)\frac{\sigma_{z}}{2}+\Omega(t)\frac{\sigma_{x}}{2}$ as in
the main text for $\text{GD}_{y}$. For the XY8 sequences,
$H_{c}(t)=\frac{1}{2}\Omega(t)\sigma_{\mu}$ ($\mu\in\\{x,y\\}$). The
parameters for the proton spin are $\gamma_{{}^{1}\text{H}}/2\pi\approx
4.2576$ kHz/G, $\sqrt{(A_{\text{H}}^{x})^{2}+(A_{\text{H}}^{y})^{2}}/2\pi=0.5$
kHz, and $A_{\text{H}}^{z}/2\pi=0.5$ kHz. We assume that one 13C nucleus (with
a natural abundance $1.1$% and $\gamma_{\text{C}}\approx 1.0705$ kHz/G) in the
diamond is coupled to the NV with
$\sqrt{(A_{\text{C}}^{x})^{2}+(A_{\text{C}}^{y})^{2}}/2\pi=100$ kHz and
$A_{\text{C}}^{z}/2\pi=28$ kHz. The magnetic field $B_{z}=100$ G. The Rabi
frequency for the $\pi$ pulse is $\Omega=2\pi\times 10$ MHz. The XY8 sequence
is repeated 50 times for a total number of $N^{\prime}=400$ $\pi$ pulses. We
change $B(t)\rightarrow B(t)+\Delta_{{\rm error}}$ to include detuning error.
We set $\Delta_{{\rm error}}=2\pi\times 2$ MHz as it is the typical error due
to unpolarized nitrogen spin of the NV center. An amplitude error $\epsilon$
will change $\Omega(t)$ to $(1+\epsilon)\Omega(t)$, where we set
$\epsilon=30\%$. The initial state of the nuclear spins are assumed to be in a
room-temperature thermal state. We measure the population of the NV qubit
sensor remaining at its initial state
$|\psi_{0}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ as the signal.

###  II.4 High-frequency AC field sensing

In Fig. 3 of the main text, the signal field $\vec{B}(t)=(B_{x},B_{y},B_{z})$
has the transverse component $B_{\perp}\equiv
B_{x}-iB_{y}=\sum_{j}b_{j}e^{i\alpha_{j}}\cos(\nu_{j}t+\theta_{j})$. In the
simulation,
$\\{(\nu_{j}-\omega_{q})/2\pi\\}=\\{-84,-68,-56,-50,-42,-2,72,90\\}$ kHz, with
the corresponding $\\{b_{j}/2\pi\\}=\\{5.3,4.6,2.7,4.2,3.4,4.5,2.2,3.1\\}$
kHz. The angles $\alpha_{j}$ and phases $\theta_{j}$ are random real numbers.
The control field has a Rabi frequency $\Omega=2\pi\times 50$ MHz. For the
results with dephasing noise $\delta_{t}$, in each run of the simulation we
model $\delta_{t}$ by a OU process with a correlation time $\tau_{B}=4$ ms and
a standard deviation $\delta_{B}=2\pi\times 0.1$ MHz for a decoherence time
$T_{2}^{*}\approx 2$ $\mu$s. The results are the average of $10^{4}$ Monte
Carlo simulations.

For the heterodyne spectroscopy using CPMG sequences, the initial state is the
superposition state $|\psi_{0}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ and the
population remained at $|\psi_{0}\rangle$ is the measured signal.

For the heterodyne spectroscopy using $\text{GD}_{z}$, the initial state is
$|\psi_{0}\rangle=|0\rangle$. For the case of NV center, $|0\rangle$ is the
state after optical illumination and we do not need to use the initial $\pi/2$
pulse to prepare the qubit state. We measure the population remained at
$|\psi_{0}\rangle$ as the signal. There are 16 cycles of geodesic evolution
where for each cycle there are $N=10$ $\pi$ pulses.

###  II.5 Robustness and Combination with composite pulses

In Fig. 4 of the main text we demonstrate the control robustness in the
absence of signals. A Rabi frequency $\Omega=2\pi\times 50$ MHz is used for
the simulations. For DD sequences (CPMG and XY8 sequences) we choose
$N^{\prime}=40$ pulses for each sequence and the pulse interval $\tau$
satisfies $1/(2\tau)=1$ MHz. Therefore each DD sequence has a time length
$T_{{\rm seq}}=2\tau\times N^{\prime}/2=40$ $\mu$s. For the protocols of
geodesic driving, we use $\omega_{{\rm scan}}/(2\pi)=1$ MHz for the same
sensing frequency as DD, $\Delta_{{\rm scan}}=0$, and the same $T_{{\rm
seq}}=40$ $\mu$s. To investigate the robustness of the protocols, we change
the Rabi frequency to $(1+\epsilon)\Omega$ by adding a relative fluctuation,
and add a detuning error $\Delta$ (i.e., adding $\Delta\frac{\sigma_{z}}{2}$
to the qubit Hamiltonian). We denote $U(\frac{\Delta}{\Omega},\epsilon)$ the
actual evolution realized by the sequences in the presence of control errors.
In the absence of control errors, the ideal evolution after the sequence is an
identity operator, $U_{{\rm ideal}}=I$. The control fidelity is calculated in
the standard way as [9]

${\rm fidelity}={\rm
Tr}\left[U\left(\frac{\Delta}{\Omega},\epsilon\right)U_{{\rm
ideal}}^{\dagger}\right]/{\rm Tr}I.$ (S28)

## References

  * [1] J.-M. Cai, B. Naydenov, R. Pfeiffer, L. P. McGuinness, K. D. Jahnke, F. Jelezklo, M. B. Plenio, and A. Retzker, Robust dynamical decoupling with concatenated continuous driving, New J. Phys. 14, 113023 (2012).
  * [2] Y. Chu, P. Yang, M. Gong, M. Yu, B. Yu, M. B. Plenio, A. Retzker, and J. Cai, Precise Spectroscopy of High-Frequency Oscillating Fields with a Single-Qubit Sensor, Phys. Rev. Applied 15, 014031 (2021).
  * [3] J. M. Boss, K. Cujia, J. Zopes, and C. L. Degen, Quantum sensing with arbitrary frequency resolution, Science 356, 83 (2017).
  * [4] S. Schmitt, T. Gefen, F. M. Stürner, T. Unden, G. Wolff, C. Müller, J. Scheuer, B. Naydenov, M. Markham, S. Pezzagna, J. Meijer, I. Schwarz, M. Plenio, A. Retzker, L. P. McGuinness, and F. Jelezko, Submillihertz magnetic spectroscopy performed with a nanoscale quantum sensor, Science 356, 832 (2017).
  * [5] C. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences (Springer-Verlag, Berlin, 2004), Chap. 3
  * [6] Z.-Y. Wang and M. B. Plenio, Necessary and sufficient condition for quantum adiabatic evolution by unitary control fields. Phys. Rev. A 93, 052107 (2016).
  * [7] K. Xu, T. Xie, F. Shi, Z.-Y. Wang, X. Xu, P. Wang, Y. Wang, M. B. Plenio, and J. Du, Breaking the quantum adiabatic speed limit by jumping along geodesics. Sci. Adv. 5, eaax3800 (2019).
  * [8] Y. Liu and Z.-Y. Wang, Shortcuts to adiabaticity with inherent robustness and without auxiliary control, arXiv:2211.02543.
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# Recovery of differential operators from a noisy Fourier transform

K. Yu. Osipenko  Moscow State University
Institute of Information Transmission Problems (Kharkevich Institute), RAS,
Moscow<EMAIL_ADDRESS>
(Date: 08.02.2024)

###### Abstract.

The paper concerns problems of the recovery of differential operators from a
noisy Fourier transform. In particular, optimal methods are obtained for the
recovery of powers of generalized Laplace operators from a noisy Fourier
transform in the $L_{2}$-metric.

