Harold Widom’s work in Toeplitz operators

Abstract

This is a survey of Harold Widom’s work in Toeplitz operators, embracing his early results on the invertibility and spectral theory of Toeplitz operators, his investigations of the eigenvalue distribution of convolution operators, and his groundbreaking research into Toeplitz and Wiener–Hopf determinants.

1. The first half of the 20th century

In 1911, Otto Toeplitz studied doubly-infinite matrices of the form

$$\begin{equation*} (a_{j-k})_{j,k=-\infty }^\infty \end{equation*}$$

and showed in particular that such a matrix generates a bounded operator on $\ell ^2({\mathbf{Z}})$ if and only if the simply-infinite matrix $(a_{j-k})_{j,k=1}^\infty$ induces a bounded operator on $\ell ^2({\mathbf{N}})$. The former are now called Laurent matrices, whereas the latter are since then referred to as (infinite) Toeplitz matrices. It turns out that, and this is implicit already in Toeplitz’s article, the matrices define bounded operators on $\ell ^2$ if and only if there is a function $a$ in $L^\infty$ defined on the unit circle ${\mathbf{T}}$ such that the entries of the matrices are just the Fourier coefficients of $a$, that is,

$$\begin{equation*} a_k=\frac{1}{2\pi }\int _0^{2\pi } a(e^{i\theta })e^{-ik\theta }\,d\theta \quad (k \in {\mathbf{Z}}). \end{equation*}$$

If such a function $a$ exists, it is unique and is called the symbol of the corresponding Laurent or Toeplitz matrix, and these, as well as the operators they induce, are denoted by $L(a)$ and $T(a)$, respectively.

A finite $n \times n$ Toeplitz matrix may be regarded as a truncation of $T(a)$, and accordingly we write $T_n(a)\coloneq (a_{j-k})_{j,k=1}^n$. For such matrices, a pioneering result goes back to Gabor Szegő, who in 1915 established his celebrated first limit theorem, which states that if $a$ is positive, then the quotient $\det T_n(a)/\det T_{n-1}(a)$ converges to $G(a)\coloneq \exp (\frac{1}{2\pi }\int _0^{2\pi } \log a(e^{i\theta })\,d\theta )$ as $n \to \infty$. This theorem implies that if $a$ is real-valued, in which case the matrices $T_n(a)$ are all Hermitian, and if we denote by $\lambda _1(T_n(a)) \le \cdots \le \lambda _n(T_n(a))$ the eigenvalues of $T_n(a)$, then

$$\begin{equation*} \lim _{n \to \infty } \frac{1}{n}\sum _{j=1}^n \varphi (\lambda _j(T_n(a)))=\frac{1}{2\pi }\int _0^{2\pi } \varphi (a(e^{i\theta }))\,d\theta \end{equation*}$$

for every test function $\varphi \in C({\mathbf{R}})$. This is a first-order asymptotic result for the collective eigenvalue distribution of Toeplitz matrices. In 1952, eventually motivated by Lars Onsager’s formula for the spontaneous magnetization of the two-dimensional Ising model, Szegő improved the result to a second-order asymptotic formula, which is now called Szegő’s strong limit theorem. We refer to the article DIK3 for an exhaustive treatment of this story. Almost at the same time, in 1953, M. Kac, W. L. Murdock, and G. Szegő succeeded in describing the behavior of the extreme eigenvalues $\lambda _j(T_n(a))$ and $\lambda _{n-j}(T_n(a))$ ($j$ fixed and $n \to \infty$).

Since $L(a)$ is unitarily equivalent to the operator of multiplication by $a$ on $L^2({\mathbf{T}})$, it follows that $L(a)$ is invertible if and only if $a$ is invertible in $L^\infty ({\mathbf{T}})$. Invertibilty of $T(a)$ is a much more delicate issue and is a problem that has been studied by many authors since the appearance of Toeplitz operators up to the present. In 1929, A. Wintner solved the problem for triangular matrices $T(a)$, and in 1954, P. Hartman and A. Wintner showed that if $a$ is real-valued ($\Leftrightarrow$ $T(a)$ is Hermitian), then the spectrum of $T(a)$ is the convex hull of the essential range of $a$.

