Approximation to continuous functions by Cesàro means of double Fourier series and conjugate series
Let f(x,y) be of period \(2\pi\) in each of x,y. given \(\alpha\),\(\beta\) with \(0<\alpha,\beta \leq 2\), we say that \(f\in Z(\alpha,\beta)\) if \[ f(x+u,y)+f(x-u,y)-2f(x,y)=O(u^{\alpha}); \] \[ f(x,y+v)+f(x,y-v)- 2f(x,y)=O(v^{\beta}); \] uniformly in x,y. The class Z(\(\alpha\),\(\beta)\) is the same as Lip(\(\alpha\),\(\beta)\) when \(0<\alpha,\beta <1\); but Z(\(\alpha\),\(\beta)\) is larger than Lip(\(\alpha\),\(\beta)\) when either \(\alpha\) or \(\beta\) (or both) is equal to 1. Let \(S_{mn}(f,x,y)\) denote the partial sum of the Fourier series of f, and let \(\sigma_{nm}^{\lambda \delta}\) denote its (C,\(\lambda\),\(\delta)\) mean; that is to say \[ \sigma_{mn}^{\lambda \delta}(f,x,y)=\frac{1}{A_ m^{\lambda}A_ n^{\delta}}\sum^{m}_{j=0}\sum^{n}_{k=0}A_{m-j}^{\lambda - 1}A_{n-k}^{\delta -1}S_{jk}(f,x,y). \] Let \(f\in Z(\alpha,\beta)\), where \(0<\alpha,\beta \leq 1\). It is proved that if \(0<\lambda,\delta\), then (uniformly) in (x,y) \[ \sigma_{mn}^{\lambda \delta}(f,x,y)- f(x,y)=O((m+1)^{-\alpha})+O((n+1)^{-\beta}), \] except that when \(\alpha =1\) the first term on the right has an extra factor \(\log (m+2)\), and when \(\beta =1\) the second term has an extra factor \(\log (n+2)\). It is to be noted that the order terms are independent of \(\lambda\),\(\delta\). Similar results are obtained for the various conjugate series and functions, but here the order terms sometimes have extra logarithmic factors. In the case of most of the results, examples are given to show that they are best possible.
scientific article; zbMATH DE number 4000673
scientific article