###### Key words and phrases:

Optimal recovery, differential operators, Fourier transform

###### 2010 Mathematics Subject Classification:

41A65, 42B10, 49N30

##  1\. Introduction

Let $X$ be a linear space, $Y,Z$ be normed linear spaces. The problem of
optimal recovery of the linear operator $\Lambda\colon X\to Z$ by inaccurately
given values of the linear operator $I\colon X\to Y$ on the set $W\subset X$
is posed as a problem of finding the value

$E(\Lambda,W,I,\delta)=\operatorname*{inf\vphantom{p}}_{\varphi\colon Y\to
Z}\sup_{\begin{subarray}{c}x\in W,\ y\in Y\\\
\|Ix-y\|_{Y}\leq\delta\end{subarray}}\|\Lambda x-\varphi(y)\|_{Z},$

called the error of optimal recovery, and the mapping $\varphi$ on which the
lower bound is attained, called the optimal recovery method (here $\delta\geq
0$ is a parameter that characterizes the error of setting the values of the
operator $I$). Initially, this problem was posed for the case when $\Lambda$
is a linear functional, $Y$ is a finite-dimensional space and the information
is known exactly ($\delta=0$), by S. A. Smolyak [1]. In fact, this statement
was a generalization of A. N. Kolmogorov’s problem about the best quadrature
formula on the class of functions [2], in which the integral and the values of
the functions are replaced by arbitrary linear functionals and there is no
condition for the linearity of the recovery method. Subsequently, much
research has been devoted to extensions of this problem (see [3]–[11], and the
references given therein).

One of the first papers in which the problem of constructing an optimal
recovery method for a linear operator was considered was the paper [4]. This
topic was further developed in the papers [12]–[20]. It turned out that in
some cases it is possible to construct a whole family of optimal recovery
methods for a linear operator. The study of such families began in [21] and
continued in [22], [23], [15], [20]. Some general approach to constructing of
family of optimal recovery methods was proposed in [24].

The aim of this paper is to construct families of optimal recovery methods for
powers of generalized Laplace operators and the Weil derivative from a noisy
Fourier transform in the $L_{2}$-metric.

##  2\. Optimal recovery methods from a noisy Fourier transform

Let $S$ be the Schwartz space of rapidly decreasing $C^{\infty}$-functions on
$\mathbb{R}^{d}$, $S^{\prime}$ be the corresponding space of distributions,
and let $F\colon S^{\prime}\to S^{\prime}$ be the Fourier transform. Set

$X_{p}=\left\\{\,x(\cdot)\in S^{\prime}:\varphi(\cdot)Fx(\cdot)\in
L_{2}(\mathbb{R}^{d}),\ Fx(\cdot)\in L_{p}(\mathbb{R}^{d})\,\right\\}.$

We define the operator $D$ as follows

$Dx(\cdot)=F^{-1}(\varphi(\cdot)Fx(\cdot))(\cdot).$

Put

(1) $\Lambda x(\cdot)=F^{-1}(\psi(\cdot)Fx(\cdot))(\cdot).$

Consider the problem of the optimal recovery of values of the operator
$\Lambda$ on the class

$W_{p}=\left\\{\,x(\cdot)\in X_{p}:\|Dx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq
1\,\right\\}$

from the noisy Fourier transform of the function $x(\cdot)$. Assume that
$\Lambda x(\cdot)\in L_{2}(\mathbb{R}^{d})$ for all $x(\cdot)\in X_{p}$. As
recovery methods we consider all possible mappings $m\colon
L_{p}(\mathbb{R}^{d})\to L_{2}(\mathbb{R}^{d})$. The error of a method $m$ is
defined by

$e_{p}(\Lambda,D,m)=\sup_{\begin{subarray}{c}x(\cdot)\in W_{p},\ y(\cdot)\in
L_{p}(\mathbb{R}^{d})\\\
\|Fx(\cdot)-y(\cdot)\|_{L_{p}(\mathbb{R}^{d})}\leq\delta\end{subarray}}\|\Lambda
x(\cdot)-m(y)(\cdot)\|_{L_{2}(\mathbb{R}^{d})}.$

The quantity

(2) $E_{p}(\Lambda,D)=\inf_{m\colon L_{p}(\mathbb{R}^{d})\to
L_{2}(\mathbb{R}^{d})}e_{p}(\Lambda,D,m)$

is called the error of optimal recovery, and the method on which the infimum
is attained, an optimal method.

It is easily checked that

(3) $E_{p}(\Lambda,D)\geq\sup_{\begin{subarray}{c}x(\cdot)\in W_{p}\\\
\|Fx(\cdot)\|_{L_{p}(\mathbb{R}^{d})}\leq\delta\end{subarray}}\|\Lambda
x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}.$

Indeed, let $x(\cdot)\in W_{p}$,
$\|Fx(\cdot)\|_{L_{p}(\mathbb{R}^{d})}\leq\delta$, and let $m\colon
L_{p}(\mathbb{R}^{d})\to L_{2}(\mathbb{R}^{d})$ be an arbitrary recovery
method. Since $x(\cdot)\in W_{p}$ and $-x(\cdot)\in W_{p}$, we have

$2\|\Lambda x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\|\Lambda
x(\cdot)-m(0)(\cdot)\|_{L_{2}(\mathbb{R}^{d})}+\|-\Lambda
x(\cdot)-m(0)(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq 2e_{p}(\Lambda,D,m).$

It follows that, for any method $m$,

$e_{p}(\Lambda,D,m)\geq\sup_{\begin{subarray}{c}x(\cdot)\in W_{p}\\\
\|Fx(\cdot)\|_{L_{p}(\mathbb{R}^{d})}\leq\delta\end{subarray}}\|\Lambda
x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}.$

Now the required inequality follows by taking the lower bound on the left over
all methods.

##  3\. Optimal recovery methods for $\Lambda_{\theta}^{\eta/2}$

Consider the polar transformation in $\mathbb{R}^{d}$

$\begin{array}[]{rcl}t_{1}&=&\rho\cos\omega_{1},\\\
t_{2}&=&\rho\sin\omega_{1}\cos\omega_{2},\\\ {3}\\\
t_{d-1}&=&\rho\sin\omega_{1}\sin\omega_{2}\ldots\sin\omega_{d-2}\cos\omega_{d-1},\\\
t_{d}&=&\rho\sin\omega_{1}\sin\omega_{2}\ldots\sin\omega_{d-2}\sin\omega_{d-1}.\end{array}$

Set $\omega=(\omega_{1},\ldots,\omega_{d-1})$. For any function $f(\cdot)$ we
put

$\widetilde{f}(\omega)=|f(\cos\omega_{1},\ldots,\sin\omega_{1}\sin\omega_{2}\ldots\sin\omega_{d-2}\sin\omega_{d-1})|.$

Note that if $|f(\cdot)|$ is a homogenous function of degree $\kappa$, then
$\widetilde{f}(\omega)=\rho^{-\kappa}|f(t)|$.

