On an orthogonal bivariate trigonometric Schauder basis for the space of continuous functions
The main result of the article is the construction of an orthogonal trigonometric Schauder basis in the space of all continuous functions on the torus \(C(T^2)\) with the smallest growth of the polynomial degree. This is achieved by using ideas involving a dyadic anisotropic periodic multiresolution analysis. Let \(T^2=[-\pi, \pi)^2\) be the two-dimensional torus and for any vector \(\mathbf{x}=(x_1, x_2)\in \mathbb R^2,\) set \[ \left\| \mathbf{x}\right\|_{\ell^q} =\begin{cases} \left( |x_1|^q +|x_2|^q \right)^{1/q}& 1\leq q< \infty \\ \max(|x_1|, |x_2| ) & q=\infty .\end{cases} \] The \(q\)-degree of a trigonometric polynomial \(t\) of two variables is defined as \[ \text{deg}_q t=\min \left\{ n\in \mathbb{N} : t(\mathbf{x}) =\sum_{\left\| \mathbf{k} \right\|_{\ell_q}\leq n }c_{\mathbf{k} } e^{ i(k_1 x_1 +k_2 x_2)}, c_{\mathbf{k} }\in \mathbb{C}, \mathbf{k} \in \mathbb Z^2 \right\} . \] For any \(f\in C(T^2),\) the operator \(S_\mu f\) is defined as \[ S_\mu f=\sum_{k=1}^\mu \langle f, t_k \rangle t_k , \] where \(t_k\) is a trigonometric polynomial, \(\mu, k\in\mathbb{N}.\) For any set \(A\subset \mathbb Z^2,\) let \[ T_A =\left\{t: t(\mathbf{x})=\sum_{\mathbf{k}\in A} c_{\mathbf{k} } e^{ i(k_1 x_1 +k_2 x_2)}, c_{\mathbf{k} }\in \mathbb{C} \right\} . \] The distance between \(f\) and \(T_A\) is defined as \[ E_A(f)_\infty =\inf_{t\in T_A} \left\| f-t\right\|_\infty. \] The authors construct a polynomial system \(\left\{ t_k: k\in \mathbb{N} \right\}\) that is an orthonormal Schauder basis in \(C(T^2)\) such that for any \(f\in C(T^2),\) the following relation holds \[ \left\| f-S_\mu f\right\|_\infty\leq \left( 1+C(\epsilon) \right) E_A(f)_\infty, \] where \(C(\epsilon)>0.\) It is also shown that the orthogonal projection operator \(S_\mu\) for the functions \(\left\{ t_k: k\in \mathbb{N} \right\}\) acting as an operator from \(C(T^2)\) into itself is uniformly bounded for all \(\mu \in \mathbb{N},\) i.e., \[ \left\|S_\mu \right\|\leq C(\epsilon) , \quad 0< C(\epsilon). \]
anisotropic periodic multiresolution analysis
scientific article; zbMATH DE number 7003505