# Dual pair and Approximate dual for continuous frames in Hilbert spaces

## TL;DR

This paper introduces the concepts of dual and approximate dual for continuous frames in Hilbert spaces, exploring their properties and relationships with continuous Riesz bases through examples.

## Contribution

It presents new definitions and properties of dual and approximate duals for continuous frames, and clarifies their relations with continuous Riesz bases.

## Key findings

- Properties of dual and approximate duals are established.
- Relations between continuous Riesz bases are illustrated with examples.
- Foundational concepts for continuous frames are expanded.

## Abstract

In this manuscript, the concept of dual and approximate dual for continuous frames in Hilbert spaces will be introduced. Some of its properties will be studied. Also, the relations between two continuous Riesz bases in Hilbert spaces will be clarified through examples.