# A shape preserving quasi-interpolation
operator based on a new transcendental RBF

Mohammad Heidari<EMAIL_ADDRESS>Maryam Mohammadi
<EMAIL_ADDRESS>Stefano De Marchi<EMAIL_ADDRESS>Faculty of
Mathematical Sciences and Computer, Kharazmi University, Teheran, Iran
Department of Mathematics "Tullio Levi-Civita", University of Padova, Italy

###### Abstract

It is well-known that the univariate Multiquadric quasi-interpolation operator
is constructed based on the piecewise linear interpolation by $|x|$. In this
paper, we first introduce a new transcendental RBF based on the hyperbolic
tangent function as an smooth approximant to $\phi(r)=r$ with higher accuracy
and better convergence properties than the MQ RBF $\sqrt{r^{2}+c^{2}}$. Then
the Wu–Schaback’s quasi-interpolation formula is rewritten using the proposed
RBF. It preserves convexity and monotonicity. We prove that the proposed
scheme converges with a rate of $O(h^{2})$. So it has a higher degree of
smoothness. Some numerical experiments are given in order to demonstrate the
efficiency and accuracy of the method.

###### keywords:

Radial basis functions (RBFs) , quasi-interpolation, hyperbolic tangent
function

##  1 Introduction

Given a set of $n$ distinct (scattered) points
$\\{x_{j}\\}_{j=0}^{n}\in\Omega\subseteq\mathbb{R}^{d}$ and corresponding data
values $\\{f_{j}\\}_{j=0}^{n}\in\mathbb{R}$, a standard way to interpolate a
function $f\in C^{1}:\Omega\rightarrow\mathbb{R}$ is by using

$\displaystyle\mathcal{L}f(x)=\sum_{j=0}^{n}\lambda_{j}\mathcal{X}(x-x_{j}),$
(1)

with the coefficients $\lambda_{j}$ determined by the interpolation conditions
$\mathcal{L}f(x_{j})=f_{j},~{}j=0,\ldots,n,$ where $\mathcal{X}(\cdot)$ is an
interpolation kernel. Many authors use Radial Basis Functions (RBFs) to solve
the interpolating problem (1), that is
$\mathcal{X}(x-x_{j})=\phi(\|x-x_{j}\|),$ ($\|\cdot\|$ is the Euclidean norm)
with $\phi:[0,\infty)\rightarrow\mathbb{R}$, is some radial function [1].
Then, the coefficients $\lambda_{j}$ are determined solving a symmetric linear
system $A\lambda=f$, where $A=\left[\phi(\|x_{i}-x_{j}\|)\right]_{0\leq
i,j\leq n}.$ RBF method provides excellent interpolants for high dimensional
scattered data sets. The corresponding theory had been extensively studied by
many researchers (see e.g [2, 3, 4, 5, 6, 7, 8, 1, 9, 10]). That is why in the
last few decades, RBFs have been widely applied in a number of fields such as
multivariate function approximation, neural networks and solution of
differential and integral equations (see e.g [11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21]). The Multiquadric (MQ) RBF

$\displaystyle\phi_{j}(x)=\sqrt{{\|x-x_{j}\|}^{2}+c^{2}},$ (2)

proposed by Hardy [22], is undoubtedly the most popular RBF that is used in
many applications and is representative of the class of global infinitely
differentiable RBFs. Hardy [23] summarized the achievement of study of MQ from
1968 to 1988 and showed that it can be applied in hydrology, geodesy,
photogrammetry, surveying and mapping, geophysics, crustal movement, geology,
mining and so on. In the survey paper [24], Franke pointed out that MQ
interpolation was the best among 29 scattered data interpolation methods in
terms of timing, storage, accuracy, visual pleasantness of surface
reconstruction and ease to implement. The existence of the solution of the
associated interpolation problem was shown later on by Micchelli [5]. Although
the MQ interpolation is always solvable, the resulting matrix quickly becomes
ill-conditioned as the number of points increases. Researchers concentrated on
a weaker form of (1), known as quasi-interpolation, that holds only for
polynomials of some low degree $m$, i.e.,

$\mathcal{L}p_{m}(x_{j})=p_{m}(x_{j}),\quad\forall p_{m}\in\Pi_{m}^{d},$

for all $0\leq j\leq n$, where $\Pi_{m}^{d}$ denotes the space of polynomials
of degree less and equal to $m$ in $\mathbb{R}^{d}$. Beatson and Powell [25,
26] first proposed a univariate quasi-interpolation formula where
$\mathcal{X}$ in (1), is a linear combination of MQ RBF and low degree
polynomials. Their idea is based on the fact that the MQ degenerates to
$|x-x_{j}|$, for $c=0$ and $d=1$, hence quasi-interpolation (1) is the usual
piecewise linear interpolation which reproduces linear polynomials as $c$
tends to zero. However, their operator requires the approximation of the
derivatives of the function at endpoints, which is not convenient for
practical purposes. Thus, Wu and Schaback [27] constructed another univariate
MQ quasi-interpolation operator with without the use of derivatives at the
endpoints. Given data

$\displaystyle a=x_{0}<x_{2}<\dotsb<x_{n}=b\qquad h:=\max_{2\leq j\leq
n}(x_{j}-x_{j-1}),$

Wu–Schaback’s MQ quasi-interpolation formula is

$\displaystyle(\mathcal{L}_{{MQ}}f)(x)=f_{0}\alpha_{0}(x)+f_{1}\alpha_{1}(x)+\sum_{j=2}^{n-2}f_{j}\psi_{j}(x)+f_{n-1}\alpha_{n-1}(x)+f_{n}\alpha_{n}(x)$
(3)

where

$\displaystyle\alpha_{0}(x)$
$\displaystyle=\dfrac{1}{2}+\dfrac{\phi_{1}(x)-(x-x_{0})}{2(x_{1}-x_{0})},$
$\displaystyle\alpha_{1}(x)$
$\displaystyle=\dfrac{\phi_{2}(x)-\phi_{1}(x)}{2(x_{2}-x_{1})}-\dfrac{\phi_{1}(x)-(x-x_{0})}{2(x_{1}-x_{0})},$
$\displaystyle\alpha_{n-1}(x)$
$\displaystyle=\dfrac{(x_{n}-x)-\phi_{n-1}(x)}{2(x_{n}-x_{n-1)}}-\dfrac{\phi_{n-1}(x)-\phi_{n-2}(x)}{2(x_{n-1}-x_{n-2})},$
$\displaystyle\alpha_{n}(x)$
$\displaystyle=\dfrac{1}{2}+\dfrac{\phi_{n-1}(x)-(x_{n}-x)}{2(x_{n}-x_{n-1})},$
$\displaystyle\psi_{j}(x)$
$\displaystyle=\dfrac{\phi_{j+1}(x)-\phi_{j}(x)}{2(x_{j+1}-x_{j})}-\dfrac{\phi_{j}(x)-\phi_{j-1}(x)}{2(x_{j}-x_{j-1)}},\qquad
2\leq j\leq n-2.$

The main advantage of this formula is that it does not require the solution of
any linear system. Instead, the formula uses the function values $f_{j}$ at
$x_{j}$ as its coefficients. The drawback is that instead of $c=O(h)$, one
needs to use a smaller shape parameter $c^{2}|\log c|=O(h^{2})$ in order to
achieve quadratic convergence, resulting in a lower smoothness. Note that for
$c=0$, the basis functions given in quasi-interpolant $\mathcal{L}_{MQ}f$ are
cardinal with respect to $\\{x_{j}\\}_{j=0}^{n}$. For a general quasi-
interpolation operator $\mathcal{L}$ we can state the following definitions.

######  Definition 1.1.

The quasi-interpolation operator $\mathcal{L}$ constructed at the data points
$\\{(x_{j},f_{j})\\}$, is called to be monotonicity-preserving, if the first
order divided difference $f[x_{j},x_{j+1}]$ is nonnegative (non-positive)
implies that $\left(\mathcal{L}f\right)^{\prime}$ is also nonnegative (non-
positive).

######  Definition 1.2.

The quasi-interpolation operator $\mathcal{L}$ constructed at the data points
$(x_{j},f_{j})$, is called to be convexity-preserving if the second order
divided difference $f[x_{j-1},x_{j},x_{j+1}]$ is nonnegative (non-positive,
zero) implies that $\left(\mathcal{L}f\right)^{\prime\prime}$ is also
nonnegative (non-positive, zero).

