On homogeneous decomposition spaces and associated decompositions of distribution spaces

Abstract

A new construction of decomposition smoothness spaces of homogeneous type is considered. The smoothness spaces are based on structured and flexible decompositions of the frequency space double-struckRd∖{0}. We construct simple adapted tight frames for L2(double-struckRd) that can be used to fully characterise the smoothness norm in terms of a sparseness condition imposed on the frame coefficients. Moreover, it is proved that the frames provide a universal decomposition of tempered distributions with convergence in the tempered distributions modulo polynomials. As an application of the general theory, the notion of homogeneous α‐modulation spaces is introduced.

On a discrete transform of homogeneous decomposition spaces

“…It can be verified that are Ḟ s p,q ( h) and Ṁs p,q ( h) are (quasi-)Banach spaces that only (up to norm equivalence) depend on h and not the particular choice of BAPU, see [1,3]. We mention that it is possible to consider other reservoirs of distributions than S ′ \P to build the function spaces, see Voigtlaender [27] for further details.…”

“…The present authors considered a general construction of homogeneous smoothness spaces, based on structured decomposition of the frequency space R d \{0}, in [1]. Adapted tight frames for L 2 (R d ) were were considered in [1] and they were shown to provide universal decompositions of tempered distributions with convergence in the tempered distributions modulo polynomials.…”

“…The present authors considered a general construction of homogeneous smoothness spaces, based on structured decomposition of the frequency space R d \{0}, in [1]. Adapted tight frames for L 2 (R d ) were were considered in [1] and they were shown to provide universal decompositions of tempered distributions with convergence in the tempered distributions modulo polynomials. Moreover, atomic decompositions of the corresponding homogeneous smoothness spaces were obtained, completely characterising the smoothness spaces by a sparseness condition on the frame coefficients, facilitating compression of the elements of such homogeneous smoothness spaces using the corresponding frame coefficients.…”

“…In the present paper, which can be considered a continuation of [1], we study additional properties of discrete representations of homogeneous decomposition spaces of Besov and Triebel-Lizorkin type. Most importantly, in Section 4, we introduce the notion of almost diagonal matrices for homogeneous decomposition spaces of Besov and Triebel-Lizorkin type and we use the tight frame introduced [1] to link such matrices to bounded operators on Besov and Triebel-Lizorkin type spaces.…”

“…As an application of the results obtained, we study various perturbation of the frame from [1] to obtain compactly supported frames for homogeneous decomposition spaces of Besov and Triebel-Lizorkin type. This is considered in Section 5.…”