Approximation of a function \(f\) of generalized Lipschitz class by its extended Legendre wavelet series
The authors propose two generalizations of the class of \(\alpha\)-Lipschitz functions on $[0,1)$. For some $\alpha \in (0,1]$, these classes are $\operatorname{Lip}_{\alpha}^{(s)}[0,1)$ and $\operatorname{Lip}\xi [0,1]$, where $f\in\operatorname{Lip}_{\alpha}^{(s)}[0,1)$ if \[ \left\vert f(x+t)-f(x)\right\vert =O\left( s(t)\left\vert t\right\vert ^{\alpha}\right),\quad 0<\alpha \leq 1,\quad x,x+t\in [0,1], \] and \(f\in\operatorname{Lip}\xi [0,1]\) if \[ \vert f(x)-f(y)\vert =O(\xi\vert x-y\vert),\quad x,y\in [0,1]. \] Here, $s$ is positive, increasing, such that $s(t)\vert t\vert ^{\alpha}\rightarrow 0$ as $t\rightarrow 0^{+}$ and, $\xi $ is increasing, such that $\xi (t)\rightarrow 0$ as $t\rightarrow 0^{+}$. Considering the generalized Legendre wavelet expansions for functions belonging to the classes $\operatorname{Lip}_{\alpha}^{(s)}[0,1)$ and \(\operatorname{Lip}\xi [0,1]\), several estimations are obtained for these functions with respect to the \(L_{2}\) norm including some corollaries for the $\operatorname{Lip}_{\alpha}^{(s)}[0,1)$ classes.
\(\operatorname{Lip}_\alpha ^{(s)}[0,1)\) class
scientific article; zbMATH DE number 6987687