Adaptive decompositions in monogenic reproducing kernel Hilbert spaces of Paley-Wiener type
The authors study in this paper an adaptive decomposition for functions in monogenic reproducing kernel Hilbert spaces of Paley-Wiener type, which include the Paley-Wiener space, the Hardy and Bergman spaces on strip domains in \(\mathbb{R}^{m+1}\). In particular, the authors study Adaptive Fourier Decomposition (AFD) in monogenic RKHSs such as the Paley-Wiener space (\(PW\left(\pi,C\ell_{0,m}(\mathbb{C})\right)\)), Hardy space on strips (\(H^{2}\left(S_{a},C\ell_{0,m}(\mathbb{C})\right)\)) and Bergman space on strips (\(A^{2}\left(S_{a},C\ell_{0,m}(\mathbb{C})\right)\)), where \(S_{a}=\left\{x=x_{0}+\underline{x}\in\mathbb{R}^{m+1} \colon |x_{0}|<a, \underline{x}\in\mathbb{R}^{m}\right\}\subset\mathbb{R}^{m+1}\) and \(C\ell_{0,m}(\mathbb{C})\) is the complex Clifford algebra generated by \(\mathbf{e}_{1},\ldots,\mathbf{e}_{m}\) over the complex field \(\mathbb{C}\). The main result of the paper is provided in the following theorem: Let \(\mathcal{H}\) be one of \(PW\left(\pi,C\ell_{0,m}(\mathbb{C})\right)\), \(H^{2}\left(S_{a},C\ell_{0,m}(\mathbb{C})\right)\) and \(A^{2}\left(S_{a},C\ell_{0,m}(\mathbb{C})\right)\). For \(f\in\mathcal{H}\), there holds \[ f(w) = \lim\limits_{n\to\infty}\sum\limits_{k=1}^{n} K(w,\overline{x}_{k})\langle R^{k}f,K(\cdot,\overline{x}_{k})\rangle_{\mathcal{H}} \] in \(\mathcal{H}\), where \[ K(w,\overline{x}) = P(w,\overline{x}) = \frac{1}{(2\pi)^{m}}\int_{\mathbb{R}^{m}}e(w+\overline{x},\underline{\xi})\chi_{B(0,\pi)}(\underline{\xi}) d\underline{\xi} \] if \(\mathcal{H}=PW\left(\pi,C\ell_{0,m}(\mathbb{C})\right)\); \[ K(w,\overline{x}) = S(w,\overline{x}) = \frac{1}{(2\pi)^{m}}\int_{\mathbb{R}^{m}}e^{-2a|\underline{\xi}|}e(w+\overline{x},\underline{\xi})d\underline{\xi} \] if \(\mathcal{H}=H^{2}\left(S_{a},C\ell_{0,m}(\mathbb{C})\right)\); \[ K(w,\overline{x}) = B(w,\overline{x}) = \frac{1}{(2\pi)^{m}}\int_{\mathbb{R}^{m}}e(w+\overline{x},\underline{\xi})\frac{2|\underline{\xi}|}{e^{2a|\underline{\xi}|}-e^{-2a|\underline{\xi}|}} d\underline{\xi} \] if \(\mathcal{H}=A^{2}\left(S_{a},C\ell_{0,m}(\mathbb{C})\right)\).
scientific article; zbMATH DE number 7597815
scientific article