Toeplitz operators are closely related to a series of other operators, namely, the operators coming from the Riemann–Hilbert boundary value problem, singular integral operators, and Wiener–Hopf integral operators. This connection was not fully understood in those times, but nowadays we know that every result on an operator belonging to one of the last three classes is a result on Toeplitz operators. Many mathematicians, including F. Noether, J. Plemelj, S. G. Mikhlin, G. Fichera, and T. Carleman, studied singular integral operators with continuous coefficients and realized that, stated in terms of Toeplitz operators with continuous symbols, for $T(a)$ to be invertible it is sufficient that $a$ have no zeros on ${\mathbf{T}}$ and that the winding number of $a$ about the origin be zero. Finally, in 1952, Israel Gohberg, by an ingenious application of the Gelfand theory of Banach algebras, was able to prove that these two conditions are also necessary for $T(a)$ to be invertible.

In 1931, Norbert Wiener and Eberhard Hopf published their paper on what is now called Wiener–Hopf factorization. This factorization amounts to factoring $T(a)$ into a product of an upper and a lower triangular matrix. However, a complete understanding of that method was gained only in the works of F. D. Gakhov in 1949 and of I. Gohberg and Mark Krein in the 1950s.

2. Invertiblity, Fredholmness, and spectra

Harold Widom entered the Toeplitz operators scene with his 1959 paper 2 jointly with Alberto Calderón and Frank Spitzer. This paper deals with Toeplitz operators $T(a)$ generated by symbols $a$ in the Wiener algebra, that is, by symbols $a$ satisfying $\sum |a_n| < \infty$. The authors consider $T(a)$ as an operator on $\ell ^\infty$ and on $\ell ^2$, and they show that in both contexts $T(a)$ is invertible if and only if $a$ has no zeros on ${\mathbf{T}}$ and has winding number zero about the origin. The approach is based on the Wiener–Hopf factorization $a(t)=a_-(t)t^\kappa a_+(t)$, which gives the inverse operator $T^{-1}(a)=T(a_+^{-1})T(a_-^{-1})$ in the case of invertibility ($\Leftrightarrow$ $\kappa =0$) and the kernel and co-kernel dimensions of $T(a)$ for $\kappa \neq 0$. The paper was submitted in May 1958, and in a note added in proof, the authors remark that a substantial part of their results are also in a 1958 paper by M. Krein. However, one theorem of 2 was not in Krein’s paper: it replaces the condition $\sum |a_n| <\infty$ by the sole requirement that $a \in L^\infty ({\mathbf{T}})$ and says that $T(a)$ is invertible on $\ell ^2$ whenever $a$ is invertible in $L^\infty ({\mathbf{T}})$ and $a/|a|=\exp (i\,\widetilde{v})$ with $v \in L^\infty ({\mathbf{T}})$ and $\widetilde{v}$ denoting the conjugate function of $v$.

In his 1960 paper 3, which was submitted in August 1958, Widom then laid the foundations for the invertibility theory of Toeplitz operators on $\ell ^2$. The paper has four theorems. In Theorems II and III, unaware of the work of Wintner and of Hartman–Wintner, he rediscovered their invertibility criteria for triangular and Hermitian Toeplitz matrices. Theorem I was a real breakthrough. It states that for $T(a)$ to be invertible, it is necessary and sufficient that $a=a_-a_+$ with $a_-^{\pm 1} \in L^2_-({\mathbf{T}})$, $a_+^{\pm 1} \in L^2_+({\mathbf{T}})$ such that the operator $f \mapsto a_+^{-1}Pa_-^{-1}f$ is bounded on $L^2({\mathbf{T}})$. Here $L^2_{\pm }({\mathbf{T}})$ are the usual Hardy spaces and $P$ is the orthogonal projection of $L^2({\mathbf{T}})$ onto $L^2_+({\mathbf{T}})$. Note that $P=(I+S)/2$, where $S$ is the Cauchy singular integral operator. It was a lucky tie of events that just at that time, in 1960, H. Helson and G. Szegő were able to characterize the weights $w$ for which $S$ is bounded on $L^2({\mathbf{T}},w)$. Combining his Theorem I and the Helson–Szegő theorem, Widom arrived at the conclusion that $T(a)$ is invertible if and only if

$$\begin{equation*} a/|a|=\exp (i(c+u+\widetilde{v})), \end{equation*}$$

where $c$ is a real constant, $u$ and $v$ are two real-valued functions in $L^\infty ({\mathbf{T}})$, and $\| u\|_\infty < \pi /2$. This beautiful result, which was published in 1960 by Widom in 5 and was rediscovered by Allen Devinatz in 1964, is referred to in the textbooks as the Widom–Devinatz theorem. We should mention that an essential generalization of Widom’s Theorem I, namely, its extension to Toeplitz operators with matrix-valued symbols on the Hardy spaces $L^p_+({\mathbf{T}})$ was independently discovered by Igor Simonenko in 1961.

