### A Characterization of Functions that Generate Wavelet and Related Expansion.

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This paper obtains a class of tight framelet packets on ${L}^{2}\left({\mathbb{R}}^{d}\right)$ from the extension principles and constructs the relationships between the basic framelet packets and the associated filters.

The construction of nonseparable and compactly supported orthonormal wavelet bases of L 2(R n); n ≥ 2, is still a challenging and an open research problem. In this paper, we provide a special method for the construction of such wavelet bases. The wavelets constructed by this method are dyadic wavelets. Also, we show that our proposed method can be adapted for an eventual construction of multidimensional orthogonal multiwavelet matrix masks, candidates for generating multidimensional multiwavelet...

We prove the existence of nonseparable, orthonormal, compactly supported wavelet bases for ${L}^{2}\left({\mathbb{R}}^{2}\right)$ of arbitrarily high regularity by using some basic techniques of algebraic and differential geometry. We even obtain a much stronger result: “most” of the orthonormal compactly supported wavelet bases for ${L}^{2}\left({\mathbb{R}}^{2}\right)$, of any regularity, are nonseparable

A given set W = W X of n-variable class C 1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum _{\chi}\left({\int}_{{\mathbb{R}}^{n}}\nabla f\xb7\nabla {W}_{\chi}^{*}\right){W}_{\chi}$ converges to f with respect to the norm ${\u2225\nabla (\xb7)\u2225}_{{L}^{2}\left({\mathbb{R}}^{n}\right)}$ . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = W x of compactly supported class C 2−ɛ functions on ℝn such that [...]...

We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family ${\psi (\xb7-n)}_{n=-\infty}^{\infty}$, which is equivalent to the weight function ${\sum}_{n=-\infty}^{\infty}\left|\psi \u0302(\xb7+n)\right|\xb2$ being > 0 a.e.

Wavelets originated in 1980's for the analysis of (seismic) signals and have seen an explosion of applications. However, almost all the material is based on wavelets over Euclidean spaces. This paper deals with an approach to the theory and algorithmic aspects of wavelets in a general separable Hilbert space framework. As examples Legendre wavelets on the interval [-1,+1] and scalar and vector spherical wavelets on the unit sphere 'Omega' are discussed in more detail.