×

Summability of trigonometric series and Calderón reproducing formulas. (English) Zbl 0886.42002

Let \(T={\mathbf R}/2\pi{\mathbf Z}\) and let \(f\in L_p(T)\). For an even function \(k\) such that \(\int_{\mathbf R} k(x)dx=1\), a summability kernel \(\varphi_a\) is defined by \(\varphi_a(t) = \sum_n \widehat{k} (an) e^{int}\), where \(\widehat{k}\) is the Fourier transform of \(k\). It is assumed that both \(k\) and \(\widehat{k}\) are continuous and satisfy a certain growth condition. Further, let \(u_f(t,a)=\varphi_a\ast f\), where \(\ast\) denotes the convolution. In the first part of the paper, the author considers \[ f_{\varepsilon}(t) =c^{-1} \int_{\varepsilon}^{\infty}\int_T u'_f(s,a)\overline{\varphi'_\alpha(s-t)} ds a da,\quad\text{where}\quad c=\int_0^{\infty} {}\widehat{k}(a){}^2a da \] and shows that \(f_{\varepsilon}\) converges to \(f\) as \(\varepsilon \rightarrow 0\), under certain additional assumptions and in different modes of convergence (in \(L_p(T)\)-norm, uniformly or almost everywhere). In the second part, some local and global properties of \(f\) are related to certain properties of \(u_f\).

MSC:

42A24 Summability and absolute summability of Fourier and trigonometric series
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
PDFBibTeX XMLCite
Full Text: EuDML