## On the exactness of the representation of conjugate functions by Cesàro 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 ${\tilde \Delta}_ n^{\beta}(\alpha)=(2^{\alpha -1}\Gamma (\beta +1)/(\sin (\alpha \pi /2)\Gamma (\beta +1-\alpha)))\cdot 1/n^{\alpha}+O(1/n^{1+\alpha}).$
conjugate Fourier series; $$Lip_ 1\alpha$$