Let $|\psi(\cdot)|$ be homogenous function of degree $\eta$ and
$|\varphi(\cdot)|$ be homogenous functions of degrees $\nu$, $\psi(t)\neq 0$
and $\varphi_{j}(t)\neq 0$ for almost all $t\in\mathbb{R}^{d}$. Set

$\displaystyle\gamma=\frac{\nu-\eta}{\nu+d(1/2-1/p)},\quad
q^{*}=\frac{1}{\gamma(1/2-1/p)},$ $\displaystyle
C_{p}(\nu,\eta)=\gamma^{-\frac{\gamma}{p}}(1-\gamma)^{-\frac{1-\gamma}{2}}\Biggl{(}\frac{B\left(q^{*}\gamma/p+1,q^{*}(1-\gamma)/2\right)}{2|\nu-\eta|}\Biggr{)}^{1/q^{*}},$

where $B(\cdot,\cdot)$ is the Euler beta-function.

It follows from [25, Theorem 6] (see also [20, Theorem 3]) the following
result

######  Theorem 1.

Let $2<p\leq\infty$, $\gamma\in(0,1)$. Assume that

(4)
$I=\int_{\Pi^{d-1}}\frac{\widetilde{\psi}^{q^{*}}(\omega)}{\widetilde{\varphi}^{q^{*}(1-\gamma)}(\omega)}J(\omega)\,d\omega<\infty,\quad\Pi^{d-1}=[0,\pi]^{d-2}\times[0,2\pi].$

Then

$E_{p}(\Lambda,D)=\frac{1}{(2\pi)^{d\gamma/2}}C_{p}(\nu,\eta)I^{1/q^{*}}\delta^{\gamma}.$

The method

$\widehat{m}(y)(t)=F^{-1}\left(\left(1-\beta\frac{|\varphi(t)|^{2}}{|\psi(t)|^{2}}\right)_{+}\psi(t)y(t)\right),$

where

$\beta=\frac{1-\gamma}{(2\pi)^{d\gamma}}C_{p}^{2}(\nu,\eta)\left(\delta
I^{1/2-1/p}\right)^{2\gamma},$

is optimal.

Moreover, the sharp inequality

$\|\Lambda
x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\frac{C_{p}(\nu,\eta)I^{1/q^{*}}}{(2\pi)^{d\gamma/2}}\|Fx(\cdot)\|_{L_{p}(\mathbb{R}^{d})}^{\gamma}\|Dx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{1-\gamma}$

holds.

Put

$\psi_{\theta}(\xi)=(|\xi_{1}|^{\theta}+\ldots+|\xi_{d}|^{\theta})^{2/\theta},\quad\theta>0.$

We denote by $\Lambda_{\theta}^{\eta/2}$ the operator $\Lambda$ which is
defined by (1) for $\psi(\cdot)=\psi_{\theta}^{\eta/2}(\cdot)$. In particular,
$\Lambda_{2}=-\Delta$, where $\Delta$ is the Laplace operator.

Consider problem (2) for $\Lambda=\Lambda_{\theta}^{\eta/2}$ and
$D=\Lambda_{\mu}^{\nu/2}$. Then for $I$ from (4) we have

(5)
$I=\int_{\Pi^{d-1}}\frac{\left(\sum_{k=1}^{d}\widetilde{t}_{k}^{\theta}(\omega)\right)^{\eta
q^{*}/\theta}}{\left(\sum_{k=1}^{d}\widetilde{t}_{k}^{\,\mu}(\omega)\right)^{\nu
q^{*}(1-\gamma)/\mu}}J(\omega)\,d\omega,$

where

$\begin{array}[]{rcl}\widetilde{t}_{1}(\omega)&=&\cos\omega_{1},\\\
\widetilde{t}_{2}(\omega)&=&\sin\omega_{1}\cos\omega_{2},\\\ {3}\\\
\widetilde{t}_{d-1}(\omega)&=&\sin\omega_{1}\sin\omega_{2}\ldots\sin\omega_{d-2}\cos\omega_{d-1},\\\
\widetilde{t}_{d}(\omega)&=&\sin\omega_{1}\sin\omega_{2}\ldots\sin\omega_{d-2}\sin\omega_{d-1}.\end{array}$

Note that

$\sum_{k=1}^{d}{\widetilde{t}_{k}}^{2}(\omega)=1.$

If $\mu\leq 2$, then

(6)
$\sum_{k=1}^{d}{\widetilde{t}_{k}}^{\mu}(\omega)\geq\sum_{k=1}^{d}{\widetilde{t}_{k}}^{2}(\omega)=1.$

For $\mu>2$ by Hölder’s inequality

$1=\sum_{k=1}^{d}{\widetilde{t}_{k}}^{2}(\omega)\leq\biggl{(}\sum_{k=1}^{d}{\widetilde{t}_{k}}^{\mu}(\omega)\biggr{)}^{\frac{2}{\mu}}d^{1-\frac{2}{\mu}}.$

Thus,

(7) $\sum_{k=1}^{d}{\widetilde{t}_{k}}^{\mu}(\omega)\geq d^{1-\frac{\mu}{2}}.$

It follows by (6) and (7) that $I<\infty$.

######  Corollary 1.

Let $2<p\leq\infty$, $\nu>\eta\geq 0$, and $\theta,\mu>0$. Then

$E_{p}(\Lambda_{\theta}^{\eta/2},\Lambda_{\mu}^{\nu/2})=\frac{1}{(2\pi)^{d\gamma/2}}C_{p}(\nu,\eta)I^{1/q^{*}}\delta^{\gamma},$

where $I$ is defined by (5). The method

$\widehat{m}(y)(t)=F^{-1}\left(\left(1-\beta\frac{\psi_{\mu}^{\nu}(t)}{\psi_{\theta}^{\eta}(t)}\right)_{+}\psi_{\theta}^{\eta/2}(t)y(t)\right),$

where

$\beta=\frac{1-\gamma}{(2\pi)^{d\gamma}}C_{p}^{2}(\nu,\eta)\left(\delta
I^{1/2-1/p}\right)^{2\gamma},$

is optimal.

Moreover, the sharp inequality

$\|\Lambda_{\theta}^{\eta/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\frac{C_{p}(\nu,\eta)I^{1/q^{*}}}{(2\pi)^{d\gamma/2}}\|Fx(\cdot)\|_{L_{p}(\mathbb{R}^{d})}^{\gamma}\|\Lambda_{\mu}^{\nu/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{1-\gamma}$

holds.

Now we consider the case when $p=2$.

