Approximation of Functions by $n$-Separate Wavelets in the Spaces ${L}^p(\mathbb{R})$, $1\leq p\leq\infty$

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.