Theorem IV of Widom’s paper 3 was another milestone. It concerns the case where $a$ is piecewise continuous with at most finitely many jumps. Consider the continuous and naturally oriented curve in the plane that arises from the essential range of $a$ by filling in line segments between the endpoints $a(t-0)$ and $a(t+0)$ of each jump. Widom proved that $T(a)$ is invertible on $\ell ^2$ if and only if this curve does not contain the origin and has winding number zero about the origin. This was the very beginning of a long and fascinating story.

The first chapter of this story was written by none other than Widom himself in 6. The $\ell ^2$ theory of Toeplitz operators bifurcates into the $\ell ^p$ and $L^p$ theories for $1 < p < \infty$. The latter two theories are based on completely different techniques although, and this is something of a mystery, in the case of piecewise continuous symbols the final results are almost identical. In 6, Widom studied Toeplitz operators $T(a)$ with piecewise continuous symbols $a$ on the Hardy space $L^p_+({\mathbf{R}})$ of the upper half-plane. These operators are defined by $f \mapsto P(af)$, where $P=(I+S)/2$ and $S$ is the Cauchy singular integral operator on $L^p({\mathbf{R}})$. (One could equally well work on $L^p_+({\mathbf{T}})$, the differences being only technical and psychological.) Widom again arrived at the boundedness of $f \mapsto a_+^{-1}Pa_-^{-1}f$ on $L^p_+({\mathbf{R}})$, understood that this is a question about the weights $w$ for which $S$ is bounded on $L^p({\mathbf{R}},w)$, and showed that $S$ is bounded if

$$\begin{equation*} w(x)=(1+|x|)^\alpha \prod _{k=1}^m |x-x_k|^{\alpha _k} \end{equation*}$$

with

$$\begin{equation*} -1/p < \alpha _k < 1/q \quad \text{and} \quad -1/p< \alpha +\sum _{k=1}^m \alpha _k < 1/q, \end{equation*}$$

where $1/p+1/q=1$. Using this insight, he was able to prove that $T(a)$ is invertible on $L^p_+({\mathbf{R}})$ if and only if a certain curve does not contain the origin and has winding number zero about the origin. This curve results from the essential range of $a$ by filling in certain circular arcs $\mathcal{A}_p(a(x-0),a(x+0))$ depending on $p$ between the endpoints of the jumps at $x \in {\mathbf{R}}$ and the arc $\mathcal{A}_q(a(+\infty ),a(-\infty ))$ for the jump at infinity. Here, for two distinct points $\alpha ,\beta \in {\mathbf{C}}$ and a number $r \in (1,\infty )$, we denote by $\mathcal{A}_r(\alpha ,\beta )$ the circular arc at the points of which the line segment $[\alpha ,\beta ]$ is seen at the angle $2\pi /\max \{r,s\}$, where $1/r+1/s=1$, and which lies on the right (resp., left) of the oriented line passing first $\alpha$ and then $\beta$ if $1<r<2$ (resp., $2<r<\infty$). For $r=2$, $\mathcal{A}_r(\alpha ,\beta )$ is simply the line segment $[\alpha ,\beta ]$. For example, if $a(x)=\text{sign}\, x$, then we have two circular arcs $\mathcal{A}_p(-1,1)$ and $\mathcal{A}_q(1,-1)$, and since $\mathcal{A}_q(1,-1)=\mathcal{A}_p(-1,1)$, it follows that $T(\text{sign})$ is invertible if and only if $p \neq 2$. Widom also computed the kernel and co-kernel dimensions of the operators if the curve has nonzero winding number. Overall, paper 6 contained the full Fredholm theory of Toeplitz operators with piecewise continuous symbols on $L^p_+({\mathbf{R}})$, including an index formula.

