Three problems in harmonic analysis and approximation theory

Abstract

In this dissertation, we deal with three problems in harmonic analysis and approximation theory. The first problem concerns the Hardy-Littlewood relations for Fourier coefficients in the two-dimensional setting, the second one is related to estimates of the coefficients of a trigonometric polynomial in different bases, and the third one refers to multidimensional integer partitions._x000D_ _x000D_ Regarding the first problem, we prove the Hardy-Littlewood theorem in two dimensions for functions whose Fourier coefficients obey general monotonicity conditions and, importantly, are not necessarily positive. The Hardy-Littlewood theorem is an analogue of Parseval's identity that establishes equivalences of norms of functions and norms of their Fourier series under some additional requirements. Results of this kind are important, in the first place, due to the fact that once such a relation is found, one becomes free to choose if it is handy to deal with functions or with coefficients in this or that case, as if having Parseval's identity. The sharpness of our result is given by a counterexample, which shows that if one slightly extends the considered class of coefficients, the Hardy-Littlewood relation fails._x000D_ _x000D_ In the second problem, we show that for any even algebraic polynomial p one can find a cosine polynomial with an arbitrary small sum of the absolute values of coefficients such that the first coefficients of its representation as an algebraic polynomial in cos x coincide with those of p. To prove the mentioned result, we consider the matrix of the coefficients of Chebyshev polynomials and derive an explicit formula for the inverse of its square submatrices. In the course of the proof of the main result, we also give some useful estimates on sums of products of binomial coefficients appearing in the expression for entries of the pseudoinverse of a Vandermonde matrix._x000D_ _x000D_ Finally, in the third problem, we obtain estimates for the number of d-dimensional integer partitions of a number n. Importantly, we show that if n is sufficiently large compared to d, then the logarithm of the number of d-dimensional partitions of n is up to an absolute constant n^(d/(d+1)). To establish the result, we introduce the notion of available subsets of the so-called lower sets (or equivalently, of the partition diagrams) and obtain sharp estimates for their cardinalities in cases of large n. This in turn allows us to estimate the number of lower subsets of lower sets. Besides, we provide estimates of the number of d-dimensional integer partitions of n for different ranges of d in terms of n, which give the asymptotics of the logarithm of this number in each case.

Date of Award	6 Mar 2024
Original language	English
Supervisor	Sergey Tikhonov (Director)