A generalized Schur–Horn theorem and optimal frame completions

Optimal completions of a frame

Given a finite sequence of vectors F0 in C we describe the spectral and geometrical structure of optimal completions of F0 obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured with respect to a general convex potential. In particular, our analysis includes the so-called Mean Square Error (MSE) and the Benedetto-Fickus’ frame potential. On a first s...

Optimal frame completions

Given a finite sequence of vectors F0 in C we describe the spectral and geometrical structure of optimal frame completions of F0 obtained by appending a finite sequence of vectors with prescribed norms, where optimality is measured with respect to a general convex potential. In particular, our analysis includes the so-called Mean Square Error (MSE) and the Benedetto-Fickus’ frame potential. On ...

Parametrizing Finite Frames and Optimal Frame Completions Dissertation

Free Vibration of a Generalized Plane Frame

This article deals with the free in-plane vibration analysis of a frame with four arbitrary inclined members by differential transform method. Based on four differential equations and sixteen boundary and compatibility conditions, the related structural eigenvalue problem will be analytically formulated. The frequency parameters and mode shapes of the frame will be calculated for various values...

Tight Frame Completions with Prescribed Norms

Let H be a finite dimensional (real or complex) Hilbert space and let {ai}i=1 be a non-increasing sequence of positive numbers. Given a finite sequence of vectors F = {fi}pi=1 in H we find necessary and sufficient conditions for the existence of r ∈ N∪{∞} and a Bessel sequence G = {gi}i=1 in H such that F ∪ G is a tight frame for H and ‖gi‖ = ai for 1 ≤ i ≤ r. Moreover, in this case we compute ...