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H. Widom

Publications and source records attributed to H. Widom.

4 recordsLinked to original sources
arXiv2006

Szegö via Jacobi

At present there exist numerous different approaches to results on Toeplitz determinants of the type of Szegö's strong limit theorem. The intention of this paper is to show that Jacobi's theorem on the minors of the inverse matrix remains one of the most comfortable tools for tackling the matter. We repeat a known proof of the Borodin-Okounkov formula and thus of the strong Szegö limit theorem that is based on Jacobi's theorem. We then use Jacobi's theorem to derive exact and asymptotic formulas for Toeplitz determinants generated by functions with nonzero winding number. This derivation is new and completely elementary

math.FA↗
arXiv2004

From Toeplitz Eigenvalues through Green's Kernels to Higher-Order Wirtinger-Sobolev Inequalities

The paper is concerned with a sequence of constants which appear in several problems. These problems include the minimal eigenvalue of certain positive definite Toeplitz matrices, the minimal eigenvalue of some higher-order ordinary differential operators, the norm of the Green kernels of these operators, the best constant in a Wirtinger-Sobolev inequality, and the conditioning of a special least squares problem. The main result of the paper gives the asymptotics of this sequence.

math.FA↗
arXiv1997

Asymptotics of a Class of Solutions to the Cylindrical Toda Equations

The small t asymptotics of a class of solutions to the 2D cylindrical Toda equations is computed. The solutions, q_k(t), have the representation q_k(t) = log det(I-lambda K_k) - log det(I-lambda K_{k-1}) where K_k are integral operators. This class includes the n-periodic cylindrical Toda equations. For n=2 our results reduce to the previously computed asymptotics of the 2D radial sinh-Gordon equation and for n=3 (and with an additional symmetry contraint) they reduce to earlier results for the radial Bullough-Dodd equation.

solv-int↗