Frames and Finite Dimensionality: Frame Transformation, Classification and Algorithms

Abstract

In this paper we will look at the connection of frames and finite dimensionality. A main focus is to present simple algorithms and make them available online. The main result is a way to 'switch' between different frames, giving an algorithm to calculate the coefficients of one frame given the analysis of another frame using their cross-Gram matrix. This is a canonical extension of the basic transformation matrix used for orthonormal bases (ONB) and is therefore called frame transformation. Furthermore we will summarize basic properties of frames in finite dimensional spaces. We will give basic algorithms to use with frames in finite dimensional spaces, useful for basic numerical experiments. Finally we will give a criteria for finite dimensional spaces using frames. Keywords: frames, discrete expansion, matrices, frame transformation, finite dimension, algorithm; MSC: 41A58, 65F30; 15A03