# E.A. Pleshcheva. Approximation of functions by $n$-separate wavelets in the spaces ${L}^p(\mathbb{R})$, $1\leq p\leq\infty$ ... P.167-176

We consider the orthonormal bases of $n$-separate MRAs and wavelets constructed by the author earlier. The classical wavelet basis of the space $L^2(\mathbb{R})$ is formed by shifts and compressions of a single function $\psi$. In contrast to the classical case, we consider a basis of $L^2(\mathbb{R})$ formed by shifts and compressions of $n$ functions $\psi^s$, $s=1,\ldots,n$. The constructed $n$-separate wavelets form an orthonormal basis of $L^2(\mathbb{R})$. In this case, the series $\sum_{s=1}^{n}\sum_{j\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}\langle f,\psi^s_{nj+s} \rangle \psi^s_{nj+s}$ converges to the function $f$ in the space $L^2(\mathbb{R})$. We write additional constraints on the functions $\varphi^s$ and $\psi^s$, $s=1,\ldots,n$, that provide the convergence of the series to the function $f$ in the spaces $L^p(\mathbb{R})$, $1 \leq p \leq \infty$, in the norm and almost everywhere.

Keywords: wavelet, scaling function, basis, multiresolution analysis

Received March 19, 2019

Revised May 15, 2019

Accepted May 20, 2019

Ekaterina Aleksandrovna Pleshcheva, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: eplescheva@gmail.com

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Cite this article as: E.A.Pleshcheva. Approximation of functions by $n$-separate wavelets in the spaces ${L}^p(\mathbb{R})$, $1\leq p\leq\infty$., Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 2, pp. 167–176.