and Arbenz [@Arbenz-91], lies in the fact that the eigenvalues of a circulant matrix are easily computable. The final bounds for $\kappa_2(A)$ are obtained by using the Cauchy interlace theorem for singular values (see [@Horn-Johnson-91] for details), to bound both singular values in (\[1.2\]).

The paper is organized as follows. In Section 2 we briefly review some basic properties of cardinal B-splines. The proof of de Boor’s conjecture for low degree ($d \leq 6$) cardinal B-splines is given in Section 3. In Section 4 we describe the embedding technique and derive the estimates for $\kappa_2(A)$.

Despite all efforts, we are unable to prove de Boor’s conjecture in this, quite probably, the easiest case. The final section contains the results of numerical testing that strongly support the validity of the conjecture, as well as some additional conjectures based on these test results.

Properties of cardinal splines
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Let $x_i = x_0 + ih$, for $i = 0, \ldots, n$, be a sequence of simple uniformly spaced knots. This sequence determines a unique sequence of normalized B-splines $N_0^d, \ldots, N_{n - d - 1}^d$ of degree $d$, such that the spline $N_i^d$ is non-trivial only on the interval $\langle x_i, x_{i + 