Since $\sqrt{x^{2}+c^{2}}$ tends to $|x|$ as $c$ tends to zero, and radial
basis interpolation (as well as the quasi-interpolation) based on $|x|$ is
piecewise linear, Wu and Schaback claimed that the shape-preserving properties
of piecewise linear interpolation can be expected to hold for quasi-
interpolation with multiquadrics, too. Actually, they first showed that the
quasi-interpolation operator of Beatson and Powell is indeed convexity
preserving. Then they proved that the quasi-interpolation operator (3) is
monotonicity and convexity preserving. In 2004, Ling [28] proposed a
multilevel quasi-interpolation operator and proved that it converges with a
rate of $O(h^{2.5})\log h$ as $c=O(h)$. In 2009, Feng and Li [29] constructed
a shape-preserving quasi-interpolation operator by shifts of cubic MQ
functions proving that it can produce an error of $O(h^{2})$ as $c=O(h)$. Wang
et al. [30] proposed an improved univariate MQ quasi-interpolation operator,
by using Hermite interpolating polynomials, with convergence rate heavily
depending on the shape parameter $c$. Jiang et al. [31] proposed two new
multilevel univariate MQ quasi-interpolation operators with higher
approximation order.

Ling proposed a multidimensional quasi-interpolation operator using the
dimension-splitting multiquadric basis function approach [32], and Wu et al.
modified their idea by using multivariate divided difference and the idea of
the superposition [33].

Gao and Wu [34] studied the quasi-interpolation for the linear functional data
rather than the discrete function values. Moreover, MQ quasi-interpolation has
been successfully applied in a wide range of fields. For example, in 2007,
Wang and Wu [35] applied the operator (3) to tackle approximate
implicitization of parametric curves. In 2011, Wu [36] presented a new
approach to construct the so-called shape preserving interpolation curves
based on MQ quasi-interpolation (3). Hon and Wu [37], Chen and Wu [38, 39],
Jiang and Wang [40], and other researches provided some successful examples
using MQ quasi-interpolation operators to solve different types of partial
differential equations.

In this paper, in the next section we introduce a new quasi-interpolation
operator based on the hyperbolic tangent function, that is the function

$g(x)=x\tanh\left({x\over c}\right),\;\;\,c>0$ (4)

which leads to a smooth and shape preserving interpolation operator with
$O(h^{2})$ rate of convergence. In section 3, we discuss its accuracy
providing an error estimate. Numerical experiments are presented in section 4
with the aim of comparing the accuracy of our quasi-interpolation scheme with
that of Wu and Schaback’s, and also verifying the convergence rate of new
quasi-interpolation operator by examples. The last section summarizes the
conclusion and some further works.

##  2 Quasi-interpolation operator based on a new transcendental RBF

In this section, we first analyse a new approximation of $|x|$ based on the
hyperbolic tangent, with better accuracy than the MQ RBF $\sqrt{x^{2}+c^{2}}$.
The general question is, are there any good approximations of the absolute
value function which are smooth? One simple approximation is MQ RBF
$\sqrt{x^{2}+c^{2}}$. Carlos Ramirez et al. [41] proved that
$\sqrt{x^{2}+c^{2}}$ is the most computationally efficient and smooth
approximation of $|x|$, while S. Voronin et al. [42] proved the following
Lemma.

######  Lemma 2.1.

The approximation of $|x|$ by the multiquadrics
$g(x)=\sqrt{x^{2}+c^{2}},c\in\mathbb{R}_{+}$ satisfies

$\displaystyle\left|\left|x\right|-\sqrt{x^{2}+c^{2}}\right|\leq c$ ,
$\displaystyle\left|x\right|\leq\sqrt{x^{2}+c^{2}}$ .

As noticed by Gauss in [43], the hyperbolic tangent can be written using the
continued fraction

$\displaystyle\tanh(x)=\frac{x}{1+\frac{x^{2}}{3+\frac{x^{2}}{5+\cdots}}}.$

This fact shows immediately that the function $\displaystyle
g(x)=x\tanh\left(\frac{x}{c}\right)$ is a nonnegative function that indeed can
be used to approximate $|x|.$

Since for the hyperbolic tangent

$\displaystyle\displaystyle\lim_{c\rightarrow
0^{+}}\tanh\left(\frac{x}{c}\right)=\left\\{\begin{array}[]{ll}1,&x>0,\\\
0,&x=0\\\ -1,&x<0.\end{array}.\right.$

we then have the approximation

$x\tanh\left(\frac{x}{c}\right)\approx|x|.$

Now, we show that the approximation of $|x|$ by
$x\tanh\left(\frac{x}{c}\right)$ is more accurate than that given by the
multiquadric.

######  Lemma 2.2.

The approximation of $|x|$ by
$g(x)=x\tanh\left(\frac{x}{c}\right),c\in\mathbb{R}_{+}$ satisfies

$\displaystyle\left|\left|x\right|-x\tanh\left({\frac{x}{c}}\right)\right|$
$\displaystyle\leq 0.28c<c,$ (6) $\displaystyle
x\tanh\left({\frac{x}{c}}\right)$ $\displaystyle\leq\left|x\right|.$ (7)

###### Proof.

The proof of (6) is trivial for $x=0$. Letting
$h(x)=\left|x\right|-x\tanh\left({\frac{x}{c}}\right)$ that, for $x>0$,
becomes $h(x)=x-x\tanh\left({\frac{x}{c}}\right).$ The maxima and minima of
$h$ are those that annihilate

$\displaystyle
h^{\prime}(x)=\left({\frac{x}{c}}\right)\left(\tanh\left(\frac{x}{c}\right)\right)^{2}-\tanh\left(\frac{x}{c}\right)+\left(1-\frac{x}{c}\right).$

Setting $\frac{x}{c}=t$, we have

$\displaystyle t\tanh^{2}(t)-\tanh(t)+\left(1-t\right)=0$

which reduces to solve $s(t)=t(\tanh(t)+1)-1=0$. The function $s$ on $t\geq 0$
is strictly increasing, with $s(0)=-1$. Hence there exits only one zero. By
numerically solving it, we find the value of $t^{*}=0.6392322714$ then
$x^{*}=0.6392322714c$. When $x<0$, ans so $t<0$, $s(t)<-1$, showing that the
value $t^{*}$ is the only extremal value of $h$. Hence,

$\displaystyle h(x^{*})=0.2784645427c\simeq 0.28c.$

To prove (7), we have

$\displaystyle x\tanh\left({\frac{x}{c}}\right)\leq\left|x\right|$
$\displaystyle\Longleftrightarrow x^{2}\tanh^{2}\left({\frac{x}{c}}\right)\leq
x^{2},$
$\displaystyle\Longleftrightarrow\tanh^{2}\left({\frac{x}{c}}\right)\leq 1.$

∎

######  Theorem 2.1.

The approximation of $|x|$ by $x\tanh(\frac{x}{c})$ is more accurate than that
with $\sqrt{x^{2}+c^{2}}$.

###### Proof.

It is clear that

$\cosh\left(\frac{x}{c}\right)>\frac{x}{c}.$

Since $\cosh(x)$ is an even function we have

$\cosh^{2}\left(\frac{x}{c}\right)>\frac{x^{2}}{c^{2}},$

then

$x^{2}\mbox{sech}^{2}\left(\frac{x}{c}\right)<c^{2},$

which in turn gives

$x^{2}-x^{2}\tanh^{2}\left(\frac{x}{c}\right)<c^{2}.$

Then

$x^{2}-x^{2}\tanh^{2}\left(\frac{x}{c}\right)<c^{2}=\left(x^{2}+c^{2}\right)-x^{2}.$

∎

Moreover, the function $x\tanh\left({\frac{x}{c}}\right)$ converges to
$\left|x\right|$ faster than $\sqrt{x^{2}+c^{2}}$ to $\left|x\right|$ by
decreasing $c$, as stated in the next Theorem.

######  Theorem 2.2.

If $c\longrightarrow 0^{+}$ then
$x\tanh\left({\frac{x}{c}}\right)-\left|x\right|=o\left(\sqrt{x^{2}+c^{2}}-\left|x\right|\right)$.

