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Computing Widom's Constant

Mathematical Physics & Probability

Speaker: Estelle Basor, American Institute of Mathematics
Related Webpage: https://www.aimath.org/~ebasor/
Location: Zoom lecture-Postponed
Start time: Wed, Nov 11 2020, 4:10PM

Finite Toeplitz matrices are matrices with entries $a_{i,j} = a_{i-j}$ where the $a_i$ are the Fourier coefficients of a function, called the symbol of the matrix. The asymptotics for the determinants of finite Toeplitz matrices for smooth symbols is well known and is of the form $G^n E$ where $n$ is the matrix size and $G$ and $E$ are constants that are easily described. In 1974 Harold Widom proved the analogue of this result for the block case, where the entries are themselves matrices of fixed size, and in 1976 described the constant $E$ as an infinite determinant. This constant is now known as Widom's constant.

Computing Widom's constant in a non-theoretical form in the matrix case is not always possible. But sometimes more can be said, and the cases where it can be made more explicit is the focus of this talk. Asymptotics for the determinants in the case where the symbols are not smooth are also in many cases well-known. The talk will also discuss recent results for the block case when the symbol has jump discontinuities.

The seminar will be rescheduled for a later date.