In this note, we investigate the performance of the PCM scheme with linear quantization rule for quantizing unit-norm tight frame expansions for R without the White Noise Hypothesis. In [4], Wang and Xu showed that for asymptotically equidistributed unit-norm tight frame the PCM quantization error has an upper bound O(δ(d+1)/2) and they conjecture the upper bound is sharp. In this note, we conf...

We consider a generator F 1⁄4 ðf1; y ;fNÞ for either a multi-frame or a super-frame generated under the action of a projective unitary representation for a discrete countable group. Examples of such frames include Gabor multi-frames, Gabor super-frames and frames for shift-invariant subspaces. We show that there exists a unique normalized tight multi-frame (resp. super-frame) generator C 1⁄4 ðc...

The introduction of vanishing moment recovery (VMR) functions in our recent work (also called “fundamental functions” in an independent paper by Daubechies, Han, Ron, and Shen) modifies the so-called “unitary extension principle” to allow the construction of compactly supported affine frames with any desirable order of vanishing moments up to the order of polynomial reproduction of the given as...

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 frame to generate a tight frame. In this paper, we analyze prefect preconditioning of frames by a diagonal operator. We derive various characterizations of functional analytic and geometric type...

Using their unitary extension principle, Ron and Shen have constructed a normalized tight frame for L2(R) consisting of spline functions with uniform knots. This paper constructs a normalized tight frame for L2((0,∞)) comprising spline functions with knots on a hybrid of uniform and geometric mesh. The construction is motivated by applications in adaptive approximation using spline functions on...

In this paper we show that any shift-invariant Bessel sequence with an at most countable number of generators can be extended to a tight frame for its closed linear span by adding another shift-invariant system with at most the same number of generators. We show that in general this result is optimal, by providing examples where it is impossible to obtain a tight frame by adding a smaller numbe...

We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. We show that any finite tight frame can be written as a union of prime tight frames. We then characterize all prime harmonic tight frames as well as all prime frames constructed from the spectral tetris method. As a byproduct of this last result, we obtain a characterization of w...

In this article we introduce the notion of J-Parseval fusion frames in a Krein space K and characterize 1-uniform J-Parseval fusion frames with ζ = √ 2. We provide some results regarding construction of new J-tight fusion frame from given J-tight fusion frames. We also characterize an uniformly J-definite subspace of a Krein space K in terms of J-fusion frame. Finally we generalize the fundamen...

We study the number of Laurent polynomials in a sum of square magnitudesof a nonnegative Laurent polynomial associated with bivariate box splines on the three-and the four-direction meshes. In addition, we study the same problem associated withtrivariate box splines. The number of Laurent polynomials in a sum of squares magnitudeof a nonnegative Laurent polynomial determines the...

In this paper we consider generalized metric spaces in the sense of Lawvere and the categorical Isbell completion construction. We show that this is an analogue of the tight span construction of classical metric spaces, and that the Isbell completion coincides with the directed tight span of Hirai and Koichi. The notions of categorical completion and cocompletion are related to the existence of...