Orthogonal Systems and Convolution Operators
Springer Science & Business Media, 2003 - Mathematics - 236 pages
The main concern of this book is the distribution of zeros of polynomials that are orthogonal on the unit circle with respect to an indefinite weighted scalar or inner product. The first theorem of this type, proved by M. G. Krein, was a far-reaching generalization of G. Szeg÷'s result for the positive definite case. A continuous analogue of that theorem was proved by Krein and H. Langer. These results, as well as many generalizations and extensions, are thoroughly treated in this book. A unifying theme is the general problem of orthogonalization with invertible squares in modules over C*-algebras. Particular modules that are considered in detail include modules of matrices, matrix polynomials, matrix-valued functions, linear operators, and others. One of the central features of this book is the interplay between orthogonal polynomials and their generalizations on the one hand, and operator theory, especially the theory of Toeplitz marices and operators, and Fredholm and Wiener-Hopf operators, on the other hand. The book is of interest to both engineers and specialists in analysis.
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apply assume Banach space block Toeplitz bounded Chapter codim coefficients column compact complex consider contains continuous analogue corresponding defined denotes dim Ker entries equals equals the number equation equivalent exists fact finite follows function Furthermore given Gohberg hence identity implies infinite inner product integral invertible squares ISBN k(t+s Krein's Theorem left invertible matrix polynomials matrix-valued module multiplicities negative eigenvalues nonzero squares norm number of negative number of zeros observe obtain operator orthogonal matrix orthogonal polynomials orthogonalization with invertible positive definite proof prove replaced respectively reverse orthogonalization satisfy scalar product Section selfadjoint sequence shows solution space span statement Suppose Szegő Taking Theorem 1.1 Theory Toeplitz matrix unit circle vector weight xr matrix