Some classes of rational functions and related Banach spaces

TL;DR

This paper studies a class of rational functions on real space, defining associated Banach spaces, and characterizes their norms, providing bounds for partial-fraction coefficients and identifying cases where the norm is explicitly achieved.

Contribution

It introduces a new Banach space framework for rational functions with specific denominator structures and characterizes the norm attainment conditions, including bounds for partial-fraction coefficients.

Findings

Norms are achieved by explicit combinations of rational functions.

Bounds for partial-fraction decomposition coefficients are established.

Characterization of the Banach space structure for these rational functions.

Abstract

For positive integers d, r, and M, we consider the class of rational functions on real d-dimensional space whose denominators are products of at most r functions of the form 1+Q(x) where each Q is a quadratic form with eigenvalues bounded above by M and below by 1/M. Each numerator is a monic monomial of the same degree as the corresponding denominator. Then we form the Banach space of countable linear combinations of such rational functions with absolutely summable coefficients, normed by the infimum of sums of absolute values of the coefficients. We show that for rational functions whose denominators are rth powers of a specific 1+Q, or differences of two such rational functions with the same numerator, the norm is achieved by and only by the obvious combination of one or two functions respectively. We also find bounds for coefficients in partial-fraction decompositions of some specific rational functions, which in some cases are quite sharp.