Finite tight frames and some applications

Abstract

A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain approaches or applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer some advantages. The use of a finite tight frame may lead to a simpler description of the symmetry transformations, to a simpler and more symmetric form of invariants or to the possibility to define new mathematical objects with physical meaning, particularly in regard with the notion of a quantization of a finite set. We present some results concerning the use of integer coefficients and frame quantization, several examples and suggest some possible applications.

28 pages, LaTeX in IOP style, New results added

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• Finite oscillator obtained through finite frame quantization
• An ETF view of Dropout regularization
• Frame-related Sequences in Chains and Scales of Hilbert Spaces
• Frame representations of qudit quantum mechanics
• Coherent States for the Manin Plane via Toeplitz Quantization