Continuous and Discrete Fourier Transforms, Extension Problems and Wiener-Hopf Equations

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The book covers a range of advanced topics in mathematical analysis and operator theory. It begins with uncertainty principles related to time-frequency operators, followed by sampling results for time-frequency transformations and uncertainty principles for Gabor and wavelet frames. The exploration of matrix-valued continuous analogues of orthogonal polynomials includes preliminary results, the study of orthogonal operator-valued polynomials, and the distribution of zeros of matrix-valued Krein functions. The discussion on band extensions delves into the real line as a limit of discrete band extensions, introducing the entropy principle and outlining key preliminaries and main results. Further, it addresses weakly positive matrix measures and generalized Toeplitz forms, highlighting their lifting properties and applications to Hankel and Hilbert transform operators. The text also tackles the reduction of the abstract four block problem to a Nehari problem, presenting main theorems and their proofs. Additionally, it examines the state space method for integro-differential equations of Wiener-Hopf type with rational matrix symbols, including an introduction, main theorems, and detailed proofs. Lastly, it discusses symbols and asymptotic expansions across various contexts, including smooth and piecewise smooth symbols on Rn and T, as well as symbols discontinuous across hyperplanes. The program of the workshop is also includ