######  Theorem 2.

Let $\nu>\eta>0$ and $0<\theta\leq\mu$. Then

(8)
$E_{2}(\Lambda_{\theta}^{\eta/2},\Lambda_{\mu}^{\nu/2})=d^{\eta(1/\theta-1/\mu)}\left(\frac{\delta}{(2\pi)^{d/2}}\right)^{1-\eta/\nu},$

and all methods

(9) $\widehat{m}(y)(t)=F^{-1}\left(a(t)\psi_{\theta}^{\eta/2}(t)y(t)\right),$

where $a(\cdot)$ are measurable functions satisfying the condition

(10)
$\psi_{\theta}^{\eta}(\xi)\left(\frac{|1-a(\xi)|^{2}}{\lambda_{2}\psi_{\mu}^{\nu}(\xi)}+\frac{|a(\xi)|^{2}}{(2\pi)^{d}\lambda_{1}}\right)\leq
1,$

in which

$\lambda_{1}=\frac{d^{2\eta(1/\theta-1/\mu)}}{(2\pi)^{d}}\left(1-\frac{\eta}{\nu}\right)\left(\frac{(2\pi)^{d}}{\delta^{2}}\right)^{\eta/\nu},\quad\lambda_{2}=d^{2\eta(1/\theta-1/\mu)}\frac{\eta}{\nu}\left(\frac{(2\pi)^{d}}{\delta^{2}}\right)^{\eta/\nu-1},$

are optimal.

The sharp inequality

(11)
$\|\Lambda_{\theta}^{\eta/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\frac{d^{\eta(1/\theta-1/\mu)}}{(2\pi)^{d(1-\eta/\nu)/2}}\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{1-\eta/\nu}\|\Lambda_{\mu}^{\nu/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{\eta/\nu}$

holds.

###### Proof.

It follows from (3) that

(12)
$E_{2}(\Lambda_{\theta}^{\eta/2},\Lambda_{\mu}^{\nu/2})\geq\sup_{\begin{subarray}{c}x(\cdot)\in
W_{2}\\\
\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\delta\end{subarray}}\|\Lambda_{\theta}^{\eta/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}.$

Consider the extremal problem

$\|\Lambda_{\theta}^{\eta/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{2}\to\max,\quad\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{2}\leq\delta^{2},\quad\|\Lambda_{\mu}^{\nu/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{2}\leq
1.$

Given $0<\varepsilon<d^{-1/\mu}(2\pi)^{d/\nu}\delta^{-2/\nu}$, we set

$\widehat{\xi}_{\varepsilon}=\frac{1}{d^{1/\mu}}\left(\frac{(2\pi)^{d}}{\delta^{2}}\right)^{\frac{1}{2\nu}}(1,\ldots,1)-(\varepsilon,\ldots,\varepsilon),\quad
B_{\varepsilon}=\\{\xi\in\mathbb{R}^{d}:|\xi-\widehat{\xi}_{\varepsilon}|<\varepsilon\,\\}.$

Consider a function $x_{\varepsilon}(\cdot)$ such that

$Fx_{\varepsilon}(\xi)=\begin{cases}\dfrac{\delta}{\sqrt{\operatorname*{mes}B_{\varepsilon}}},&\xi\in
B_{\varepsilon},\\\ 0,&\xi\notin B_{\varepsilon}.\end{cases}$

Then $\|Fx_{\varepsilon}(\cdot)\|^{2}_{L_{2}(\mathbb{R}^{d})}=\delta^{2}$ and

$\|\Lambda_{\mu}^{\nu/2}x(\cdot)\|^{2}_{L_{2}(\mathbb{R}^{d})}=\frac{\delta^{2}}{(2\pi)^{d}\operatorname*{mes}B_{\varepsilon}}\int_{B_{\varepsilon}}(|\xi_{1}|^{\mu}+\ldots+|\xi_{d}|^{\mu})^{2\nu/\mu}\,d\xi\leq
1.$

By virtue of (12) we have

$E_{2}^{2}(\Lambda_{\theta}^{\eta/2},\Lambda_{\mu}^{\nu/2})\geq\|\Lambda_{\theta}^{\eta/2}x_{\varepsilon}(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{2}\\\
=\frac{\delta^{2}}{(2\pi)^{d}\operatorname*{mes}B_{\varepsilon}}\int_{B_{\varepsilon}}\psi_{\theta}^{\eta}(\xi)\,d\xi=\frac{\delta^{2}}{(2\pi)^{d}}\psi_{\theta}^{\eta}(\widetilde{\xi}_{\varepsilon}),\quad\widetilde{\xi}_{\varepsilon}\in
B_{\varepsilon}.$

Letting $\varepsilon\to 0$ we obtain the estimate

(13) $E_{2}^{2}(\Lambda_{\theta}^{\eta/2},\Lambda_{\mu}^{\nu/2})\geq
d^{2\eta(1/\theta-1/\mu)}\left(\frac{\delta^{2}}{(2\pi)^{d}}\right)^{1-\eta/\nu}.$

We will find optimal methods among methods (9). Passing to the Fourier
transform we have

$\|\Lambda_{\theta}^{\eta/2}x(\cdot)-\widehat{m}(y)(\cdot)\|^{2}_{L_{2}(\mathbb{R}^{d})}\\\
=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\psi_{\theta}^{\eta}(\xi)\left|Fx(\xi)-a(\xi)y(\xi)\right|^{2}\,d\xi.$

We set $z(\cdot)=Fx(\cdot)-y(\cdot)$ and note that

$\int_{\mathbb{R}^{d}}|z(\xi)|^{2}\,d\xi\leq\delta^{2},\quad\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\psi_{\mu}^{\nu}(\xi)|Fx(\xi)|^{2}\,d\xi\leq
1.$

Then

$\|\Lambda_{\theta}^{\eta/2}x(\cdot)-\widehat{m}(y)(\cdot)\|^{2}_{L_{2}(\mathbb{R}^{d})}=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\psi_{\theta}^{\eta}(\xi)\left|\left(1-a(\xi)\right)Fx(\xi)+a(\xi)z(\xi)\right|^{2}\,d\xi.$

We write the integrand as

$\psi_{\theta}^{\eta}(\xi)\left|\frac{(1-a(\xi))\sqrt{\lambda_{2}}\psi_{\mu}^{\nu/2}(\xi)Fx(\xi)}{\sqrt{\lambda_{2}}\psi_{\mu}^{\nu/2}(\xi)}+\frac{a(\xi)}{(2\pi)^{d/2}\sqrt{\lambda_{1}}}(2\pi)^{d/2}\sqrt{\lambda_{1}}z(\xi)\right|^{2}.$

Applying the Cauchy-Bunyakovskii-Schwarz inequality we obtain the estimate

$\|\Lambda_{\theta}^{\eta/2}x(\cdot)-\widehat{m}(y)(\cdot)\|^{2}_{L_{2}(\mathbb{R}^{d})}\\\
\leq\operatorname*{vraisup}_{\xi\in\mathbb{R}^{d}}S(\xi)\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\left(\lambda_{2}\psi_{\mu}^{\nu}(\xi)|Fx(\xi)|^{2}+(2\pi)^{d}\lambda_{1}|z(\xi)|^{2}\right)\,d\xi,$

where

$S(\xi)=\psi_{\theta}^{\eta}(\xi)\left(\frac{|1-a(\xi)|^{2}}{\lambda_{2}\psi_{\mu}^{\nu}(\xi)}+\frac{|a(\xi)|^{2}}{(2\pi)^{d}\lambda_{1}}\right).$

If we assume that $S(\xi)\leq 1$ for almost all $\xi$, then taking into
account (13), we get

(14)
$e^{2}_{2}(\Lambda_{\theta}^{\eta/2},\Lambda_{\mu}^{\nu/2},\widehat{m})\\\
\leq\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\left(\lambda_{2}\psi_{\mu}^{\nu}(\xi)|Fx(\xi)|^{2}+(2\pi)^{d}\lambda_{1}|z(\xi)|^{2}\right)\,d\xi\leq\lambda_{2}+\lambda_{1}\delta^{2}\\\
=d^{2\eta(1/\theta-1/\mu)}\left(\frac{\delta^{2}}{(2\pi)^{d}}\right)^{1-\eta/\nu}\leq
E_{2}^{2}(\Lambda_{\theta}^{\eta/2},\Lambda_{\mu}^{\nu/2}).$

This proves (8) and shows that the methods under consideration are optimal.

