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On the exactness of the representation of conjugate functions by CesĂ ro sums. (Russian) Zbl 0566.42007

Let \({\tilde \sigma}_ n^{\beta}(f,x)=(1/A_ n^{\beta})\sum^{n}_{\nu =0}A_{n-\nu}^{\beta -1}\tilde S_{\nu}(f,x),\) where \(\tilde S_{\nu}(f,x)\) are the partial sums of the conjugate Fourier series of a \(2\pi\)-periodic function f and \(A_ n^{\beta}=\left( \begin{matrix} n+\beta \\ n\end{matrix} \right)\), \(\beta \neq - 1,-2,...\). Suppose also that \(f\in Lip_ 1\alpha\) (i.e. \(| f(x_ 1)-f(x_ 2)| \leq | x_ 1-x_ 2|^{\alpha}\), \(0<\alpha \leq 1\) for every \(x_ 1,x_ 2\in [-\pi,\pi]\)) and \(\tilde f(x)=(- 1/2\pi)\int^{\pi}_{0}\{f(x+t)-f(x-t)\} ctg t/2 dt\) is the conjugate function.
In the paper the following statement is proved. theorem. Let \(2<\beta <3\) and \[ {\tilde \Delta}_ n^{\beta}(\alpha)=\sup_{f\in Lip_ 1\alpha}\max_{x\in [-\pi,\pi]}| {\tilde \sigma}_ n^{\beta}(f,x)-\tilde f(x)|, \] then \[ {\tilde \Delta}_ n^{\beta}(\alpha)=(2^{\alpha -1}\Gamma (\beta +1)/(\sin (\alpha \pi /2)\Gamma (\beta +1-\alpha)))\cdot 1/n^{\alpha}+O(1/n^{1+\alpha}). \]
Reviewer: T.Ahobadze

MSC:

42A50 Conjugate functions, conjugate series, singular integrals
42A24 Summability and absolute summability of Fourier and trigonometric series
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