In fact,

$\displaystyle\lim_{c\longrightarrow
0^{+}}\dfrac{x\tanh\left({\frac{x}{c}}\right)-\left|x\right|}{\sqrt{x^{2}+c^{2}}-\left|x\right|}=0$
(8)

In order to illustrate the superiority of the new hyperbolic approximation to
$|x|$, $L_{\infty}$ error norm

$\max_{1\leq i\leq n}|g(x_{i})-|x_{i}||,$

and the rate of convergence

$r_{c}=\frac{\log\left(\frac{E_{c_{i}}}{E_{c_{i-1}}}\right)}{\log\left(\frac{c_{i}}{c_{i-1}}\right)},$

for both approximants $x\tanh\left(\frac{x}{c}\right)$ and
$\sqrt{x^{2}+c^{2}}$ are reported in Table 1, for $n=100,200,400$ equally
spaced points in $[-10,10]$. Table 1 shows that
$x\tanh\left(\frac{x}{c}\right)$ approximates $|x|$ much better than
$\sqrt{x^{2}+c^{2}}$ while Table 1 and the logarithmic scale plots 1 show that
the approximant $x\tanh\left(\frac{x}{c}\right)$ has exponential rate of
convergence to $|x|$ as $c\rightarrow 0$ instead of $O(c^{2})$ provided by
$\sqrt{x^{2}+c^{2}}$.

Table 1: $L_{\infty}$ errors and convergence rates for both approximants of $|x|$ for different values of $c$. |  | $\left|\left|x\right|-\sqrt{x^{2}+c^{2}}\right|$ | $\left|\left|x\right|-x\tanh\left({\frac{x}{c}}\right)\right|$
---|---|---|---
$n$ | $c$ |  $L_{\infty}$ error | $r_{c}$ |  $L_{\infty}$ error | $r_{c}$
100 | 0.1 | 4.1127e-02 | — | 2.3656e-02 | —
0.05 | 1.1698e-02 | 1.813823944 | 3.4922e-03 | 2.759998057
0.025 | 3.0478e-03 | 1.940421754 | 6.2490e-05 | 5.804367034
0.0125 | 7.7050e-04 | 1.983901373 | 1.9342e-08 | 11.65767264
| 0.00625 | 1.9317e-04 | 1.995923901 | 1.8457e-15 | 23.32106557
200 | 0.1 | 6.1665e-02 | — | 2.6930e-02 | —
0.05 | 2.0637e-02 | 1.579218611 | 1.1875e-02 | 1.181286716
0.025 | 5.8753e-03 | 1.812498837 | 1.7723e-03 | 2.744232777
0.0125 | 1.5314e-03 | 1.939811357 | 3.2376e-05 | 5.774554268
| 0.00625 | 3.8718e-04 | 1.983774825 | 1.0436e-08 | 11.59914019
400 | 0.1 | 7.8030e-02 | — | 2.7348e-02 | —
0.05 | 3.0867e-02 | 1.337963627 | 1.3456e-02 | 1.023185719
0.025 | 1.0337e-02 | 1.578247724 | 5.9488e-03 | 1.177579030
0.0125 | 2.9442e-03 | 1.811869965 | 8.9273e-04 | 2.736302862
| 0.00625 | 7.6754e-04 | 1.939561834 | 1.6476e-05 | 5.759785971

(a)

(b)

(c)

Figure 1: $\log|\mbox{error}|$ versus $\log\left(1/c\right)$ for $n=100$ (a),
$n=200$ (b), and $n=400$ (c).

###  2.1 New transcendental RBF

Let us introduce the following globally supported and infinitely
differentiable transcendental RBF

$\displaystyle\phi(r)=r\tanh\left({\frac{r}{c}}\right),$

abbreviated by RTH, where $r=\|x-x_{j}\|$ and $\|\cdot\|$ is the Euclidean
norm $\mathbb{R}^{d}$.

The parameter $c>0$ is called shape parameter whose optimal value for getting
accurate numerical solutions and good conditioning of the collocation matrix,
can be found usually numerically.

######  Theorem 2.3.

The RTH RBF is conditionally negative definite of order $1$ on every
$\mathbb{R}^{d}$.

###### Proof.

We show that $\psi(r)=-\phi(r)$ is conditionally positive definite of order
$1$. We have $\psi(r)=f(s)=-\sqrt{s}\tanh\left(\frac{\sqrt{s}}{c}\right)$,
where $s=r^{2}.$ Now for

$g(s)=-f^{\prime}(s)=\frac{1}{2}s^{-\frac{1}{2}}\tanh\left(\frac{\sqrt{s}}{c}\right)+\frac{1}{2c}\left(1-\tanh^{2}\left(\frac{\sqrt{s}}{c}\right)\right),$

we have

$(-1)^{l}g^{(l)}(s)\geq
0,\quad\mbox{for}~{}\mbox{all}~{}l\in\mathbb{N}_{0}~{}\mbox{and}~{}\mbox{all}~{}s>0.$

So $-f^{\prime}(s)$ is completely monotone on $(0,\infty)$. Now, since
$f\not\in\Pi_{m}^{d}$ , the claim is proved according to Micchelli’s theorem
[5]. ∎

######  Remark 2.1.

Since $\phi$ is conditionally negative definite of order $1$ and $\phi(0)=0$,
then the matrix $A=\left[\phi(\|x_{i}-x_{j}\|)\right]_{1\leq i,j\leq n}$ has
one positive and $n-1$ negative eigenvalues and in particular it is
invertible.

In the sequel, we consider $d=1$, since our work is confined to the univariate
case. We have seen before that the RTH RBF is an smooth approximant to
$\tau(r)=r$ with higher accuracy and better convergence properties than the MQ
RBF $\sqrt{r^{2}+c^{2}},$ by decreasing shape parameter $c$. In Figure 2, we
have plotted both RTH basis

$\displaystyle\phi_{j}(x)=\left(x-x_{j}\right)\tanh\left(\frac{x-x_{j}}{c}\right),$
(9)

and MQ basis (2) centered at $x_{j}=0$. It can be noted from Figure 2 that the
RTH RBF approaches to $|x|$ faster than the MQ RBF, even with larger shape
parameters. Moreover, in RTH RBF $\phi_{j}(x_{j})=0$ independent of the value
of $c$, but MQ requires that $c=0$. This property of the RTH RBF leads to
getting more accurate results in corresponding quasi-interpolants.

Figure 2: Plots of Multiquadric RBF (left), and RTH RBF (right) for different
values of shape parameter $c$.

The first and second derivatives of the RTH RBF (9) are of the form

$\displaystyle\phi^{{}^{\prime}}_{j}(x)$
$\displaystyle=\tanh\left({\frac{x-x_{j}}{c}}\right)+{\frac{(x-x_{j})}{c}\left(1-\left(\tanh\left({\frac{x-x_{j}}{c}}\right)\right)^{2}\right)},$
$\displaystyle\phi^{{}^{\prime\prime}}_{j}(x)$
$\displaystyle=2\,{\frac{1}{c}\left(1-\left(\tanh\left({\frac{x-x_{j}}{c}}\right)\right)^{2}\right)}-2\,{\frac{(x-x_{j})}{{c}^{2}}\tanh\left({\frac{x-x_{j}}{c}}\right)\left(1-\left(\tanh\left({\frac{x-x_{j}}{c}}\right)\right)^{2}\right)},$

and are plotted in Figure 3 for $c=1$.

(a)

(b)

(c)

Figure 3: RTH RBF (a), first derivative (b), and second derivative of RTH RBF
(c) with shape parameter $c=1$

In Tables 2 and 3, we summarized the properties of both MQ and RTH RBFs, where
$\xi=1.199678640,$ is obtained numerically by calculating the roots of the
second derivative.

