17 research outputs found
Characterizing completions of finite frames
Publication in the conference proceedings of SampTA, Bremen, Germany, 201
A recursive construction of a class of finite normalized tight frames
Abstract Finite normalized tight frames are interesting because they provide decompositions in applications and some physical interpretations. In this article, we give a recursive method for constructing them. c (2014) Wavelets and Linear Algebr
Optimal frame designs for multitasking devices with weight restrictions
Let d=(d_j)_j∈I_m ∈ N^m be a finite sequence (of dimensions) and α=(α_i)_i∈ I_n be a sequence of positive numbers (of weights), where I_k={1,...,k} for k ∈ N. We introduce the (α , d)-designs i.e., m-tuples Φ=( F_j)_j ∈ I_m such that F_j={ f_ij}_i∈ I_n is a finite sequence in C^{d_j}, j ∈ I_m, and such that the sequence of non-negative numbers (||f_ij||^2)_j ∈ I_m forms a partition of α_i, i ∈ I_n. We characterize the existence of (α , d)-designs with prescribed properties in terms of majorization relations. We show, by means of a finite-step algorithm, that there exist (α , d)-designs Φ^ op =(F_j^op)_j∈I_m that are universally optimal; that is, for every convex function φ:[0,∞)→ [0,∞) then Φ^ op minimizes the joint convex potential induced by φ among (α , d)-designs, namely Σ_{j ∈ I_m} P_φ( F_j^op) ≤ Σ_{j ∈ I_m} P_φ( F_j) for every (α , d)$-design Φ=( F_j)_{j∈ I_m}, where P_φ(F)=tr(φ(S_F)); in particular, Φ^ op minimizes both the joint frame potential and the joint mean square error among (α , d)-designs. We show that in this case F_j^op is a frame for C^{d_j}, for j ∈ I_m. This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.Fil: Benac, Maria Jose. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentina. Universidad Nacional de Santiago del Estero. Facultad de Ciencias Exactas y TecnologÃas. Departamento de Matemática; ArgentinaFil: Massey, Pedro Gustavo. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Ruiz, Mariano Andres. Universidad Nacional de La Plata. Facultad de Ciencias Exactas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Stojanoff, Demetrio. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin
Optimal dual frames and frame completions for majorization
In this paper we consider two problems in frame theory. On the one hand,
given a set of vectors we describe the spectral and geometrical
structure of optimal completions of by a finite family of vectors
with prescribed norms, where optimality is measured with respect to
majorization. In particular, these optimal completions are the minimizers of a
family of convex functionals that include the mean square error and the
Bendetto-Fickus' frame potential. On the other hand, given a fixed frame
we describe explicitly the spectral and geometrical structure of
optimal frames that are in duality with and such that
the Frobenius norms of their analysis operators is bounded from below by a
fixed constant. In this case, optimality is measured with respect to
submajorization of the frames operators. Our approach relies on the description
of the spectral and geometrical structure of matrices that minimize
submajorization on sets that are naturally associated with the problems above.Comment: 29 pages, with modifications related with the exposition of the
materia
Tetris Tight Frames Construction via Hadamard Matrices
We present a new method to construct unit norm tight frames by applying altered Hadamard matrices. Also we determine an elementary construction which can be used to produce a unit norm frame with prescribed spectrum of frame operator
Optimal frame completions with prescribed norms for majorization
Given a finite sequence of vectors F0 in a d-dimensional complex Hilbert space H we characterize in a complete and explicit way the optimal completions of F0 obtained by appending a finite sequence of vectors with prescribed norms, where optimality is measured with respect to majorization (of the eigenvalues of the frame operators of the completed sequences). Indeed, we construct (in terms of a fast algorithm) a vector?that depends on the eigenvalues of the frame operator of the initial sequence F0 and the sequence of prescribed norms?that is a minimum for majorization among all eigenvalues of frame operators of completions with prescribed norms. Then, using the eigenspaces of the frame operator of the initial sequence F0 we describe the frame operators of all optimal completions for majorization. Hence, the concrete optimal completions with prescribed norms can be obtained using recent algorithmic constructions related with the Schur-Horn theorem.Fil: Massey, Pedro Gustavo. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Ruiz, Mariano Andres. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Stojanoff, Demetrio. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin