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On scalable $K$-frames and a version of Lax-Milgram theorem

F. Javadi, M. J. Mehdipour

Abstract

In this paper, we first prove a theorem by a little modification on the Lax-Milgram theorem. Then, using $K$-frames, we obtain lower and upper bounds for the results obtained from this theorem. Also, we present some methods for the characterization of scalable $K$-frames. Finally, we introduce piecewise scalable $K$-frames and give necessary and sufficient conditions for a $K$-frame to be piecewise scalable.

On scalable $K$-frames and a version of Lax-Milgram theorem

Abstract

In this paper, we first prove a theorem by a little modification on the Lax-Milgram theorem. Then, using -frames, we obtain lower and upper bounds for the results obtained from this theorem. Also, we present some methods for the characterization of scalable -frames. Finally, we introduce piecewise scalable -frames and give necessary and sufficient conditions for a -frame to be piecewise scalable.
Paper Structure (5 sections, 14 theorems, 63 equations)

This paper contains 5 sections, 14 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Generalization of Lax-Milgram Theorem
  3. Construction of Scalable $K$-Frames
  4. Piecewise Scalable $K$-Frames
  5. Compliance with ethical standards

Key Result

Theorem 2.1

Let $\sigma$ be a continuous coercive bilinear map on ${\cal H}$, and $C$ be a non-empty closed convex subset of ${\cal H}$. Then for every bounded linear functional $L$ on ${\cal H}$, there exist unique $u_0 \in C$ and $f_0 \in {\cal H}$ such that for every $v\in {\cal H}$ Moreover, if $\sigma$ is symmetric, then $u_0$ is characterized by the property

Theorems & Definitions (24)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • Theorem 2.4
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 14 more