Multipliers in Hardy and Bergman spaces, and Riesz decomposition
Abstract
Multipliers' methods have proven to be an efficient tool in virtually any area of Analysis. Many linear operators act as multipliers on Taylor series, Fourier series, Fourier integrals, etc., of a function. This means the operators introduce some multiplicative factors to the series or integrals. As a consequence, conditions on boundedness of multipliers imply important inequalities in Analysis, in particular, in Approximation Theory. We consider series multipliers in Hardy and Bergman spaces in the unit disk D of the complex plane C, as well as multipliers of Fourier integrals in Hardy spaces in tubes over open cones (in C^n). Obtained conditions are used to derive some inequalities, e.g., Bernstein and Nikolskii type inequalities for entire functions. Some of the multiplier conditions are surprisingly sharp. As an example, a critical index for Bochner-Riesz means of Fourier integrals in Hardy spaces in tubes has been found. For the Hadamard product of two polynomials (again, a multiplier-type operator), we obtain sharp inequalities for its Mahler measure. They imply several sharp inequalities used in Approximation Theory. We conclude the thesis by the Riesz Decomposition result for m-superharmonic functions in R^n, (2m is strictly less than n), which generalizes work of K. Kitaura and Y. Mizuta for super-biharmonic functions.
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- OSU Dissertations [11222]