Strong summability of Fourier series on classes of periodic functions
Let f be a summable \(2\pi\)-periodic function (f\(\in L(0,2\pi))\) and let \[ (1)\quad S[f]=a_ 0/2+\sum^{\infty}_{j=1}(a_ j \cos jx+b_ j \sin jx) \] be its Fourier series. Let \(\psi\) (k) be a positive sequence of numbers, \(\beta\) a fixed number, and let the series \[ \sum^{\infty}_{k=1}\frac{1}{\psi (k)}[a_ k \cos (kx+\beta \pi /2)+b_ k \sin (kx+\beta \pi /2)] \] be the Fourier series of some function from L(0,2\(\pi\)). We denote this function by \(f^{\psi}_{\beta}\) and the set of all such functions by \(L^{\psi}_{\beta}\). We introduce the spaces \[ C^{\psi}_{\beta,\infty}=\{f\in C\cap L^{\psi}_{\beta};\quad ess \sup | f^{\psi}_{\beta}| \leq 1\},\quad C^{\psi}_{\beta}H_{\omega}=\{f\in C\cap L^{\psi}_{\beta};f^{\psi}_{\beta}\in H_{\omega}\} \] where C is the set of continuous \(2\pi\)-periodic functions and \(H_{\omega}=\{f\in C;\sup_{| x-y| \leq t}| f(x)-f(y)| \leq \omega (t)\) (\(\omega\) is a given modulus of continuity)\(\}\). Everything has been introduced in \textit{A. I. Stepanets} [Izv. Akad. Nauk SSSR, Ser. Mat. 50, No.1, 101-136 (1986; Zbl 0615.42005)]. Let \(S_ k(f;x)\) be a partial sum of the series (1) and let \(\rho_ k(f;x)=f(x)-S_{k-1}(f;x).\) We define the functional \[ H_ n^{(p)}(f;x,\lambda)=\sum^{\infty}_{k=n}\lambda_ k| \rho_ k(f;x)|^ p \] where \(p>0\) and \(\lambda\) is a positive sequence of numbers. In this paper there are obtained estimates of the quantities \(\| H_ n^{(p)}(f;x,\lambda)\|_ C\) on the classes \(C^{\psi}_{\beta}H_{\omega}\). One of the main estimates is \[ \sup_{f\in C^{\psi}_{\beta}H_{\omega}}\| H_ n^{(p)}(f;x,\lambda)\|_ C\leq B_ p\{\lambda_ n\psi^ p(n)\omega^ p(1/n)(n+1)+\sum^{\infty}_{k=n+1}\lambda_ k\psi^ p(k)\omega^ p(1/k)\} \] for all \(\beta\in R\), \(p>0\) and certain \(\psi\) and \(\lambda\). Some estimates analogous to that ones obtained by \textit{A. I. Stepanets} and the author [On strong summability of Fourier series (Russian) (Problems of Summation of Fourier Series, 3-13, Preprint 85.61, Inst. Math., Acad. Sci. UkrSSR (Kiev) (1985)] are obtained. Other related questions are also considered.
scientific article; zbMATH DE number 4165637