Mathematical Sciences: Applications of Fourier Analysis to Banach Space Theory

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DMS-9531594 Alex Koldobsky University of Texas @ San Antonio The study of subspaces of classical Banach spaces remains one of the most important directions of the Banach space theory. An approach based on the connection between embedding of Banach spaces into Lp, positive definite functions and stable measures was discovered by P. Levy in the 30's, and since then has been under intensive development. The author's contributions to this direction include a solution to the 1938 Schoenberg's problem on positive definite functions, construction of common subspaces of Lp-spaces, a Fourier transform criterion of isometric embedding into Lp contributing to the 1937 P. Levy's problem, a solution to the isometric version of the 1969 Kwapien's problem on Banach subspaces of Lp with p less than 1. All these results were obtained by using methods of Fourier analysis. The project addresses open problems in this direction. The problems include the description of positive definite norm dependent function, the related question on the existence of isotropic measures which are not mixtures of stable measures, characterization of subspaces of Lp and zonoids, isomorphic versions of Kwapien's problem and the problem on common subspaces, application of the Fourier transform to computing the Banach-Mazur distance, connections between the Fourier transform of norms and duality of Banach spaces, and between the Fourier transform and the volume of sections of the unit ball. The problem of extending the Fourier transform technique to certain infinite dimensional situations has applications to stochastic processes and electrical engineering. The Fourier transform, as a method of decomposing a signal into a combination of simple frequencies, has proved to be an important tool in different areas of mathematics, physics, engineering and statistics. The author has been working on applications of the Fourier transform to the Banach space theory for more than ten years, and his contributions include a solution to the Schoenberg problem on positive definite functions which had been open since 1938, extension of the Fourier transform technique to infinite dimensional situations, construction of common subspaces of Banach spaces and other results. Work in this direction provides important connections between functional analysis, harmonic analysis and probability theory. For example, calculating the Fourier transform of a norm one can find the volume of sections of convex bodies in spaces of arbitrary dimension, or construct random variables which are stable under summation. Situations where one has to calculate the Fourier transform of the norm (distance) of a Banach space occur in different applied areas. For example, if it is impossible to measure a signal directly, and one only has the information about the distance from the signal to fixed functions in the norm of a certain Banach space, then one of the ways to identify the signal is to solve a certain convolution equation involving the norm, which can usually be solved using the Fourier transform technique. This happens when, because of noise, one can only see the maximum of a signal (the supremum-norm), which often happens in electrical engineering. In the study of the strength of fibers, it is important to find the frequencies of the main defects while one can not measure the strength at every point and only knows the weakest points of certain pieces of the material. Finding the spectral function of the strength is an example of how the Fourier transform of norms works for the study of materials.