In different language, particular cases of the Fredholm results of 6 were already evident in papers by B. V. Hvedelidze since 1947. The characterization of the weights $w$ for which $S$ is bounded on $L^p(\Gamma ,w)$ has a long history, starting with G. H. Hardy and J. E. Littlewood and culminating with work by R. Hunt, B. Muckenhoupt, R. Wheeden (1973), A. Calderón (1977), and G. David (1984). In the late 1960s and the 1970s, I. Gohberg and N. Krupnik introduced their local principle by means of which they could not only give a simpler proof of Widom’s result but also consider Lyapunov curves $\Gamma$ with power weights $w$, the case of matrix-valued symbols, and Banach algebras generated by Toeplitz operators with piecewise continuous symbols. In 1972, R. Duduchava settled matters for Toeplitz operators on $\ell ^p$. The theory reached a certain final stage only in the 1990s by work of I. Spitkovsky (general weights $w$) and Yu. I. Karlovich and the second author (general curves $\Gamma$ and general weights $w$). In these more general situations, Harold Widom’s circular arcs undergo a metamorphosis into horns, logarithmic spirals, logarithmic horns, and eventually into logarithmic leaves with a halo BK.

The invertibility and Fredholm criteria for Toeplitz operators with analytic, real-valued, or piecewise continuous symbols imply a description of the spectrum and of the essential spectrum of the operators. (The essential spectrum of an operator $T$ is the set of all complex $\lambda$ for which $T-\lambda I$ is not Fredholm, that is, not invertible modulo compact operators.) In all known cases, the spectrum and essential spectrum turned out to be connected sets, and in 1963, Paul Halmos posed the question whether the spectrum of $T(a)$ is connected for every $a \in L^\infty ({\mathbf{T}})$. In 10, submitted in April 1963, Widom proved that the answer is Yes for the spectrum of Toeplitz operators on $\ell ^2$, and in his paper 12 of 1966, he performed the same feat for Toeplitz operators on $L^p_+({\mathbf{T}})$. In 1972, Ronald Douglas established the connectedness of the essential spectrum of Toeplitz operators on $\ell ^2$, and only in 2009, A. Yu. Karlovich and I. Spitkovsky KS were able to prove that both the spectrum and the essential spectrum of Toeplitz operators are always connected on $L^p_+(\Gamma ,w)$ for $1<p<\infty$ and general curves $\Gamma$ and weights $w$.

We cannot conclude this section without mentioning that several basic results on Toeplitz operators, which nowadays appear on the first pages of the textbooks, were established just around 1960, and that tracing back to the sources of these results is a subtle matter. For example, the Brown–Halmos theorem, according to which the spectrum of $T(a)$ is a subset of the convex hull of the essential range of $a$, though explicitly published for the first time by P. Halmos and A. Brown in 1963, was known to at least Widom and I. B. Simonenko already in 1960. As for Widom, the theorem is in his article 11, which is based on lectures at the IAS in 1960. We also remark that in the very early 1960s, Simonenko Sim1Sim2 already had the results of 14 on locally sectorial symbols and the theorem that a Toeplitz operator is invertible if and only if it is Fredholm of index zero, which was published by Lewis Coburn in 1967 and has been known as Coburn’s lemma since then. Those years were indeed turbulent times.

3. Extreme eigenvalues of convolution operators

Another topic of Harold Widom’s early work is extreme eigenvalues of integral operators of the form

$$\begin{equation*} (C_\tau f)(x)=\int _{-\tau }^\tau k(x-y)f(y)\,dy, \quad x \in (-\tau ,\tau ), \end{equation*}$$

considered on $L^2(-\tau ,\tau )$. These operators are the continuous analogue of finite Toeplitz matrices. Since the kernel of the operator is translation invariant, we may change integration over $(-\tau ,\tau )$ to integration over $(0,2\tau )$ and therefore think of $C_\tau$ as the compression to $L^2(0,2\tau )$ of the Wiener–Hopf operator with the kernel $k(x-y)$, in which integration goes from $0$ to $\infty$. The symbol of such operators is the Fourier transform of the function $k$, $\hat{k}(\xi )\coloneq \int _{-\infty }^\infty k(x)e^{i\xi x}\,dx$. Of interest is the case in which the function $k$ is real-valued and even and in $L^1({\mathbf{R}})$. In that case $C_\tau$ is a compact Hermitian operator and we may label the eigenvalues as $\lambda _1(C_\tau ) \ge \lambda _2(C_\tau ) \ge \cdots .$ As predicted by Kac, Murdock, and Szegő, who studied the extreme eigenvalues of Hermitian Toeplitz matrices, the asymptotic behavior of $\lambda _j(C_\tau )$ for fixed $j$ and for $\tau \to \infty$ depends heavily on the behavior of the symbol $\hat{k}$ near its maximum. Suppose that the maximal value is $1$ and that it is attained at $\xi =0$ and only there. Under the assumption that $\hat{k}(\xi )=1-c|\xi |^\alpha +o(|\xi |^\alpha )$ as $\xi \to 0$ and that some more minor technical conditions are satisfied, Widom proved that