Table 2: Comparing both RBFs. Name | $\phi_{j}(x)$ | $\displaystyle\lim_{x\rightarrow x_{j}}\phi_{j}(x)$ | $\displaystyle\lim_{c\rightarrow 0}\phi_{j}(x)$ | $\displaystyle\lim_{x\rightarrow\pm\infty}\phi^{{}^{\prime}}_{j}(x)$ | condition
---|---|---|---|---|---
MQ RBF | $\sqrt{c^{2}+(x-x_{j})^{2}}$ | $c$ | $\lvert x-x_{j}\rvert$ | $\pm 1$ | $x\in(-\infty,\infty)$
RTH RBF | $\left(x-x_{j}\right)\tanh\left(\dfrac{x-x_{j}}{c}\right)$ | $0$ | $\lvert x-x_{j}\rvert$ | $\pm 1$ | $x\in(-\infty,\infty)$
Table 3: Comparing both RBFs. Name | $\phi_{j}(x)$ | $\phi^{{}^{\prime}}_{j}(x)$ | $\phi^{{}^{\prime\prime}}_{j}(x)$ | condition
---|---|---|---|---
MQ RBF | $\sqrt{c^{2}+(x-x_{j})^{2}}$ | Strictly increasing | $\geq 0$ | $x\in(-\infty,\infty)$
RTH RBF | $\left(x-x_{j}\right)\tanh\left(\dfrac{x-x_{j}}{c}\right)$ | Strictly increasing | $\geq 0$ | $x\in[-c\xi,c\xi]$

###  2.2 Quasi-interpolation operator

The quasi-interpolation operator of a function $f:[a,b]\rightarrow\mathbb{R}$
with RTH RBF on the scattered points

$\displaystyle a=x_{0}<x_{2}<\dotsb<x_{n}=b\qquad h:=\max_{2\leq j\leq
n}(x_{j}-x_{j-1}),$ (10)

has the form

$(\mathcal{L}_{RTH}f)(x)=f_{0}\alpha_{0}(x)+f_{1}\alpha_{1}(x)+\sum_{j=2}^{n-2}f_{j}\psi_{j}(x)+f_{n-1}\alpha_{n-1}(x)+f_{n}\alpha_{n}(x)$
(11)

where

$\displaystyle\alpha_{0}(x)$
$\displaystyle=\dfrac{1}{2}+\dfrac{\phi_{1}(x)-(x-x_{0})}{2(x_{1}-x_{0})},$
$\displaystyle\alpha_{1}(x)$
$\displaystyle=\dfrac{\phi_{2}(x)-\phi_{1}(x)}{2(x_{2}-x_{1})}-\dfrac{\phi_{1}(x)-(x-x_{0})}{2(x_{1}-x_{0})},$
$\displaystyle\alpha_{n-1}(x)$
$\displaystyle=\dfrac{(x_{n}-x)-\phi_{n-1}(x)}{2(x_{n}-x_{n-1)}}-\dfrac{\phi_{n-1}(x)-\phi_{n-2}(x)}{2(x_{n-1}-x_{n-2})},$
$\displaystyle\alpha_{n}(x)$
$\displaystyle=\dfrac{1}{2}+\dfrac{\phi_{n-1}(x)-(x_{n}-x)}{2(x_{n}-x_{n-1})},$
$\displaystyle\phi_{j}(x)$
$\displaystyle=\left(x-x_{j}\right)\tanh\left(\frac{x-x_{j}}{c}\right),\quad
j=1,\ldots,n-1,~{}c\in\mathbb{R}_{+},$ $\displaystyle\psi_{j}(x)$
$\displaystyle=\dfrac{\phi_{j+1}(x)-\phi_{j}(x)}{2(x_{j+1}-x_{j})}-\dfrac{\phi_{j}(x)-\phi_{j-1}(x)}{2(x_{j}-x_{j-1)}},\qquad
2\leq j\leq n-2.$

The formula (11) can be rewritten as

$\displaystyle\left(\mathcal{L}_{RTH}f\right)(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{j=1}^{n-1}f[x_{j-1},x_{j},x_{j+1}](x_{j+1}-x_{j-1})\phi_{j}(x)+$
$\displaystyle\frac{f_{0}+f_{n}}{2}+\frac{1}{2}f[x_{0},x_{1}](x-x_{0})-\frac{1}{2}f[x_{n-1},x_{n}](x_{n}-x).$

Let
$\phi_{-1}(x)=|x-x_{-1}|,~{}\phi_{0}(x)=|x-x_{0}|,~{}\phi_{n}(x)=|x-x_{n}|$
and $\phi_{n+1}(x)=|x-x_{n+1}|$, then for $x\in[x_{0},x_{n}]$, the operator
$\mathcal{L}_{RTH}$ can be rearranged as

$\displaystyle(\mathcal{L}_{RTH}f)(x)=\sum_{j=0}^{n}f_{j}\psi_{j}(x),$ (13)

where

$\displaystyle\psi_{j}(x)=\frac{\phi_{j+1}(x)-\phi_{j}(x)}{2(x_{j+1}-x_{j})}-\frac{\phi_{j}(x)-\phi_{j-1}(x)}{2(x_{j}-x_{j-1})},\quad
j=0,\ldots,n,$

and $x_{-1}<x_{0},~{}x_{n+1}>x_{n}.$

######  Remark 2.2.

From relation (2.2), it is clear that the quasi-interpolation operator
$\mathcal{L}_{RTH}$ reproduces the linear polynomials on $[x_{0},x_{n}]$, that
is

$\displaystyle\sum_{j=0}^{n}(ax_{j}+b)\psi_{j}(x)=ax+b,\quad
a,b\in\mathbb{R}\,,$ (14)

from which we also get $\displaystyle\sum_{j=0}^{n}\psi_{j}(x)=1$ at any point
$x\in[x_{0},x_{n}]$.

In order to prove the shape-preserving property of the quasi-interpolation
operator (11), we give some important definitions and theorems from
differential geometry (cf. e.g. [44]).

######  Definition 2.1.

A differentiable plane curve $\alpha:(a,b)\rightarrow\mathbb{R}^{2}$ is said
to be regular if its derivative never vanishes. That is

$\displaystyle\forall
t\in(a,b),\quad\quad\quad\quad\alpha^{\prime}(t)=\left(\frac{d\alpha_{1}}{dt},\frac{d\alpha_{2}}{dt}\right)\neq(0,0).$

######  Theorem 2.4.

Let $C$ be a regular plane curve given by $\alpha(t)$. Then the curvature
$\kappa$ of $C$ at $t$ is given by

$\kappa[\alpha](t)=\left\|\alpha^{\prime}(t)\times\alpha^{\prime\prime}(t)\right\|/\left\|\alpha^{\prime}(t)\right\|^{3}.$

######  Definition 2.2.

Let $f\in C^{2}[a,b]$. The curvature of the plane curve $y=f(x)$ is given by

$\kappa(x)=\dfrac{\lvert
f^{\prime\prime}(x)\rvert}{(1+(f^{\prime}(x))^{2})^{\frac{3}{2}}}.$

######  Theorem 2.5 (Fundamental theorem of plane curves).

Let $\alpha,\gamma:(a,b)\rightarrow\mathbb{R}^{2}$ be regular plane curves
such that $\kappa[\alpha](t)=\kappa[\gamma](t)$ for all $t\in(a,b)$. Then
there is an orientation-preserving Euclidean motion
$F:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ such that $\gamma=F~{}o~{}\alpha$.

######  Corollary 2.5.1.

Two unit-speed plane curves which have the same curvature differ only by a
Euclidean motion.

######  Theorem 2.6.

The quasi-interpolation operator $\mathcal{L}_{RTH}$ constructed by data
points $\\{(x_{j},f_{j})\\}$, is monotonicity and convexity-preserving for $c$
small enough.

###### Proof.

According to the Corollary 2.5.1, it suffices to show that

$\displaystyle\displaystyle\lim_{c\rightarrow
0}\lvert\kappa_{\mathcal{L}_{MQ}}(x)-\kappa_{\mathcal{L}_{RTH}}(x)\rvert=0.$

Let $x\neq x_{j}$, otherwise both quasi-interpolants (3) and (11) do not have
first and second derivatives as $c$ approaches $0$. Now, according to
definition 2.2, we have

$\displaystyle\kappa_{{\mathcal{L}_{MQ}}}(x)=\dfrac{\lvert({\mathcal{L}_{MQ}}f)^{\prime\prime}(x)\rvert}{\left(1+\left(({\mathcal{L}_{MQ}}f)^{\prime}(x)\right)^{2}\right)^{\frac{3}{2}}}.$

