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Frames which are tight might be considered optimally conditioned in the sense of their numerical stability.This leads to the question of perfect preconditioning of frames,i.e., modification of a given...
Finite frames can be viewed as mass points distributed in N-dimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic fram...
PRECONDITIONING TECHNIQUES IN FRAME THEORY AND PROBABILISTIC FRAMES
Parseval frame Scalable frame Fritz John Theorem Probabilistic frames frame potential continuous frames
2015/12/10
In this chapter we survey two topics that have recently been investigated in frame theory. First, we give an overview of the class of scalable frames.These are (finite) frames with the property that e...
The recently introduced and characterized scalable frames can be considered as those frames which allow for perfect preconditioning in the sense that the frame vectors can be rescaled to yield a tight...
Prime tight frames
divisible frames equiangular tight frames frames harmonic tight frames prime frames spectral tetris frames tight frames
2015/12/10
We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. In particular, we show that any finite tight frame can be written as a union of pri...
MULTI-WINDOW GABOR FRAMES IN AMALGAM SPACES
Wiener amalgam space Gabor frame Wiener’s Lemma
2015/12/10
We show that multi-window Gabor frames with windows in the Wiener algebra W(L∞, `1) are Banach frames for all Wiener amalgam spaces. As a by-product of our results we prove thecanonical dual of a Gabo...
Scalable Frames and Convex Geometry
Scalable frames tight frames preconditioning Farkas’s lemma
2015/12/10
The recently introduced and characterized scalable frames can be considered as those frames which allow for perfect preconditioning in the sense that the frame vectors can be rescaled to yield a tight...
FINITE TWO-DISTANCE TIGHT FRAMES
Spherical two-distance sets finite tight frames strongly regular graphs spherical 2-designs spherical designs of harmonic index 2
2015/12/10
A finite collection of unit vectors S ⊂ Rn is called a spherical two-distance set if there are two numbers a and b such that the inner products of distinct vectors from S are either a or b. We p...
The objective of this paper is the linear reconstruction of a vector, up to a unimodular constant, when all phase information is lost, meaning only the magnitudes of frame coefficients are known. Reco...
A Nonlinear Reconstruction Algorithm from Absolute Value of Frame Coefficients for Low Redundancy Frames
Nonlinear Reconstruction Algorithm Absolute Value of Frame Coefficients Low Redundancy Frames
2015/9/29
In this paper we present a signal reconstruction algorithm from absolute value of frame coefficients that requires arelatively low redundancy. The basic idea is to use a nonlinear embedding of the inp...
Stability Theorems for Fourier Frames and Wavelet Riesz Bases
Frames Riesz basis nonharmonic series wavelets
2015/9/29
In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type co...
Deficits and Excesses of Frames
Bessel sequences deficit excess frames Gabor systems Riesz bases wavelets Weyl–Heisenberg systems
2015/9/29
The excess of a sequence in a Hilbert space is the greatest number of elements that can be removed yet leave a set with the same closed span. We study the excess and the dual concept of the deficit of...
Density,Overcompleteness,and Localization of Frames.I.Theory
Density excess frames Gabor systems modulation spaces overcompleteness Riesz bases wavelets Weyl–Heisenberg systems
2015/9/29
Frames have applications in numerous fields of mathematics and engineering.The fundamental property of frames which makes them so useful is their overcompleteness.In most applications, it is this over...
Density, Overcompleteness,and Localization of Frames.II.Gabor Systems
Density excess frames Gabor systems modulation spaces overcompleteness Riesz bases wavelets Weyl–Heisenberg systems
2015/9/29
This work develops a quantitative framework for describing the overcompleteness of a large class of frames. A previous article introduced notions of localization and approximation between two frames F...
A NEW IDENTITY FOR PARSEVAL FRAMES
Bessel sequence Frame Hilbert space Parseval frame Parseval Frame Identity
2015/9/29
In this paper we establish a surprising new identity for Parseval frames in a Hilbert space. Several variations of this result are given, including an extension to general frames. Finally, we discuss ...