Sequence space representations for spaces of smooth functions and distributions via Wilson bases

TL;DR

This paper introduces explicit sequence space representations for test function and distribution spaces using Wilson bases, revealing their structure as unconditional Schauder bases and unifying the Valdivia-Vogt structure tables.

Contribution

It provides a novel explicit construction of sequence space representations and demonstrates that Wilson bases serve as unconditional Schauder bases for these spaces.

Findings

Wilson bases generate explicit sequence space representations.

Wilson bases are unconditional Schauder bases for the spaces.

The Valdivia-Vogt structure tables can be viewed as a unified diagram.

Abstract

We provide explicit sequence space representations for the test function and distribution spaces occurring in the Valdivia-Vogt structure tables by making use of Wilson bases generated by compactly supported smooth windows. Furthermore, we show that these kind of bases are common unconditional Schauder bases for all separable spaces occurring in these tables. Our work implies that the Valdivia-Vogt structure tables for test functions and distributions may be interpreted as one large commutative diagram.