It remains to verify that the set of functions $a(\cdot)$ satisfying (10) is
nonempty. Put

$a(\xi)=\frac{(2\pi)^{d}\lambda_{1}}{(2\pi)^{d}\lambda_{1}+\lambda_{2}\psi_{\mu}^{\nu}(\xi)}.$

Then

$S(\xi)=\frac{\psi_{\theta}^{\eta}(\xi)}{(2\pi)^{d}\lambda_{1}+\lambda_{2}\psi_{\mu}^{\nu}(\xi)}.$

Since $\theta\leq\mu$ by Hölder’s inequality

$\sum_{j=1}^{d}|\xi_{j}|^{\theta}\leq\biggl{(}\sum_{j=1}^{d}|\xi_{j}|^{\mu}\biggr{)}^{\theta/\mu}d^{1-\theta/\mu}.$

Putting $\rho=(|\xi_{1}|^{\theta}+\ldots+|\xi_{d}|^{\theta})^{1/\theta}$, we
obtain

$\sum_{j=1}^{d}|\xi_{j}|^{\mu}\geq\rho^{\mu}d^{1-\mu/\theta}.$

Thus,

$S(\xi)\leq\frac{\rho^{2\eta}}{(2\pi)^{d}\lambda_{1}+\lambda_{2}\rho^{2\nu}d^{2\nu(1/\mu-1/\theta)}}.$

It is easily checked that the function
$f(\rho)=(2\pi)^{d}\lambda_{1}+\lambda_{2}\rho^{2\nu}d^{2\nu(1/\mu-1/\theta)}-\rho^{2\eta}$
reaches a minimum on $[0,+\infty)$ at

$\rho_{0}=d^{1/\theta-1/\mu}\left(\frac{(2\pi)^{d}}{\delta^{2}}\right)^{1/(2\nu)}.$

Moreover, $f(\rho_{0})=0$. Consequently, $f(\rho)\geq 0$ for all $\rho\geq 0$.
Hence $S(\xi)\leq 1$ for all $\xi$.

Let $x(\cdot)\in X_{2}$ for $\varphi(\cdot)=\psi_{\mu}^{\nu/2}(\cdot)$. Put
$A=\|\Lambda_{\mu}^{\nu/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}+\varepsilon$,
$\varepsilon>0$. Consider $\widehat{x}(\cdot)=x(\cdot)/A$. Put
$\delta=\|F\widehat{x}(\cdot)\|_{L_{2}(\mathbb{R}^{d})}$. It follows from (14)
that

(15) $\sup_{\begin{subarray}{c}x(\cdot)\in W_{2}\\\
\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\delta\end{subarray}}\|\Lambda_{\theta}^{\eta/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}=E_{2}(\Lambda_{\theta}^{\eta/2},\Lambda_{\mu}^{\nu/2})=d^{\eta(1/\theta-1/\mu)}\left(\frac{\delta}{(2\pi)^{d/2}}\right)^{1-\eta/\nu}.$

Thus,

$\|\Lambda_{\theta}^{\eta/2}\widehat{x}(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq
d^{\eta(1/\theta-1/\mu)}\left(\frac{\delta}{(2\pi)^{d/2}}\right)^{1-\eta/\nu}.$

Consequently,

$\|\Lambda_{\theta}^{\eta/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\frac{d^{\eta(1/\theta-1/\mu)}}{(2\pi)^{d(1-\eta/\nu)/2}}\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{1-\eta/\nu}\left(\|\Lambda_{\mu}^{\nu/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}+\varepsilon\right)^{\eta/\nu}.$

Letting $\varepsilon\to 0$ we obtain (11).

If there exists a

$C<\frac{d^{\eta(1/\theta-1/\mu)}}{(2\pi)^{d(1-\eta/\nu)/2}}$

for which

$\|\Lambda_{\theta}^{\eta/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq
C\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{1-\eta/\nu}\|\Lambda_{\mu}^{\nu/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{\eta/\nu},$

then

$\sup_{\begin{subarray}{c}x(\cdot)\in W_{2}\\\
\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\delta\end{subarray}}\|\Lambda_{\theta}^{\eta/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq
C\delta^{1-\eta/\nu}<\frac{d^{\eta(1/\theta-1/\mu)}}{(2\pi)^{d(1-\eta/\nu)/2}}\delta^{1-\eta/\nu}.$

This contradicts with (15). ∎

Let $\alpha=(\alpha_{1},\ldots,\alpha_{d})\in\mathbb{R}_{+}^{d}$. We define
the operator $D^{\alpha}$ (the derivative of order $\alpha$) by

$D^{\alpha}x(\cdot)=F^{-1}((i\xi)^{\alpha}Fx(\xi))(\cdot),$

where $(i\xi)^{\alpha}=(i\xi_{1})^{\alpha_{1}}\ldots(i\xi_{d})^{\alpha_{d}}$.
For $\mu=2\nu$ we have

$\|\Lambda_{2\nu}^{\nu/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{2}=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\left(|\xi_{1}|^{2\nu}+\ldots+|\xi_{d}|^{2\nu}\right)|Fx(\xi)|^{2}\,d\xi\\\
=\sum_{j=1}^{d}\|D^{\nu e_{j}}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{2},$

where $e_{j}$, $j=1\ldots,d$, is a standard basis in $\mathbb{R}^{d}$.

From Theorem 2 we obtain the following result.

######  Corollary 2.

Let $\nu>\eta>0$ and $0<\theta\leq 2\nu$. Then

$E_{2}(\Lambda_{\theta}^{\eta/2},\Lambda_{2\nu}^{\nu/2})=d^{\eta(1/\theta-1/(2\nu))}\left(\frac{\delta}{(2\pi)^{d/2}}\right)^{1-\eta/\nu},$

and all methods

$\widehat{m}(y)(t)=F^{-1}\left(a(t)\psi_{\theta}^{\eta/2}(t)y(t)\right),$

where $a(\cdot)$ are measurable functions satisfying the condition

$\psi_{\theta}^{\eta}(\xi)\left(\frac{|1-a(\xi)|^{2}}{\lambda_{2}\psi_{2\nu}^{\nu}(\xi)}+\frac{|a(\xi)|^{2}}{(2\pi)^{d}\lambda_{1}}\right)\leq
1,$

in which

$\displaystyle\lambda_{1}$
$\displaystyle=\frac{d^{2\eta(1/\theta-1/(2\nu))}}{(2\pi)^{d}}\left(1-\frac{\eta}{\nu}\right)\left(\frac{(2\pi)^{d}}{\delta^{2}}\right)^{\eta/\nu},$
$\displaystyle\quad\lambda_{2}$
$\displaystyle=d^{2\eta(1/\theta-1/(2\nu))}\frac{\eta}{\nu}\left(\frac{(2\pi)^{d}}{\delta^{2}}\right)^{\eta/\nu-1},$

are optimal.