Since for MQ RBF,

$\displaystyle\phi_{j}^{\prime\prime}(x)=\frac{c^{2}}{\left(c^{2}+(x-x_{j})^{2}\right)^{3/2}},$

then

$\displaystyle\lim_{c\longrightarrow 0}\phi_{j}^{\prime\prime}(x)=0.$

Moreover

$\displaystyle(\mathcal{L}_{MQ}f)^{\prime\prime}(x)=\dfrac{1}{2}\sum_{j=1}^{n-1}\left[\dfrac{f_{j+1}-f_{j}}{x_{j+1}-x_{j}}-\dfrac{f_{j}-f_{j-1}}{x_{j}-x_{j-1}}\right]\phi_{j}^{\prime\prime}(x),$

then

$\displaystyle\lim_{c\longrightarrow 0}\kappa_{\mathcal{L}_{{{MQ}}}}(x)=0,$

which leads to

$\displaystyle\forall\epsilon>0\quad\exists\delta_{1}>0;\quad\lvert\,c\,\rvert<\delta_{1}\Rightarrow\lvert\kappa_{\mathcal{L}_{{{MQ}}}}(x)\rvert<\epsilon.$

Similarly, for RTH RBF, we have

$\displaystyle\lim_{c\longrightarrow 0}\phi_{j}^{\prime\prime}(x)=0,$

then

$\displaystyle\lim_{c\longrightarrow 0}\kappa_{\mathcal{L}_{{{RTH}}}}(x)=0,$

which leads to

$\displaystyle\forall\epsilon>0\quad\exists\delta_{2}>0;\quad\lvert\,c\,\rvert<\delta_{2}\Rightarrow\lvert\kappa_{\mathcal{L}_{{{RTH}}}}(x)\rvert<\epsilon.$

The proof completes by considering $\delta=\min\\{\delta_{1},\delta_{2}\\}$. ∎

##  3 Accuracy of the quasi-interpolation operator $\mathcal{L}_{RTH}$

In this section, we give an approximation order for the quasi-interpolation
operator $\mathcal{L}_{RTH}.$

######  Theorem 3.1.

Assume $f^{\prime\prime}$ is Lipschitz continuous. The quasi-interpolation
operator $\mathcal{L}_{RTH}f$, at the point set (10) as $h\rightarrow 0$,
converges as follows

$\displaystyle\|f-\mathcal{L}_{RTH}f\|_{\infty}\leq kh^{2},$ (15)

where $k$ is independent of $h$ and $c$.

###### Proof.

Let $t(y)$ be the local Taylor approximation of $f$ at $y$, that is

$\displaystyle t(y)=f(x)+f^{\prime}(x)(y-x)\,,\;\;x\in[a,b]$

According to Remark 2.2, we get

$\displaystyle\sum_{j=0}^{n}(x-x_{j})\psi_{j}(x)=0,\quad\sum_{j=0}^{n}\psi_{j}(x)=1.$

Then we get

$\displaystyle\sum_{j=0}^{n}t(x_{j})\psi_{j}(x)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\left[f(x)+f^{\prime}(x)(x_{j}-x)\right]\psi_{j}(x)$
$\displaystyle=$ $\displaystyle
f(x)\sum_{j=0}^{n}\psi_{j}(x)+f^{\prime}(x)\sum_{j=0}^{n}(x-x_{j})\psi_{j}(x)$
$\displaystyle=$ $\displaystyle f(x).$

Since $f^{\prime\prime}(x)$ is Lipschitz continuous, then for every
$x_{1},~{}x_{2}\in[a,b]$,
$|f^{\prime\prime}(x_{1})-f^{\prime\prime}(x_{2})|\leq c_{0}|x_{1}-x_{2}|,$
where $\displaystyle 0<c_{0}=\mbox{ess sup}_{a\leq x\leq
b}|f^{\prime\prime\prime}(x)|$. Now according to (2.2), we have

$\displaystyle|\mathcal{L}_{RTH}f(x)-f(x)|$ $\displaystyle=$
$\displaystyle\left|\sum_{j=0}^{n}\left(f(x_{j})-t(x_{j})\right)\psi_{j}(x)\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\left|\sum_{j=1}^{n-1}\left(f[x_{j-1},x_{j},x_{j+1}]-t[x_{j-1},x_{j},x_{j+1}]\right)(x_{j+1}-x_{j-1})\phi_{j}(x)\right|$
$\displaystyle+c_{1}(x-x_{0})^{2}+c_{2}(x_{n}-x)^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{4}\sum_{j=1}^{n-1}|f^{\prime\prime}(\xi)-f^{\prime\prime}(\eta)||\phi_{j}(x)\,(x_{j+1}-x_{j-1})|,\quad(\xi,\eta\in(x_{j-1},x_{j+1}))$
$\displaystyle+c_{1}(x-x_{0})^{2}+c_{2}(x_{n}-x)^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}c_{0}h\sum_{j=1}^{n-1}|x-x_{j}|(x_{j+1}-x_{j-1})+c_{1}(x-x_{0})^{2}+c_{2}(x_{n}-x)^{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}c_{0}h\sum_{\displaystyle\mathop{j=1}_{|x-x_{j}|\leq
h}}^{n-1}|x-x_{j}|(x_{j+1}-x_{j-1})+\frac{1}{2}c_{0}h\sum_{\displaystyle\mathop{j=1}_{|x-x_{j}|>h}}^{n-1}|x-x_{j}|(x_{j+1}-x_{j-1})$
$\displaystyle+c_{1}(x-x_{0})^{2}+c_{2}(x_{n}-x)^{2}$ $\displaystyle\leq$
$\displaystyle
4c_{0}h^{3}+c_{0}h\left(\int_{|x-t|>h}|x-t|~{}dt+O(h)\right)+c_{1}(x-x_{0})^{2}+c_{2}(x_{n}-x)^{2}$
$\displaystyle\leq$ $\displaystyle k_{1}h^{3}+k_{2}h^{2}+k_{3}{(b-a)}^{2}$
$\displaystyle\leq$ $\displaystyle kh^{2}$

∎

##  4 Numerical results

In this section, we compare the accuracy of the quasi-interpolation operator
$\mathcal{L}_{RTH}$ with that of Wu and Schaback, $\mathcal{L}_{MQ}$ (defined
in (3)) for the approximation of five functions. We take equidistant center
points and choose different shape parameters $c$ and also different step sizes
$h$. The maximum absolute error norm is then computed for comparing
approximation accuracy. The rate of convergence is also computed by

$r_{h}=\frac{\ln\left(\frac{E_{h_{i}}}{E_{h_{i-1}}}\right)}{\ln\left(\frac{h_{i}}{h_{i-1}}\right)},$

where $E_{h_{i}}$ indicates the error of the quasi-interpolant
$\mathcal{L}_{RTH}f$ corresponding to the parameter $h_{i}$. In all tests, we
chose $m=200$ equidistant evaluation points.

###  4.1 Test problem 1

In the first test problem, we apply the RTH quasi-interpolation to approximate
the function (cf. [45])

$\displaystyle f_{1}(x)=\frac{\sinh(x)}{1+\cosh(x)},\quad x\in[-3,3].$

The results are shown in Tables 4-6. In Tables 4, 5, and 6, we set
$h=0.1,0.01,0.001$, respectively, and $c=2h,h,0.5h,0.2h,0.1h$, then we compute
the $\|\mathcal{L}_{RTH}f-f\|_{\infty}$ and
$\|\mathcal{L}_{MQ}f-f\|_{\infty}$. In Table 7, we set $c=0.01$,
$h=0.2,0.1,0.05,0.025,0.0125$, to observe the convergence rate $r_{h}$ of
$\mathcal{L}_{RTH}f$ with the variation of $h$.

