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Gabor meets Littlewood-Paley: Gabor expansions in L p ( d )

Karlheinz Gröchenig, Christopher Heil (2001)

Studia Mathematica

It is known that Gabor expansions do not converge unconditionally in L p and that L p cannot be characterized in terms of the magnitudes of Gabor coefficients. By using a combination of Littlewood-Paley and Gabor theory, we show that L p can nevertheless be characterized in terms of Gabor expansions, and that the partial sums of Gabor expansions converge in L p -norm.

Generalized Calderón conditions and regular orbit spaces

Hartmut Führ (2010)

Colloquium Mathematicae

The construction of generalized continuous wavelet transforms on locally compact abelian groups A from quasi-regular representations of a semidirect product group G = A ⋊ H acting on L²(A) requires the existence of a square-integrable function whose Plancherel transform satisfies a Calderón-type resolution of the identity. The question then arises under what conditions such square-integrable functions exist. The existing literature on this subject leaves a gap between sufficient and necessary criteria....

Good-λ inequalities for wavelets of compact support

Sarah V. Cook (2004)

Colloquium Mathematicae

For a wavelet ψ of compact support, we define a square function S w and a maximal function NΛ. We then obtain the L p equivalence of these functions for 0 < p < ∞. We show this equivalence by using good-λ inequalities.

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