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From Toeplitz eigenvalues through Green’s kernels to higher-order Wirtinger-Sobolev inequalities. (English) Zbl 1118.47016

Dritschel, Michael A. (ed.), The extended field of operator theory. Containing lectures of the 15th international workshop on operator theory and its applications, IWOTA 2004, Newcastle, Uk, July 12–16, 2004. Basel: Birkhäuser (ISBN 978-3-7643-7979-7/hbk; 978-3-7643-7980-3/e-book). Operator Theory: Advances and Applications 171, 73-87 (2006).
Let \(\alpha\) be a natural number. The authors consider a sequence of constants \(c_\alpha\) appearing in the following five problems.
(1) Let \(T_n(a)\) be the \(n\times n\) Toeplitz matrix generated by \(a(t)=| 1-t| ^{2\alpha}b(t)\), where \(b\) is a positive smooth function. The smallest eigenvalue of \(T_n(a)\) satisfies \(\lambda_{\text{min}}(T_n(a))\sim c_\alpha b(1)/n^{2\alpha}\) as \(n\to\infty\).
(2) Consider the boundary value problem \((-1)^\alpha u^{(2\alpha)}(x)=v(x)\) for \(x\in[0,1]\) subject to the boundary value condition \(u(j)=u'(j)=\dots=u^{(\alpha-1)}(j)=0\) for \(j=0,1\). The minimal eigenvalue of this boundary value problem is \(c_\alpha\).
(3) Let \(G_\alpha(x,y)\) be the Green kernel of the boundary value problem (2). Let \((K_\alpha v)(x)=\int_0^1 G_\alpha(x,y)v(y)\,dy\). Then \(1/c_\alpha=\| K_\alpha\| \).
(4) The best constant in the Wirtinger-Sobolev inequality \(\int_0^1| u(x)| ^2\,dx\leq C\int_0^1| u^{(\alpha)}(x)| ^2dx\) subject to the boundary value conditions of problem (2) is \(C=1/c_\alpha\).
(5) Suppose we are given \(n\) complex numbers \(y_1,\dots,y_n\) and we want to know whether there exists a polynomial \(p\) of degree at most \(\alpha-1\) such that \(p(j)=y_j\) for \(1\leq j\leq n\). Such a polynomial exists if and only if \(\delta_k:=y_k-\binom {\alpha}{1}y_{k+1}+ \binom {\alpha}{2}y_{k+2}-\dots+(-1)^\alpha y_{k+\alpha}=0\) for \(1\leq k\leq n-\alpha\). If \(D(y_1,\dots,y_n)=(\sum_{k=1}^{n-\alpha}\delta_k^2)^{1/2}\), \(E(y_1,\dots,y_n)=\min(\sum_{j=1}^n| y_j-p(j)| ^2)^{1/2}\), where the minimum is taken over all polynomials of degree at most \(\alpha-1\), then \[ \max_{D(y_1,\dots,y_n)\neq 0}\frac{E(y_1,\dots,y_n)}{D(y_1,\dots,y_n)}\sim\frac{n^\alpha}{\sqrt{c_\alpha}}. \] The main result of the paper under review is the asymptotic formula \(c_\alpha=\sqrt{8\pi\alpha}(4\alpha/e)^{2\alpha}[1+O(\alpha^{-1/2})]\) as \(\alpha\to\infty\). Upper and lower bounds for \(c_\alpha\) are also given.
For the entire collection see [Zbl 1105.47002].

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
26D15 Inequalities for sums, series and integrals
34B27 Green’s functions for ordinary differential equations
39A12 Discrete version of topics in analysis
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