Table 4: Comparison of approximation accuracy of RTH and MQ quasi-interpolation for the test problem 1. $c$ | $0.2$ | $0.1$ | $0.05$ | $0.02$ | $0.01$
---|---|---|---|---|---
$h$ | $0.1$ | $0.1$ | $0.1$ | $0.1$ | $0.1$
$\|\mathcal{L}_{MQ}f-f\|_{\infty}$ | $9.3\times 10^{-3}$ | $3.1\times 10^{-3}$ | $1.1\times 10^{-3}$ | $3.8\times 10^{-4}$ | $2.8\times 10^{-4}$
$\|\mathcal{L}_{RTH}f-f\|_{\infty}$ | $2.9\times 10^{-3}$ | $6.2\times 10^{-4}$ | $7.1\times 10^{-5}$ | $2.3\times 10^{-4}$ | $2.4\times 10^{-4}$
Table 5: Comparison of approximation accuracy of RTH and MQ quasi-interpolation for the test problem 1. $c$ | $0.02$ | $0.01$ | $0.005$ | $0.002$ | $0.001$
---|---|---|---|---|---
$h$ | $0.01$ | $0.01$ | $0.01$ | $0.01$ | $0.01$
$\|\mathcal{L}_{MQ}f-f\|_{\infty}$ | $1.8\times 10^{-4}$ | $5.3\times 10^{-5}$ | $1.6\times 10^{-5}$ | $3.7\times 10^{-6}$ | $1.4\times 10^{-6}$
$\|\mathcal{L}_{RTH}f-f\|_{\infty}$ | $3.0\times 10^{-5}$ | $6.3\times 10^{-6}$ | $7.2\times 10^{-7}$ | $1.7\times 10^{-9}$ | $7.9\times 10^{-14}$
Table 6: Comparison of approximation accuracy of RTH and MQ quasi-interpolation for the test problem 1. $c$ | $0.002$ | $0.001$ | $0.0005$ | $0.0002$ | $0.0001$
---|---|---|---|---|---
$h$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | $0.001$
$\|\mathcal{L}_{MQ}f-f\|_{\infty}$ | $2.7\times 10^{-6}$ | $7.5\times 10^{-7}$ | $2.1\times 10^{-7}$ | $4.6\times 10^{-8}$ | $1.6\times 10^{-8}$
$\|\mathcal{L}_{RTH}f-f\|_{\infty}$ | $3.0\times 10^{-7}$ | $6.3\times 10^{-8}$ | $7.2\times 10^{-9}$ | $1.7\times 10^{-11}$ | $1.1\times 10^{-15}$
Table 7: Convergence rates of $\mathcal{L}_{RTH}f$ by using $c=0.01$, $h=0.2,0.1,0.05,0.025,0.0125$ for the test problem 1. $c$ | $0.01$ | $0.01$ | $0.01$ | $0.01$ | $0.01$
---|---|---|---|---|---
$h$ | $0.2$ | $0.1$ | $0.05$ | $0.025$ | $0.0125$
$\|\mathcal{L}_{RTH}f-f\|_{\infty}$ | $9.5\times 10^{-4}$ | $2.4\times 10^{-4}$ | $5.4\times 10^{-5}$ | $5.1\times 10^{-6}$ | $1.0\times 10^{-6}$
$r_{h}$ | - | $1.9855$ | $2.1657$ | $3.4056$ | $2.3028$

###  4.2 Test problem 2

In this experiment we apply the RTH quasi-interpolation to approximate the
function (again considered in [45])

$\displaystyle f_{2}(x)=\sin\left(\frac{x}{2}\right)-2\cos(x)+4\sin(\pi
x),\quad x\in[-4,4].$ (16)

The comparison results are shown in Tables 8-10. In Tables 8, 9, and 10, we
set $h=0.1,0.01,0.001$, respectively, and $c=2h,h,0.5h,0.2h,0.1h$, then we
compute the $\|\mathcal{L}_{RTH}f-f\|_{\infty}$ and
$\|\mathcal{L}_{MQ}f-f\|_{\infty}$. In Table 11, we set $c=0.01$,
$h=0.2,0.1,0.05,0.025,0.0125$, to observe the convergence rate $r_{h}$ of
$\mathcal{L}_{RTH}f$ with the variation of $h$.

Table 8: Comparison of approximation accuracy of RTH and MQ quasi-interpolation for the test problem 2. $c$ | $0.2$ | $0.1$ | $0.05$ | $0.02$ | $0.01$
---|---|---|---|---|---
$h$ | $0.1$ | $0.1$ | $0.1$ | $0.1$ | $0.1$
$\|\mathcal{L}_{MQ}f-f\|_{\infty}$ | $1.2$ | $4.5\times 10^{-1}$ | $1.7\times 10^{-1}$ | $7.1\times 10^{-2}$ | $5.4\times 10^{-2}$
$\|\mathcal{L}_{RTH}f-f\|_{\infty}$ | $4.5\times 10^{-1}$ | $1.2\times 10^{-1}$ | $1.4\times 10^{-2}$ | $4.5\times 10^{-2}$ | $4.9\times 10^{-2}$
Table 9: Comparison of approximation accuracy of RTH and MQ quasi-interpolation for the test problem 2. $c$ | $0.02$ | $0.01$ | $0.005$ | $0.002$ | $0.001$
---|---|---|---|---|---
$h$ | $0.01$ | $0.01$ | $0.01$ | $0.01$ | $0.01$
$\|\mathcal{L}_{MQ}f-f\|_{\infty}$ | $3.0\times 10^{-2}$ | $9.2\times 10^{-3}$ | $2.9\times 10^{-3}$ | $7.1\times 10^{-4}$ | $2.8\times 10^{-4}$
$\|\mathcal{L}_{RTH}f-f\|_{\infty}$ | $6.4\times 10^{-3}$ | $1.4\times 10^{-3}$ | $1.5\times 10^{-4}$ | $3.7\times 10^{-7}$ | $1.7\times 10^{-11}$
Table 10: Comparison of approximation accuracy of RTH and MQ quasi-interpolation for the test problem 2. $c$ | $0.002$ | $0.001$ | $0.0005$ | $0.0002$ | $0.0001$
---|---|---|---|---|---
$h$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | $0.001$
$\|\mathcal{L}_{MQ}f-f\|_{\infty}$ | $4.9\times 10^{-4}$ | $1.4\times 10^{-4}$ | $4.1\times 10^{-5}$ | $9.0\times 10^{-6}$ | $3.3\times 10^{-6}$
$\|\mathcal{L}_{RTH}f-f\|_{\infty}$ | $6.4\times 10^{-5}$ | $1.4\times 10^{-5}$ | $1.5\times 10^{-6}$ | $3.7\times 10^{-9}$ | $1.7\times 10^{-13}$
Table 11: Convergence rates of $\mathcal{L}_{RTH}f$ by using $c=0.01$, $h=0.2,0.1,0.05,0.025,0.0125$ for the test problem 2. $c$ | $0.01$ | $0.01$ | $0.01$ | $0.01$ | $0.01$
---|---|---|---|---|---
$h$ | $0.2$ | $0.1$ | $0.05$ | $0.025$ | $0.0125$
$\|\mathcal{L}_{RTH}f-f\|_{\infty}$ | $2.0\times 10^{-1}$ | $4.9\times 10^{-2}$ | $1.1\times 10^{-2}$ | $1.1\times 10^{-3}$ | $2.7\times 10^{-4}$
$r_{h}$ | - | $2.0101$ | $2.0899$ | $3.4262$ | $2.0034$

###  4.3 Test problem 3

Consider the function (see again [45])

$\displaystyle f_{3}(x)=10e^{-x^{2}}+x^{2},\quad x\in[-3,3],$ (17)

for approximating by the RTH quasi-interpolation operator. The comparison
results are shown in Tables 12-14. In Tables 12, 13, and 14, we set
$h=0.1,0.01,0.001$, respectively, and $c=2h,h,0.5h,0.2h,0.1h$, then we compute
the $\|\mathcal{L}_{RTH}f-f\|_{\infty}$ and
$\|\mathcal{L}_{MQ}f-f\|_{\infty}$. In Table 15, we set $c=0.01$,
$h=0.2,0.1,0.05,0.025,0.0125$, to observe the convergence rate $r_{h}$ of
$\mathcal{L}_{RTH}f$ on varying $h$.

