Local convergence of Fourier series with respect to periodized wavelets
The paper investigates the behavior of wavelet Fourier series of \(f\in L([0,1]^m)\) at Lebesgue and so-called strong Lebesgue points. Here \(x\) is said to be a Lebesgue point of \(f\) if \[ \lim_{h\rightarrow + 0}{1\over h^m}\int\limits_{[-h,h]^m}| f(x+t)-f(x) | dt=0. \] The wavelet Fourier basis of \(L_2([0,1]^m)\) is constructed by means of a tensor product of a one-dimensional multiresolution analysis of \(L^2([0,1])\) generated by a suitable orthogonal scaling function \(\varphi\). It is shown that the wavelet Fourier series converges at each strong Lebesgue point. Further, if \(\varphi\) and a corresponding orthogonal wavelet \(\psi\) satisfy a suitable decay condition then convergence at each Lebesgue point is found. The results can be transferred to the non-periodic case.
scientific article; zbMATH DE number 1214022