$$\begin{equation*} \lambda _j(C_\tau )=1-\frac{c}{\mu _{j,\alpha }}\,\frac{1}{\tau ^\alpha }+o\left(\frac{1}{\tau ^\alpha }\right)\quad \text{as}\quad \tau \to \infty , \end{equation*}$$

where the $\mu _{j,\alpha }$ are certain constants. For $\alpha =2$, this was done in his 1958 paper 1, where he even improved the $o(1/\tau ^2)$ to $\nu _{j,\alpha }/\tau ^3+o(1/\tau ^3)$. Papers 7 and 8 of 1961 are for general $\alpha \in (0,\infty )$. The constants $\mu _{j,\alpha }$ are shown to be the eigenvalues of a certain positive definite integral operator with some kernel $K_\alpha (x,y)$ on $L^2(-1,1)$. If $\alpha =2k$ is an even natural number, then $K_\alpha (x,y)$ is Green’s function of the differential operator $u \mapsto (-1)^ku^{(2k)}$ on $(-1,1)$ with the boundary conditions $u^{(\ell )}(-1)=u^{(\ell )}(1)=0$ for $0 \le \ell \le k-1$.

To prove these results, Widom derives a formula for the determinants of banded Toeplitz matrices and some kind of an analogue of this formula for integral operators. These formulas are of interest by themselves and the starting point of yet another story. Subtracting $\lambda I$ and setting the resulting determinants zero, he gets the eigenvalues, and a clever approximation argument then yields the desired result. Widom’s 1963 paper 9 is devoted to the extreme eigenvalues of the compressions of convolution operators on $L^2({\mathbf{R}}^n)$ to $L^2(\tau \Omega )$ as $\tau \to \infty$.

Extreme eigenvalues of Hermitian Toeplitz matrices, whose symbol has a prescribed behavior near the maximum that is governed by a parameter $\alpha \in (0,\infty )$, were thoroughly studied by Seymor Parter in the 1960s. As Harold told us, there was an agreement between Parter and him that Parter should focus on the Toeplitz case while he would embark on the Wiener–Hopf case.

This is also the right place for another story. In May 2008, the second author received a (handwritten!) letter from Peter Dörfler with the question whether there are results on the large $n$ behavior of the maximal singular value ($=$ spectral norm) of the $(n+1) \times (n+1)$ triangular Toeplitz matrices,

$$\begin{equation*} T_n=(-1)^\nu \left(\begin{array}{ccccc} 0 & \binom{0}{\nu -1} & \binom{1}{\nu -1} & \cdots & \binom{n-1}{\nu -1}\\[5.0pt] & 0 & \binom{0}{\nu -1} & \cdots & \binom{n-2}{\nu -1}\\[5.0pt] & & & \ddots & \vdots \\[5.0pt] & & & & \binom{0}{\nu -1}\\[5.0pt] & & & & 0 \end{array} \right), \end{equation*}$$

composed of binomial coefficients with an integer $\nu \ge 1$. The matrix $T_n$ is the representation of the operator taking the $\nu$th derivative, $f \mapsto D^\nu f$, in the orthonormal basis of Laguerre polynomials in the space $\mathcal{P}_n$ of algebraic polynomials of degree at most $n$ with the Laguerre norm given by $\| f\|^2=\int _0^\infty |f(x)|^2e^{-x}\,dx$. Thus, the norm $\| T_n\|$ is just the best constant for which the so-called Markov-type inequality $\|D^\nu f\|\le M \|f\|$ holds for all $f \in \mathcal{P}_n$.

