Construction of Banach frames and atomic decompositions of anisotropic Besov spaces
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Construction of Banach frames and atomic decompositions of anisotropic Besov spaces Dimitri Bytchenkoff1,2 Received: 12 June 2020 / Accepted: 12 September 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We construct generalised shift-invariant systems of functions of several real variables for anisotropic Besov spaces that can be generated by the decomposition method using any given expansive matrix and establish the conditions on those systems under which they will constitute Banach frames or sets of atoms for the anisotropic homo- or inhomogeneous Besov spaces. Keywords Anisotropic Besov spaces · Anisotropic wavelets · Banach frames · Atomic decompositions Mathematics Subject Classification 42B35 · 46E35 · 42C15 · 42C40
1 Introduction Besov spaces, originally constructed by the approximation method [1], play an extremely important role in the theory of differentiable functions of several real variables as they, on the one hand, constitute a closed system with respect to embedding theorems and are, on the other hand, closely related to Sobolev spaces [13]. Along with Sobolev, Besov spaces are an integral part of the embedding theory, which studies connexions between differential properties of functions in different metrics [2, 12, 13]. Harmonic analysis uses either bases or frames and sets of atoms to decompose functions of a function space into basic building blocks or synthesise them from those blocks. The isotropic Besov spaces are known [14] to have orthonormal bases made of wavelets [7]. Herein, we shall use the innovative approach reported in [15, 16] to construct Banach frames and sets of atoms for the anisotropic homoand inhomogeneous Besov spaces introduced in [4] as decomposition spaces [8]. Similar results, although formulated and achieved rather differently, were reported in [4] and [3]. * Dimitri Bytchenkoff dimitri.bytchenkoff@univ‑lorraine.fr 1
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
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Laboratoire d’Energétique et de Mécanique Théorique et Appliquée, Université de Lorraine, 2 avenue de la Forêt de Haye, 54505 Vandoeuvre‑lès‑Nancy, France
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D. Bytchenkoff
The work [4] was, in its turn, an ingenious generalisation of the ideas developed in [9] and [10]. This report is structured as follows: the elements of the decomposition method essential to the present work are outlined at the beginning of Sect. 2. The definitions of the anisotropic homo- and inhomogeneous Besov spaces viewed as decompositions spaces follow in Sects. 2.1 and 2.2. The notion of the Banach frame and that of the set of atoms for the decomposition space are reminded at the beginning of Sect. 3. In the same section, we give the statements of the two theorems that we shall use to construct Banach frames and atomic decompositions of anisotropic Besov spaces. In Sect. 4.1 of Sect. 4, the set of anisotropic homogeneous Besov wavelets is defined as a generalised s
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