The sharp inequality

(16)
$\|\Lambda_{\theta}^{\eta/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\frac{d^{\eta(1/\theta-1/(2\nu))}}{(2\pi)^{d(1-\eta/\nu)/2}}\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{1-\eta/\nu}\biggl{(}\sum_{j=1}^{d}\|D^{\nu
e_{j}}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{2}\biggr{)}^{\eta/(2\nu)}$

holds.

For integer $\nu$ inequality (16) can be rewritten in the form

$\|\Lambda_{\theta}^{\eta/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\frac{d^{\eta(1/\theta-1/(2\nu))}}{(2\pi)^{d(1-\eta/\nu)/2}}\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{1-\eta/\nu}\biggl{(}\sum_{j=1}^{d}\biggl{\|}\frac{\partial^{\nu}x}{\partial
t_{j}^{\nu}}(\cdot)\biggr{\|}_{L_{2}(\mathbb{R}^{d})}^{2}\biggr{)}^{\eta/(2\nu)}.$

##  4\. Optimal recovery methods for $D^{\alpha}$

Now we consider problem (2) for $\Lambda=D^{\alpha}$ and
$D=\Lambda_{\mu}^{\nu/2}$. Then for $I$ from (4) we have

(17)
$I=\int_{\Pi^{d-1}}\frac{\left(\widetilde{t}_{1}^{\alpha_{1}}(\omega)\ldots\widetilde{t}_{d}^{\alpha_{d}}(\omega)\right)^{q_{1}}}{\left(\sum_{k=1}^{d}\widetilde{t}_{k}^{\,\mu}(\omega)\right)^{\nu
q_{1}(1-\gamma_{1})/\mu}}J(\omega)\,d\omega,$

where

$\gamma_{1}=\frac{\nu-|\alpha|}{\nu+d(1/2-1/p)},\quad
q_{1}=\frac{1}{\gamma_{1}(1/2-1/p)},\quad|\alpha|=\alpha_{1}+\ldots+\alpha_{d}.$

It follows by (6) and (7) that $I<\infty$.

######  Corollary 3.

Let $2<p\leq\infty$, $\nu>|\alpha|\geq 0$, and $\mu>0$. Then

$E_{p}(D^{\alpha},\Lambda_{\mu}^{\nu/2})=\frac{1}{(2\pi)^{d\gamma_{1}/2}}C_{p}(\nu,|\alpha|)I^{1/q_{1}}\delta^{\gamma_{1}},$

where $I$ is defined by (17). The method

$\widehat{m}(y)(t)=F^{-1}\left(\left(1-\beta\frac{\psi_{\mu}^{\nu}(t)}{t_{1}^{2\alpha_{1}}\ldots
t_{d}^{2\alpha_{d}}}\right)_{+}(it)^{\alpha}y(t)\right),$

where

$\beta=\frac{1-\gamma_{1}}{(2\pi)^{d\gamma_{1}}}C_{p}^{2}(\nu,|\alpha|)\left(\delta
I^{1/2-1/p}\right)^{2\gamma_{1}},$

is optimal.

Moreover, the sharp inequality

$\|D^{\alpha}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\frac{C_{p}(\nu,|\alpha|)I^{1/q_{1}}}{(2\pi)^{d\gamma_{1}/2}}\|Fx(\cdot)\|_{L_{p}(\mathbb{R}^{d})}^{\gamma_{1}}\|\Lambda_{\mu}^{\nu/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{1-\gamma_{1}}$

holds.

Consider the case when $p=2$.

######  Theorem 3.

Let $2\nu\geq\mu>0$ and $0<|\alpha|<\nu$. Then

(18)
$E_{2}(D^{\alpha},\Lambda_{\mu}^{\nu/2})=\left(\frac{\delta}{(2\pi)^{d/2}}\right)^{1-|\alpha|/\nu}|\alpha|^{-|\alpha|/\mu}\prod_{\begin{subarray}{c}j=1\\\
\alpha_{j}\neq 0\end{subarray}}^{d}\alpha_{j}^{\alpha_{j}/\mu},$

and all methods

(19) $\widehat{m}(y)(t)=F^{-1}\left(a(t)(it)^{\alpha}y(t)\right),$

where $a(\cdot)$ are measurable functions satisfying the condition

(20)
$\psi_{\theta}^{\eta}(\xi)\left(\frac{|1-a(\xi)|^{2}}{\lambda_{2}\psi_{\mu}^{\nu}(\xi)}+\frac{|a(\xi)|^{2}}{(2\pi)^{d}\lambda_{1}}\right)\leq
1,$

in which

$\displaystyle\lambda_{1}$
$\displaystyle=\frac{|\alpha|^{-2|\alpha|/\mu}}{(2\pi)^{d}}\left(1-\frac{|\alpha|}{\nu}\right)\left(\frac{(2\pi)^{d}}{\delta^{2}}\right)^{|\alpha|/\nu}\prod_{\begin{subarray}{c}j=1\\\
\alpha_{j}\neq 0\end{subarray}}^{d}\alpha_{j}^{2\alpha_{j}/\mu},$
$\displaystyle\lambda_{2}$
$\displaystyle=\frac{|\alpha|^{-2|\alpha|/\mu+1}}{\nu}\left(\frac{(2\pi)^{d}}{\delta^{2}}\right)^{|\alpha|/\nu-1}\prod_{\begin{subarray}{c}j=1\\\
\alpha_{j}\neq 0\end{subarray}}^{d}\alpha_{j}^{2\alpha_{j}/\mu},$

are optimal.

The sharp inequality

(21)
$\|D^{\alpha}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\frac{|\alpha|^{-|\alpha|/\mu}}{(2\pi)^{d(1-|\alpha|/\nu)/2}}\prod_{\begin{subarray}{c}j=1\\\
\alpha_{j}\neq
0\end{subarray}}^{d}\alpha_{j}^{\alpha_{j}/\mu}\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{1-|\alpha|/\nu}\|\Lambda_{\mu}^{\nu/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{|\alpha|/\nu}$

holds.