Table 12: Comparison of approximation accuracy of RTH and MQ quasi-interpolation for the test problem 3. $c$ | $0.2$ | $0.1$ | $0.05$ | $0.02$ | $0.01$
---|---|---|---|---|---
$h$ | $0.1$ | $0.1$ | $0.1$ | $0.1$ | $0.1$
$\|\mathcal{L}_{MQ}f-f\|_{\infty}$ | $4.9\times 10^{-1}$ | $2.0\times 10^{-1}$ | $7.4\times 10^{-2}$ | $3.1\times 10^{-2}$ | $2.4\times 10^{-2}$
$\|\mathcal{L}_{RTH}f-f\|_{\infty}$ | $2.2\times 10^{-1}$ | $5.5\times 10^{-2}$ | $6.4\times 10^{-3}$ | $2.0\times 10^{-2}$ | $2.1\times 10^{-2}$
Table 13: Comparison of approximation accuracy of RTH and MQ quasi-interpolation for the test problem 3. $c$ | $0.02$ | $0.01$ | $0.005$ | $0.002$ | $0.001$
---|---|---|---|---|---
$h$ | $0.01$ | $0.01$ | $0.01$ | $0.01$ | $0.01$
$\|\mathcal{L}_{MQ}f-f\|_{\infty}$ | $1.3\times 10^{-2}$ | $4.0\times 10^{-3}$ | $1.3\times 10^{-3}$ | $3.1\times 10^{-4}$ | $1.2\times 10^{-4}$
$\|\mathcal{L}_{RTH}f-f\|_{\infty}$ | $2.8\times 10^{-3}$ | $5.9\times 10^{-4}$ | $6.7\times 10^{-5}$ | $1.6\times 10^{-7}$ | $7.4\times 10^{-12}$
Table 14: Comparison of approximation accuracy of RTH and MQ quasi-interpolation for the test problem 3. $c$ | $0.002$ | $0.001$ | $0.0005$ | $0.0002$ | $0.0001$
---|---|---|---|---|---
$h$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | $0.001$
$\|\mathcal{L}_{MQ}f-f\|_{\infty}$ | $2.1\times 10^{-4}$ | $6.0\times 10^{-5}$ | $1.8\times 10^{-5}$ | $3.9\times 10^{-6}$ | $1.4\times 10^{-6}$
$\|\mathcal{L}_{RTH}f-f\|_{\infty}$ | $2.8\times 10^{-5}$ | $5.9\times 10^{-6}$ | $6.7\times 10^{-7}$ | $1.6\times 10^{-9}$ | $7.5\times 10^{-14}$
Table 15: Convergence rates of $\mathcal{L}_{RTH}f$ by using $c=0.01$, $h=0.2,0.1,0.05,0.025,0.0125$; Test Problem 3. $c$ | $0.01$ | $0.01$ | $0.01$ | $0.01$ | $0.01$
---|---|---|---|---|---
$h$ | $0.2$ | $0.1$ | $0.05$ | $0.025$ | $0.0125$
$\|\mathcal{L}_{RTH}f-f\|_{\infty}$ | $8.6\times 10^{-2}$ | $2.1\times 10^{-2}$ | $5.0\times 10^{-3}$ | $4.7\times 10^{-4}$ | $2.6\times 10^{-4}$
$r_{h}$ | - | $2.0085$ | $2.0943$ | $3.4210$ | $2.0419$

######  Remark 4.1.

By analyzing the results in Tables 4-6, 8-10, and 12-14, we see that the
accuracy of the RTH quasi-interpolation scheme is dependent on the shape
parameter $c$ and on step size $h$. Furthermore, the accuracy of the RTH
quasi-interpolation operator is better than that of MQ for the same values of
$c$ and $h$. From Tables 7, 11, 15, we see that the convergence rate of
$\mathcal{L}_{RTH}$ reaches up to $2$ which justifies our theoretical findings
of Section 3. By these numerical experiments, we can say that the quasi-
interpolation $\mathcal{L}_{RTH}$ is a very attractive alternative, in terms
of accuracy and convergence, to $\mathcal{L}_{MQ}$.

###  4.4 Test problem 4 (Runge function)

Let us consider the Runge function on $[-1,1]$, that is $\displaystyle
f_{4}(x)=\frac{1}{1+25x^{2}}$. Figure 4 shows the exact and approximate values
of $f_{4}$ for $c=0.01,~{}h=0.1,~{}0.02$. In Figure 4, we see that the Runge
phenomenon has disappeared by decreasing $h$. Relative errors are shown in
Figure 5 using the RTH quasi-interpolation operator.

(a)

(b)

Figure 4: RTH quasi-interpolation of $f_{4}(x)=\frac{1}{1+25x^{2}};$ $h=0.1$
(a), $h=0.02$ (b), and $c=0.01.$

(a)

(b)

(c)

Figure 5: Relative errors: $c=0.1$ (a), $c=0.01$ (b), $c=0.001$ (c), and
$h=0.02;$ for the test problem 4.

###  4.5 Test problem 5 (Gibbs Phenomenon)

It is well-known that any global or high order approximation method suffers
from the Gibbs phenomenon if the function has a jump discontinuity in the
given domain. In this test problem, we show that the RTH quasi-interpolation
operator substantially mitigates the Gibbs phenomenon (cf. [46]).

$\displaystyle f_{5}(x)=\left\\{\begin{array}[]{ll}\frac{10}{3}x,&0\leq x\leq
0.3,\\\ 1,&0.3\leq x\leq 0.6,\\\ 0,&0.6<x\leq 1.\end{array}\right.$

Figure 6 shows the exact and approximate values of $f_{5}$. In Figure 6, we
see that the Gibbs oscillations are considerably attenuated by decreasing $c$.
Relative errors are reported in Figure 7.

(a)

(b)

(c)

Figure 6: Approximations of $f_{5}$ with RTH quasi-interpolation; $c=0.1$ (a),
$c=0.01$ (b), $c=0.001$ (c), and $h=0.01.$

(a)

(b)

(c)

Figure 7: Relative errors: $c=0.1$ (a), $c=0.01$ (b), $c=0.001$ (c), and
$h=0.01;$ Test problem 5.

###  4.6 Test problem 6 (A piecewise analytic function)

As a final example, we consider the piecewise analytic function (cf. [47])

$\displaystyle f_{6}(x)=\left\\{\begin{array}[]{ll}\sin(x),&x<0,\\\
\cos(x),&x>0,\end{array}\right.$

with $x\in[-1,1]$. Figure 8 shows the exact and approximate values of $f_{6}$,
where Gibbs oscillations are considerably attenuated by decreasing $c$.
Relative errors are shown in Figure 9.

(a)

(b)

(c)

Figure 8: RTH quasi-interpolation of the piecewise analytic function $f_{6}$;
$c=0.1$ (a), $c=0.01$ (b), $c=0.001$ (c), and $h=0.02.$

(a)

(b)

(c)

Figure 9: Relative errors: $c=0.1$ (a), $c=0.01$ (b), $c=0.001$ (c), and
$h=0.02;$ Test problem 6.

##  5 Conclusion

In this paper, an efficient shape preserving quasi-interpolation operator with
high degree of smoothness and very accurate results is proposed. It is based
on the reformulation of Wu–Schaback’s quasi-interpolation operator by a new
transcendental RBF of the form $\phi(r)=r\tanh\left({r\over c}\right)$. The
quasi-interpolation operator, called ${\mathcal{L}}_{RTH}$ has nice
convergence properties, being $\|{\mathcal{L}}_{RTH}-f\|_{\infty}\leq
k\,h^{2}$, with $h$ being the step size and $k$ a positive constant
independent on the shape parameter $c$ and the step size $h$ (cf. Theorem
3.1). Numerical experiments reveal that the proposed quasi-interpolation
operator not only gives very accurate results but also it does not suffer of
the Runge and Gibbs phenomena (see Test problems 4-6).
As a future work we are working in the application of the operator to real
worlds problems, in particular to irregular surfaces approximation and image
segmentation.

Acknowledgments. The third author thanks for the support the GNCS-INdAM. His
research has been done within the Italian Network on Approximation (RITA) and
the thematic group on "Approximation Theory and Applications" of the Italian
Mathematical Union.

