# On inequivalences of sequences of characters

## TL;DR

This paper proves that certain bijections between trigonometric and Walsh functions cannot be extended to isomorphisms in Lp spaces for 1<p<infinity, highlighting fundamental inequivalences in sequence representations.

## Contribution

It establishes that bijections between trigonometric and Walsh functions do not extend to isomorphisms in Lp spaces, revealing fundamental differences in these function systems.

## Key findings

- Bijections between trigonometric and Walsh functions do not extend to Lp isomorphisms for 1<p<infinity.
- Fundamental inequivalences exist between sequences of characters in different function systems.
- The results clarify limitations in representing functions via Walsh versus trigonometric bases.

## Abstract

We establish various results including the following: if $1<p<\infty$ and $\sigma$ is a bijection between the trigonometric functions on $[0,1)$ and the Walsh functions on $[0,1)$. Then $\sigma$ does not extend to an isomorphism $L_p[0,1) \rightarrow L_p[0,1)$.