Generalized upper and lower Legendre conjugates for weight functions

TL;DR

This paper introduces generalized Legendre conjugates for weight functions, exploring their properties and effects on growth indices, with applications to associated weight sequences and weighted spaces.

Contribution

It generalizes known conjugates, studies their impact on growth indices, and connects these transformations to operations on weight sequences, enhancing understanding of weighted spaces.

Findings

Generalized Legendre conjugates are introduced and analyzed.

Transformations affect growth indices of weight functions.

Applications include weighted spaces and sequence operations.

Abstract

We introduce and study new transformations between two functions satisfying some basic growth properties and generalize the known lower and upper Legendre conjugate (or envelope). We also investigate how these transformations modify recently defined growth indices for weight functions. A special but important and useful situation, to which the knowledge is then applied, is when considering associated weight functions which are expressed in terms of an underlying weight sequence. In this case these transformations precisely correspond to the point-wise product resp. point-wise division of the given sequences. Therefore, the new approach studied in this work illustrates the genuineness and importance and suggests applications for weighted spaces in different directions.