Summation methods of trigonometric Fourier series defined by the Zak transform. (Q2721680)

Let \(f\in C_{2\pi}\) and let \(\omega_k(f;\delta)\) denote the \(k\)th modulus of continuity of \(f\). Then ist Fourier series at \(x\) is given by NEWLINE\[NEWLINE{1\over 2} a_0+ \sum^\infty_{k=1} (a_k\cos kx+ b_k\sin kx)= \sum^\infty_{k=0} A_k(f; x).NEWLINE\]NEWLINE In this paper, a new-type of summation method \(U^z_n\) (see \textit{F. Schipp} and \textit{L. Szili} [Bolyai Soc. Math. Stud. 5, 307--320 (1996; Zbl 0866.41017)]), which is based on the Zak transform is used. First, the authors deduce from known theorems due to \textit{A. Kivinukk} [Bolyai Soc. Math. Stud. 5, 237--245 (1996; Zbl 0862.42002)] and \textit{A. Timan} [``Theory of approximation of functions of a real variable'' (1963; Zbl 0117.29001)] two results which include a result on the order of approximation. Later, they discuss two special cases of the \(U^z_n\) transform to obtain the order of approximation. We mention here one such result:NEWLINENEWLINE Theorem 1. The summation method NEWLINE\[NEWLINEU^{z,4}_n(f; x)= \sum^\infty_{m=0} \Biggl(\sum^n_{k=0} \text{sinc}^4\Biggl(m+{k\over n}\Biggr)\, A_k(f; x)\Biggr)\cos mnxNEWLINE\]NEWLINE generated by the Zak transform of the \(\text{sinc}^4\) function forms a uniformly bounded linear transformation of \(C_{2\pi}\) into \(C_{2\pi}\). Moreover, there exists an absolute constant \(M> 0\) such that for all \(f\in C_{2\pi}\) NEWLINE\[NEWLINE\| f- U^{z,4}_n(f)\|_{C_{2\pi}}\leq M\,\omega_2\Biggl(f; {1\over n}\Biggr).NEWLINE\]NEWLINE The authors also use Lanczo's transform (see \textit{A. Kivinukk} [Uch. Zap. Tartu. Gos. Univ. 448, Tr. Mat. Mekh. 21, 31--39 (1978; Zbl 0406.42005)]) of the Fourier series of \(f\in C_{2\pi}\) defined by NEWLINE\[NEWLINEL_n(f; x)={1\over 2} a_0+ \sum^n_{k=1} \text{sinc}\Biggl({k\over n}\Biggr)\, A_k(f; x)NEWLINE\]NEWLINE and obtain the following result of the order of approximation:NEWLINENEWLINE Theorem. For some \(M> 0\) and for all \(f\in C_{2\pi}\) NEWLINE\[NEWLINE\| f- L_n(f)\|_{C_{2\pi}}\leq M\,\omega_2\Biggl(f; {1\over n}\Biggr).NEWLINE\]