###### Proof.

It follows from (3) that

(22)
$E_{2}(D^{\alpha},\Lambda_{\mu}^{\nu/2})\geq\sup_{\begin{subarray}{c}x(\cdot)\in
W_{2}\\\
\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\leq\delta\end{subarray}}\|D^{\alpha}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}.$

Consider the extremal problem

$\|D^{\alpha}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{2}\to\max,\quad\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{2}\leq\delta^{2},\quad\|\Lambda_{\mu}^{\nu/2}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{2}\leq
1.$

Given

$0<\varepsilon<\min\left\\{|\alpha|^{-1/\mu}\left(\frac{(2\pi)^{d}}{\delta^{2}}\right)^{1/(2\nu)}\alpha_{j}^{1/\mu}:\alpha_{j}>0,\
j=1,\ldots,d\right\\},$

we set

$\widehat{\xi}_{\varepsilon}=|\alpha|^{-1/\mu}\left(\frac{(2\pi)^{d}}{\delta^{2}}\right)^{1/(2\nu)}(\alpha_{1}^{1/\mu},\ldots,\alpha_{d}^{1/\mu})-(\varepsilon_{1},\ldots,\varepsilon_{d}),\quad\varepsilon_{j}=\begin{cases}\varepsilon,&\alpha_{j}>0,\\\
0,&\alpha_{j}=0,\end{cases},\\\
B_{\varepsilon}=\\{\xi\in\mathbb{R}^{d}:|\xi-\widehat{\xi}_{\varepsilon}|<\varepsilon\,\\}.$

Consider a function $x_{\varepsilon}(\cdot)$ such that

$Fx_{\varepsilon}(\xi)=\begin{cases}\dfrac{\delta}{\sqrt{\operatorname*{mes}B_{\varepsilon}}}\left(1+d\varepsilon^{\nu}\left(\dfrac{\delta^{2}}{(2\pi)^{d}}\right)^{\mu/(2\nu)}\right)^{-1/2},&\xi\in
B_{\varepsilon},\\\ 0,&\xi\notin B_{\varepsilon}.\end{cases}$

Then

$\|Fx_{\varepsilon}(\cdot)\|^{2}_{L_{2}(\mathbb{R}^{d})}=\delta^{2}\left(1+d\varepsilon^{\nu}\left(\dfrac{\delta^{2}}{(2\pi)^{d}}\right)^{\mu/(2\nu)}\right)^{-1}\leq\delta^{2}$

and

$\|\Lambda_{\mu}^{\nu/2}x(\cdot)\|^{2}_{L_{2}(\mathbb{R}^{d})}\\\
=\frac{\delta^{2}}{(2\pi)^{d}\operatorname*{mes}B_{\varepsilon}}\left(1+d\varepsilon^{\nu}\left(\dfrac{\delta^{2}}{(2\pi)^{d}}\right)^{\mu/(2\nu)}\right)^{-1}\int_{B_{\varepsilon}}(|\xi_{1}|^{\mu}+\ldots+|\xi_{d}|^{\mu})^{2\nu/\mu}\,d\xi\\\
\leq\frac{\delta^{2}}{(2\pi)^{d}}\left(1+d\varepsilon^{\nu}\left(\dfrac{\delta^{2}}{(2\pi)^{d}}\right)^{\mu/(2\nu)}\right)^{-1}\left(\left(\frac{(2\pi)^{d}}{\delta^{2}}\right)^{\mu/(2\nu)}|\alpha|^{-1}\sum_{j=1}^{d}\alpha_{j}+d\varepsilon^{\mu}\right)^{2\nu/\mu}\\\
=1.$

By virtue of (22) we have

$E_{2}^{2}(D^{\alpha},\Lambda_{\mu}^{\nu/2})\geq\|D^{\alpha}x_{\varepsilon}(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{2}\\\
=\frac{\delta^{2}}{(2\pi)^{d}\operatorname*{mes}B_{\varepsilon}}\left(1+d\varepsilon^{\nu}\left(\dfrac{\delta^{2}}{(2\pi)^{d}}\right)^{\mu/(2\nu)}\right)^{-1}\int_{B_{\varepsilon}}|\xi_{1}|^{2\alpha_{1}}\ldots|\xi_{d}|^{2\alpha_{d}}\,d\xi\\\
=\frac{\delta^{2}}{(2\pi)^{d}}\left(1+d\varepsilon^{\nu}\left(\dfrac{\delta^{2}}{(2\pi)^{d}}\right)^{\mu/(2\nu)}\right)^{-1}|\widetilde{\xi}_{1}|^{2\alpha_{1}}\ldots|\widetilde{\xi}_{d}|^{2\alpha_{d}},\quad(\widetilde{\xi}_{1},\ldots,\widetilde{\xi}_{d})\in
B_{\varepsilon}.$

Letting $\varepsilon\to 0$ we obtain the estimate

(23)
$E_{2}^{2}(D^{\alpha},\Lambda_{\mu}^{\nu/2})\geq\left(\frac{\delta^{2}}{(2\pi)^{d}}\right)^{1-|\alpha|/\nu}|\alpha|^{-2|\alpha|/\mu}\prod_{\begin{subarray}{c}j=1\\\
\alpha_{j}\neq 0\end{subarray}}^{d}\alpha_{j}^{2\alpha_{j}/\mu}.$

We will find optimal methods among methods (19). Passing to the Fourier
transform we have

$\|D^{\alpha}x(\cdot)-\widehat{m}(y)(\cdot)\|^{2}_{L_{2}(\mathbb{R}^{d})}\\\
=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}|\xi_{1}|^{2\alpha_{1}}\ldots|\xi_{d}|^{2\alpha_{d}}\left|Fx(\xi)-a(\xi)y(\xi)\right|^{2}\,d\xi.$

We set $z(\cdot)=Fx(\cdot)-y(\cdot)$ and note that

$\int_{\mathbb{R}^{d}}|z(\xi)|^{2}\,d\xi\leq\delta^{2},\quad\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\psi_{\mu}^{\nu}(\xi)|Fx(\xi)|^{2}\,d\xi\leq
1.$

Then

$\|D^{\alpha}x(\cdot)-\widehat{m}(y)(\cdot)\|^{2}_{L_{2}(\mathbb{R}^{d})}\\\
=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}|\xi_{1}|^{2\alpha_{1}}\ldots|\xi_{d}|^{2\alpha_{d}}\left|\left(1-a(\xi)\right)Fx(\xi)+a(\xi)z(\xi)\right|^{2}\,d\xi.$

We write the integrand as

$|\xi_{1}|^{2\alpha_{1}}\ldots|\xi_{d}|^{2\alpha_{d}}\left|\frac{(1-a(\xi))\sqrt{\lambda_{2}}\psi_{\mu}^{\nu/2}(\xi)Fx(\xi)}{\sqrt{\lambda_{2}}\psi_{\mu}^{\nu/2}(\xi)}+\frac{a(\xi)}{(2\pi)^{d/2}\sqrt{\lambda_{1}}}(2\pi)^{d/2}\sqrt{\lambda_{1}}z(\xi)\right|^{2}.$

Applying the Cauchy-Bunyakovskii-Schwarz inequality we obtain the estimate

$\|D^{\alpha}x(\cdot)-\widehat{m}(y)(\cdot)\|^{2}_{L_{2}(\mathbb{R}^{d})}\\\
\leq\operatorname*{vraisup}_{\xi\in\mathbb{R}^{d}}S(\xi)\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\left(\lambda_{2}\psi_{\mu}^{\nu}(\xi)|Fx(\xi)|^{2}+(2\pi)^{d}\lambda_{1}|z(\xi)|^{2}\right)\,d\xi,$

where

$S(\xi)=|\xi_{1}|^{2\alpha_{1}}\ldots|\xi_{d}|^{2\alpha_{d}}\left(\frac{|1-a(\xi)|^{2}}{\lambda_{2}\psi_{\mu}^{\nu}(\xi)}+\frac{|a(\xi)|^{2}}{(2\pi)^{d}\lambda_{1}}\right).$

