Soft Frames in Soft Hilbert Spaces

Osmin Ferrer,Arley Sierra,José Sanabria

doi:10.3390/math9182249

Abstract

In this paper, we use soft linear operators to introduce the notion of discrete frames on soft Hilbert spaces, which extends the classical notion of frames on Hilbert spaces to the context of algebraic structures on soft sets. Among other results, we show that the frame operator associated to a soft discrete frame is bounded, self-adjoint, invertible and with a bounded inverse. Furthermore, we prove that every element in a soft Hilbert space satisfies the frame decomposition theorem. This theoretical framework is potentially applicable in signal processing because the frame coefficients serve to model the data packets to be transmitted in communication networks.

Highlights

Discrete frame theory on Hilbert spaces is relatively recent
The main advantage is that a good discrete frame can behave almost like an orthonormal basis; with an additional, they do not require the uniqueness of the coefficients when writing a Hilbert space vector as a linear combination of the frame elements
We have introduced the concept of discrete frames in soft Hilbert spaces

Summary

Introduction

Discrete frame theory on Hilbert spaces is relatively recent. This theory was introduced by Duffin and Schaeffer in 1952 [1] as a tool to study problems related to non-harmonic Fourier series. Let {αn}n∈N be an orthonormal sequence in a soft Hilbert space Hhaving a finite set of parameters A.

Results

Conclusion