On Retro Frame Associated With Measurable Space

TL;DR

This paper introduces a new duality concept for retro frames in measurable spaces, providing conditions for dual existence and discussing a class that always admits a dual frame, extending frame theory beyond Hilbert spaces.

Contribution

It defines $\Omega_0$-type duality for retro $(\Omega,\mu)$-frames and establishes necessary and sufficient conditions for their duals, expanding the theoretical framework of frames.

Findings

Necessary and sufficient conditions for dual existence are derived.

A class of retro $(\Omega,\mu)$-frames always admits a dual.

Extension of frame duality concepts to measurable space contexts.

Abstract

Frames are redundant system which are useful in the reconstruction of certain classes of spaces. Duffin and Schaeffer introduced frames for Hilbert spaces, while addressing some deep problems in non harmonic Fourier series. The dual of a frame (Hilbert) always exists and can be obtained in a natural way. In this paper we introduce the notion of $\Omega_0$-type duality of retro $(\Omega,\mu)$-frames are given. Necessary and sufficient conditions for the existence of the dual of retro $(\Omega,\mu)$-frames are obtained. A special class of retro $(\Omega,\mu)$-frames which always admit a dual frame is discussed.