ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 47B30, 42A75, 47G20 Spitkovsky, Ilya, The College of William & Mary, USA Almost periodic factorization and its applications We consider an almost periodic analogue of the Wiener-Hopf factorization, that is, possible representations of n × n matrix functions G in the form

G(t) = G_{+}(t)Λ(t)G_{−}(t),  t ∈ ℝ,  (1)

where G_{±} are almost periodic, analytic and invertible in the upper/lower half plane, Λ(t) = diag [e^{iλ_{1}t}, …, e^{iλ_{n}t}], and λ_{1}, …, λ_{n} ∈ ℝ. Existence criteria and explicit construction of the factorization (1) will be given, mostly for block triangular matrix functions

G(t) = [ \begin{matrix}e^{iλt}I & 0 \\ f(t) & e^{−iλt}I \end{matrix} ] .  (2)

Several applications of the factorization (1) for matrices (2) will be considered, including convolution integro-difference equations on finite intervals, Carathéodory-Toeplitz and Nehari problems for (matrix) functions of several variables, Sarason interpolation and (Toeplitz) corona problem for almost periodic matrix functions.