On the rate of convergence of Shannon wavelet expansion for functions of local \(\Lambda\)BV (Q1271948)

The main purpose of the paper is to investigate the convergence of the partial sum \(S_n(f,x)\) of the Shannon wavelet expansion for functions of local \(\Lambda\)BV and to estimate the rate of convergence. The author considered the orthonormal bases of wavelets in \(L^2(R)\) and assumed only the standard conditions on the wavelets; the scaling function \(\varphi(x)\) whose translates \(\{\varphi(t-u)\}\) are orthogonal, the mother wavelet \(\psi(x)\) based on \(\varphi(x)\) and the space \(S_r\) of \(r\)-regular functions with a suitable topology. Based on the previous study of G. Walter, who considered the convergence of \(f_n(x)\) to \(f(x)\) and proved that if \(\varphi\in S_r\) and \(f\in L^1\cap L^2\) is continuous in \((a,b)\) then \(f_n(x)\) converges uniformly to \(f(x)\) on compact subsets of \((a,b)\), the author improved these results and proved that if \(f\in L^2\) is continuous in \((a,b)\) and \(\varphi(x)\) satisfies certain conditions, then \(f_n(x)\) converges uniformly to \(f(x)\) on a compact subset of \((a,b)\) and established an estimate of the rate of convergence.