If we assume that $S(\xi)\leq 1$ for almost all $\xi$, then taking into
account (23), we get

(24) $e^{2}_{2}(D^{\alpha},\Lambda_{\mu}^{\nu/2},\widehat{m})\\\
\leq\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}\left(\lambda_{2}\psi_{\mu}^{\nu}(\xi)|Fx(\xi)|^{2}+(2\pi)^{d}\lambda_{1}|z(\xi)|^{2}\right)\,d\xi\leq\lambda_{2}+\lambda_{1}\delta^{2}\\\
=\left(\frac{\delta^{2}}{(2\pi)^{d}}\right)^{1-|\alpha|/\nu}|\alpha|^{-2|\alpha|/\mu}\prod_{\begin{subarray}{c}j=1\\\
\alpha_{j}\neq 0\end{subarray}}^{d}\alpha_{j}^{2\alpha_{j}/\mu}\leq
E_{2}^{2}(D^{\alpha},\Lambda_{\mu}^{\nu/2}).$

This proves (18) and shows that the methods under consideration are optimal.

It remains to verify that the set of functions $a(\cdot)$ satisfying (20) is
nonempty. Put

$a(\xi)=\frac{(2\pi)^{d}\lambda_{1}}{(2\pi)^{d}\lambda_{1}+\lambda_{2}\psi_{\mu}^{\nu}(\xi)}.$

Then

$S(\xi)=\frac{|\xi_{1}|^{2\alpha_{1}}\ldots|\xi_{d}|^{2\alpha_{d}}}{(2\pi)^{d}\lambda_{1}+\lambda_{2}\psi_{\mu}^{\nu}(\xi)}.$

Consider the function

$H(t)=-1+(2\pi)^{d}\lambda_{1}e^{-2(\alpha,t)}+\lambda_{2}\biggl{(}\sum_{j=1}^{d}e^{\mu
t_{j}-\frac{\mu}{\nu}(\alpha,t)}\biggr{)}^{2\nu/\mu},$

where $(\alpha,t)=\alpha_{1}t_{1}+\ldots\alpha_{d}t_{d}$. It is easy to prove
that $H(\cdot)$ is a convex function. Moreover, $H(\widehat{t})=0$ and the
derivative of $H(\cdot)$ at the point $\widehat{t}$ is also zero, where

$\widehat{t}=\left(\frac{1}{\mu}\ln\frac{\alpha_{1}}{|\alpha|}+\frac{1}{2\nu}\ln\frac{(2\pi)^{d}}{\delta^{2}},\ldots,\frac{1}{\mu}\ln\frac{\alpha_{d}}{|\alpha|}+\frac{1}{2\nu}\ln\frac{(2\pi)^{d}}{\delta^{2}}\right).$

Consequently, $H(t)\geq 0$ for all $t\in\mathbb{R}^{d}$. It means that

$e^{-2(\alpha,t)}\leq(2\pi)^{d}\lambda_{1}+\lambda_{2}\biggl{(}\sum_{j=1}^{d}e^{\mu
t_{j}}\biggr{)}^{2\nu/\mu}.$

Put $|\xi_{j}|=e^{t_{j}}$, $j=1,\ldots,d$. Then we obtain

$|\xi_{1}|^{2\alpha_{1}}\ldots|\xi_{d}|^{2\alpha_{d}}\leq(2\pi)^{d}\lambda_{1}+\lambda_{2}\psi_{\mu}^{\nu}(\xi).$

Thus, $S(\xi)\leq 1$.

The proof of (21) is similar to the proof of (11). ∎

For $\mu=2\nu$ we obtain

######  Corollary 4.

Let $0<|\alpha|<\nu$. Then

$E_{2}(D^{\alpha},\Lambda_{2\nu}^{\nu/2})=\left(\frac{\delta}{(2\pi)^{d/2}}\right)^{1-|\alpha|/\nu}|\alpha|^{-|\alpha|/(2\nu)}\prod_{\begin{subarray}{c}j=1\\\
\alpha_{j}\neq 0\end{subarray}}^{d}\alpha_{j}^{\alpha_{j}/(2\nu)},$

and all methods

$\widehat{m}(y)(t)=F^{-1}\left(a(t)(it)^{\alpha}y(t)\right),$

where $a(\cdot)$ are measurable functions satisfying the condition

$\psi_{\theta}^{\eta}(\xi)\left(\frac{|1-a(\xi)|^{2}}{\lambda_{2}\psi_{2\nu}^{\nu}(\xi)}+\frac{|a(\xi)|^{2}}{(2\pi)^{d}\lambda_{1}}\right)\leq
1,$

in which

$\displaystyle\lambda_{1}$
$\displaystyle=\frac{1}{(2\pi)^{d}}\left(1-\frac{|\alpha|}{\nu}\right)\left(\frac{(2\pi)^{d}}{|\alpha|\delta^{2}}\right)^{|\alpha|/\nu}\prod_{\begin{subarray}{c}j=1\\\
\alpha_{j}\neq 0\end{subarray}}^{d}\alpha_{j}^{\alpha_{j}/\nu},$
$\displaystyle\lambda_{2}$
$\displaystyle=\frac{|\alpha|}{\nu}\left(\frac{(2\pi)^{d}}{|\alpha|\delta^{2}}\right)^{|\alpha|/\nu-1}\prod_{\begin{subarray}{c}j=1\\\
\alpha_{j}\neq 0\end{subarray}}^{d}\alpha_{j}^{\alpha_{j}/\nu},$

are optimal.

The sharp inequality

(25) $\|D^{\alpha}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\\\
\leq\frac{|\alpha|^{-|\alpha|/(2\nu)}}{(2\pi)^{d(1-|\alpha|/\nu)/2}}\prod_{\begin{subarray}{c}j=1\\\
\alpha_{j}\neq
0\end{subarray}}^{d}\alpha_{j}^{\alpha_{j}/(2\nu)}\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{1-|\alpha|/\nu}\biggl{(}\sum_{j=1}^{d}\|D^{\nu
e_{j}}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{2}\biggr{)}^{|\alpha|/(2\nu)}$

holds.

For integer $\nu$ inequality (25) can be rewritten in the form

$\|D^{\alpha}x(\cdot)\|_{L_{2}(\mathbb{R}^{d})}\\\
\leq\frac{|\alpha|^{-|\alpha|/(2\nu)}}{(2\pi)^{d(1-|\alpha|/\nu)/2}}\prod_{\begin{subarray}{c}j=1\\\
\alpha_{j}\neq
0\end{subarray}}^{d}\alpha_{j}^{\alpha_{j}/(2\nu)}\|Fx(\cdot)\|_{L_{2}(\mathbb{R}^{d})}^{1-|\alpha|/\nu}\biggl{(}\sum_{j=1}^{d}\biggl{\|}\frac{\partial^{\nu}x}{\partial
t_{j}^{\nu}}(\cdot)\biggr{\|}_{L_{2}(\mathbb{R}^{d})}^{2}\biggr{)}^{|\alpha|/(2\nu)}.$

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