## References

  * Wendland [2004] Holger Wendland.  _Scattered data approximation_ , volume 17.  Cambridge university press, 2004.
  * Buhmann [2003] Martin D Buhmann.  _Radial basis functions: theory and implementations_ , volume 12.  Cambridge university press, 2003.
  * Madych [1992] WR Madych.  Miscellaneous error bounds for multiquadric and related interpolators.  _Computers & Mathematics with Applications_, 24(12):121–138, 1992.
  * Madych and Nelson [1989] WR Madych and SA Nelson.  Error bounds for multiquadric interpolation.  _Approximation Theory VI_ , 2:413–416, 1989.
  * Micchelli [1986] Charles A Micchelli.  Interpolation of scattered data: distance matrices and conditionally positive definite functions.  _Constructive approximation_ , 2(1):11–22, 1986\.
  * Powell [1992a] Michael JD Powell.  The theory of radial basis function approximation in 1990.  _Advances in numerical analysis_ , pages 105–210, 1992a.
  * Powell [1992b] Michael JD Powell.  The theory of radial basis function approximation in 1990.  _Advances in numerical analysis_ , pages 105–210, 1992b.
  * Wendland [1995] Holger Wendland.  Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree.  _Advances in computational Mathematics_ , 4(1):389–396, 1995.
  * Wu [1995] Zongmin Wu.  Compactly supported positive definite radial functions.  _Advances in computational mathematics_ , 4(1):283–292, 1995.
  * Wu and Schaback [1993] Zong-min Wu and Robert Schaback.  Local error estimates for radial basis function interpolation of scattered data.  _IMA journal of Numerical Analysis_ , 13(1):13–27, 1993.
  * Fasshauer [1996] Gregory E Fasshauer.  Solving partial differential equations by collocation with radial basis functions.  In _Proceedings of Chamonix_ , volume 1997, pages 1–8. Citeseer, 1996\.
  * Fasshauer [2007] Gregory E Fasshauer.  _Meshfree approximation methods with MATLAB_ , volume 6.  World Scientific, 2007.
  * Franke and Schaback [1998] Carsten Franke and Robert Schaback.  Convergence order estimates of meshless collocation methods using radial basis functions.  _Advances in computational mathematics_ , 8(4):381–399, 1998.
  * Golberg et al. [1996] MA Golberg, CS Chen, and SR Karur.  Improved multiquadric approximation for partial differential equations.  _Engineering Analysis with boundary elements_ , 18(1):9–17, 1996.
  * Hon and Mao [1998] Yiu-Chung Hon and XZ Mao.  An efficient numerical scheme for Burgers’ equation.  _Applied Mathematics and Computation_ , 95(1):37–50, 1998.
  * Kansa [1990a] Edward J Kansa.  Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—i surface approximations and partial derivative estimates.  _Computers & Mathematics with applications_, 19(8-9):127–145, 1990a.
  * Kansa [1990b] Edward J Kansa.  Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—ii solutions to parabolic, hyperbolic and elliptic partial differential equations.  _Computers & mathematics with applications_, 19(8-9):147–161, 1990b.
  * Mohammadi et al. [2014] M Mohammadi, R Mokhtari, and R Schaback.  A meshless method for solving the 2d brusselator reaction-diffusion system.  _Comput. Model. Eng. Sci.(CMES)_ , 101:113–138, 2014.
  * Saberi Zafarghandi et al. [2019] Fahimeh Saberi Zafarghandi, Maryam Mohammadi, and Robert Schaback.  On the fractional derivatives of radial basis functions: Theories and applications.  _Mathematical Methods in the Applied Sciences_ , 42(11):3877–3899, 2019.
  * Wendland [1999] Holger Wendland.  Meshless galerkin methods using radial basis functions.  _Mathematics of computation_ , 68(228):1521–1531, 1999.
  * Zafarghandi et al. [2019] Fahimeh Saberi Zafarghandi, Maryam Mohammadi, Esmail Babolian, and Shahnam Javadi.  Radial basis functions method for solving the fractional diffusion equations.  _Applied Mathematics and Computation_ , 342:224–246, 2019\.
  * Hardy [1971] Rolland L Hardy.  Multiquadric equations of topography and other irregular surfaces.  _Journal of geophysical research_ , 76(8):1905–1915, 1971.
  * Hardy [1990] Rolland L Hardy.  Theory and applications of the multiquadric-biharmonic method 20 years of discovery 1968–1988.  _Computers & Mathematics with Applications_, 19(8-9):163–208, 1990.
  * Franke [1982] Richard Franke.  Scattered data interpolation: tests of some methods.  _Mathematics of computation_ , 38(157):181–200, 1982.
  * Beatson and Powell [1992] RK Beatson and MJD Powell.  Univariate multiquadric approximation: quasi-interpolation to scattered data.  _Constructive approximation_ , 8(3):275–288, 1992\.
  * Powell [1990] Michael James David Powell.  Univariate multiquadric approximation: reproduction of linear polynomials.  In _Multivariate Approximation and Interpolation_ , pages 227–240. Springer, 1990.
  * Wu and Schaback [1994] Zongmin Wu and Robert Schaback.  Shape preserving properties and convergence of univariate multiquadric quasi-interpolation.  _Acta Mathematicae Applicatae Sinica_ , 10(4):441–446, 1994.
  * Ling [2004] Leevan Ling.  A univariate quasi-multiquadric interpolationwith better smoothness.  _Computers & Mathematics with Applications_, 48(5-6):897–912, 2004.
  * Feng and Li [2009] Renzhong Feng and Feng Li.  A shape-preserving quasi-interpolation operator satisfying quadratic polynomial reproduction property to scattered data.  _Journal of computational and applied mathematics_ , 225(2):594–601, 2009.
  * Wang et al. [2010] Ren-Hong Wang, Min Xu, and Qin Fang.  A kind of improved univariate multiquadric quasi-interpolation operators.  _Computers & mathematics with applications_, 59(1):451–456, 2010.
  * Jiang et al. [2011] Zi-Wu Jiang, Ren-Hong Wang, Chun-Gang Zhu, and Min Xu.  High accuracy multiquadric quasi-interpolation.  _Applied Mathematical Modelling_ , 35(5):2185–2195, 2011.
  * Ling [2005] Leevan Ling.  Multivariate quasi-interpolation schemes for dimension-splitting multiquadric.  _Applied mathematics and computation_ , 161(1):195–209, 2005.
  * Wu et al. [2015] Ruifeng Wu, Tieru Wu, and Huilai Li.  A family of multivariate multiquadric quasi-interpolation operators with higher degree polynomial reproduction.  _Journal of Computational and Applied Mathematics_ , 274:88–108, 2015.
  * Gao and Wu [2012] Wenwu Gao and Zongmin Wu.  Quasi-interpolation for linear functional data.  _Journal of Computational and Applied Mathematics_ , 236(13):3256–3264, 2012.
  * Wang and Wu [2007] Renhong Wang and Jinming Wu.  Approximate implicitization based on rbf networks and mq quasi-interpolation.  _Journal of Computational Mathematics_ , pages 97–103, 2007.
  * Wu and Zhang [2014] Jinming Wu and Xiaolei Zhang.  A new approach for approximate implicitization of parametric curves.  _Computational and Applied Mathematics_ , 33(2):399–409, 2014.
  * Hon and Wu [2000] YC Hon and Zongmin Wu.  A quasi-interpolation method for solving stiff ordinary differential equations.  _International Journal for Numerical Methods in Engineering_ , 48(8):1187–1197, 2000.
  * Chen and Wu [2006] Ronghua Chen and Zongmin Wu.  Applying multiquadric quasi-interpolation to solve burgers’ equation.  _Applied mathematics and computation_ , 172(1):472–484, 2006.
  * Chen and Wu [2007] Ronghua Chen and Zongmin Wu.  Solving partial differential equation by using multiquadric quasi-interpolation.  _Applied Mathematics and Computation_ , 186(2):1502–1510, 2007.
  * Jiang and Wang [2012] Zi-Wu Jiang and Ren-Hong Wang.  Numerical solution of one-dimensional sine–gordon equation using high accuracy multiquadric quasi-interpolation.  _Applied Mathematics and Computation_ , 218(15):7711–7716, 2012.
  * Ramirez et al. [2013] Carlos Ramirez, Reinaldo Sanchez, Vladik Kreinovich, and Miguel Argaez.  $\sqrt{(}x^{2}+\mu)$ is the most computationally efficient smooth approximation to |x|: a proof.  2013\.
  * Voronin et al. [2014] Sergey Voronin, Gorkem Ozkaya, and Davis Yoshida.  Convolution based smooth approximations to the absolute value function with application to non-smooth regularization.  _arXiv preprint arXiv:1408.6795_ , 2014.
  * Wall [2018] Hubert Stanley Wall.  _Analytic theory of continued fractions_.  Courier Dover Publications, 2018.
  * O’neill [2006] Barrett O’neill.  _Elementary differential geometry_.  Elsevier, 2006.
  * Fasshauer and McCourt [2012] Gregory E Fasshauer and Michael J McCourt.  Stable evaluation of gaussian radial basis function interpolants.  _SIAM Journal on Scientific Computing_ , 34(2):A737–A762, 2012.
  * Chen et al. [2009] Ronghua Chen, Xuli Han, and Zongmin Wu.  A multiquadric quasi-interpolation with linear reproducing and preserving monotonicity.  _Journal of computational and applied mathematics_ , 231(2):517–525, 2009.
  * Jung [2007] Jae-Hun Jung.  A note on the gibbs phenomenon with multiquadric radial basis functions.  _Applied numerical mathematics_ , 57(2):213–229, 2007.