Lipschitz Analysis of g-Phase Retrievable Frames

Abstract

A g-phase retrievable frame is a $\lambda$-phase retrievable frame in finite dimensional Hilbert space $\mathcal{H}_n$, where $\lambda$ is an special function, which is called phase coefficient function. In this paper we study the Lipschitz analysis of the nonlinear map $\alpha_{\lambda,{\mathcal{F}}}:\widehat{\mathcal{H}_n}\longrightarrow\mathbb{F}^m, \ \ \ \alpha_{\lambda,{\mathcal{F}}}(\hat{x}):=\begin{bmatrix}\lambda\left( \left\langle {x,f_k}\right\rangle\right)\end{bmatrix}_{1\leq k\leq m}$, where $\widehat{\mathcal{H}_n}$ is the quotient space corresponding to a special equivalence relation on $\mathcal{H}_n$ with respect to phase coefficient function $\lambda$, $\mathcal{F}=\{f_k\}_{k=1}^m$ is a $\lambda$-phase retrievable frame for $\mathcal{H}_n$, $\mathbb{F}=\mathbb{R}$ for real Hilbert space $\mathcal{H}_n$ and $\mathbb{F}=\mathbb{C}$ for complex Hilbert space $\mathcal{H}_n$.

Keywords

• frame
• phase coefficient function
• phase retrievable frame
• $\lambda$-phase retrievable frame
• g-phase retrievable frame
• lipschitz continuous function