# Generalized Bessel and Frame Measures

## TL;DR

This paper introduces and explores generalized notions of Bessel and frame measures in $L^p$ spaces with respect to a measure, providing new constructions, properties, and examples of such measures.

## Contribution

It defines $(p,q)$-Bessel and frame measures, extends classical concepts, and offers methods to construct and analyze these measures in a broad setting.

## Key findings

- Every finite Borel measure is a $(p,q)$-Bessel measure for some measure.
- Many measures admit infinite $(p,q)$-Bessel measures.
- $(p,q)$-Bessel/frame measures are $\sigma$-finite and non-unique.

## Abstract

Considering a finite Borel measure $ \mu $ on $ \mathbb{R}^d $, a pair of conjugate exponents $ p, q $, and a compatible semi-inner product on $ L^p(\mu) $, we introduce $ (p,q) $-Bessel and $ (p,q) $-frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we define notions of $ q $-Bessel and $ q$-frame in the semi-inner product space $ L^p(\mu) $. Every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for a finite measure $ \mu $. We construct a large number of examples of finite measures $ \mu $ which admit infinite $ (p,q) $-Bessel measures $ \nu $. We show that if $ \nu $ is a $ (p,q) $-Bessel/frame measure for $ \mu $, then $ \nu $ is $ \sigma $-finite and it is not unique. In fact, by using convolutions of probability measures, one can obtain other $ (p,q) $-Bessel/frame measures for $ \mu $. We present a general way of constructing a $ (p,q) $-Bessel/frame measure for a given measure.