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On the construction of frames for Triebel-Lizorkin and Besov spaces. (English) Zbl 1089.42018

In the paper, a general method for the construction of frames for homogeneous Triebel-Lizorkin and Besov spaces is presented. This method is not based on the idea of Multiresolution Analysis. Instead, a perturbation principle is proposed.
Let \( S \) be the frame operator \( S f = \sum_{\psi \in \Psi}\langle f, \psi \rangle \psi. \) Start with a frame \( \Phi := \{\varphi_I\}_{I \in D} \) consisting of translations and dilations of a sufficiently smooth and rapidly decaying function, such as the Frazier-Jawerth frame. Then take the new frame \( \Psi := \{ \psi_I\}_{I \in D},\) where \( \psi \) satisfies for certain parameters \(\varepsilon , r, M, k\) the conditions \[ | \partial^\alpha \varphi(x) - \partial^\alpha \psi (x) | \leq \varepsilon (1 + | x| )^{-M} , \qquad | \alpha | \leq r , \] and \[ \int_{\mathbb R^d} x^{\alpha} \psi(x) dx = 0, \qquad | \alpha | \leq k . \] The paper answers, how to choose the parameters in order to ensure that \( \Psi \) forms a frame for Besov and Triebel-Lizorkin spaces, whether \( S^{-1}\Psi \) is a frame itself and the representation \[ f = \sum_{I \in D}\langle f, S^{-1}\psi_I \rangle \psi_I \] holds true.

MSC:

42C15 General harmonic expansions, frames
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
41A63 Multidimensional problems
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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