This question reminded the second author of an ingenious trick used in Harold Widom’s 1966 paper 13 (and employed independently also by Lawrence Shampine in Sha). Let us consider an $n \times n$ matrix $A_n=(a_{jk})_{j,k=0}^{n-1}$ and denote by $H_n$ the integral operator on $L^2(0,1)$ with the piecewise constant kernel $h_n(x,y)=a_{[nx],[ny]}$, where $[\cdot ]$ stands for the integral part. Widom and Shampine proved that $\|A_n\|=n \| H_n\|$, thus transferring consideration of $A_n$ on the sequence $\{\mathbb{C}^n\}$ of increasing spaces to the consideration of a sequence $\{H_n\}$ of operators in one and the same space $L^2(0,1)$. If one could show that after appropriate scaling the operators $H_n$ converge in the operator norm to some nonzero operator $H$, that is, $n^{-\mu } H_n \to H$ in norm, it would follow that $n^{-\mu }\|H_n\| \to \|H\|$ and hence $\|A_n\|\sim \|H\| n^{\mu +1}.$

To compute $\| T_n\|$, we may ignore the factor $(-1)^\nu$ and the diagonal of zeros. In the resulting $n \times n$ matrix, the $j,k$ entry is equal to $\binom{k-j}{\nu -1}$ for $j <k$. Consequently, if $x < y$, then the kernel of the scaled integral operator $n^{-(\nu -1)}H_n$ is

$$\begin{equation*} \begin{split} \frac{1}{n^{\nu -1}}a_{[nx],[ny]}&=\frac{1}{n^{\nu -1}}\binom{[ny]-[nx]}{\nu -1}\\ & = \frac{1}{(\nu -1)!}\,\frac{[ny]-[nx]}{n}\,\frac{[ny]-[nx]-1}{n}\,\cdots \,\frac{[ny]-[nx]-\nu +2}{n}, \end{split} \end{equation*}$$

which converges uniformly to $(y-x)^{\nu -1}/(\nu -1)!$ as $n \to \infty$ and thus yields the asymptotics $\|T_n\|=\|L_\nu \|n^\nu (1+o(1))$, where $L_\nu$ is the Volterra integral operator on $L^2(0,1)$ given by

$$\begin{equation*} (L_\nu f)(x)=\frac{1}{(\nu -1)!}\int _x^1 (y-x)^{\nu -1} f(y)\,dy. \end{equation*}$$

Clearly, $L_\nu =L_1^\nu$ and $\|L_\nu \|=\|L_\nu ^*\|$ with

$$\begin{equation*} (L_\nu ^* f)(x)=\frac{1}{(\nu -1)!}\int _0^x (x-y)^{\nu -1} f(y)\,dy. \end{equation*}$$

Note that $\|L_1\|=2/\pi$. For more on the subject and, in particular, for more about pieces of the amazing story around the norms of the Volterra operators $L_\nu$, we refer to BD.

4. Eigenvalue distribution

Widom made several fundamental contributions to the collective eigenvalue distribution of truncated Toeplitz and Wiener–Hopf operators and their generalizations, such as pseudodifferential operators. In this section, we focus our attention on two of his papers on this topic.

In his 1980 paper 24 with Henry Landau, he investigated the positive definite operator given on $L^2(-\tau ,\tau )$ by

$$\begin{equation*} (C_\tau f)(x)=\frac{\gamma }{2\pi i}\int _{-\tau }^\tau \frac{e^{-i\alpha (x-y)}-e^{-i\beta (x-y)}}{x-y}\,f(y)\,dy, \quad x \in (-\tau ,\tau ). \end{equation*}$$

This operator is of crucial interest in random matrix theory and in laser theory. For example, as observed by H. Brunner, A. Iserles, and S. Nørsett BIN, if $\gamma = \pi$, $\alpha =-2$, $\beta =2$, in which case the operator is convolution by $\sin (2t)/t$, the eigenvalues of $C_\tau$ are the singular values of the famous Fox–Li operator. The symbol of $C_\tau$ is $\gamma \chi _{(\alpha ,\beta )}$, and hence it has two jumps. No general result of the type of Szegő’s strong limit theorem delivered a second-order trace formula in this situation. By an extremely ingenious argument, Landau and Widom nevertheless succeeded in establishing a second-order result for the eigenvalues, which confirmed a conjecture by D. Slepian of 1965. The result says that if $\varphi$ is in $C^\infty ({\mathbf{R}})$ and $\varphi (0)=0$, then