Approximation of conjugate functions by Fourier sums in \(L^ p_{2\pi}\)
The author proves the following theorem: If \(1<p<\infty\), then \[ {\tilde \rho}_ n(p,\epsilon)=\sup_{f\in L^ p_{2\pi}(\epsilon)}\| \tilde f-S_ n(\tilde f)\|_ p\asymp \sum^{\min (n,\mu^ 2)}_{k=0}(k+1)^{-1}\epsilon_{n+k}+ \] \[ \sum^{\mu^ 2}_{k=n}(k+1)^{-1}\epsilon_ k, \] where \(\mu =\max \{k\in N:k\leq \exp (\theta)\}\), \(\theta =(p-1)^{-1}\) for \(1<p\leq 2\) and \(\theta =p-1\) for \(2\leq p<\infty\), and \(L^ p_ 2(\epsilon)=\{f\in L^ p_{2\pi}:E_ n(f)\leq \epsilon_ n(n\in N)\}\) for an arbitrary sequence \(\epsilon_ n\) such that \(\epsilon_ n\searrow 0(n\nearrow \infty).\) This result extends the author's and Oskolkov's investigations relating to the cases \(p=1,\infty\) and the quantity \(\rho_ n(p,\epsilon)=\sup_{f}\| f-S_ n(f)\|_ p\) [the author, Mat. Zametki 27, 33-48 (1980; Zbl 0445.41005); ibid. 34, No.1, 77-90 (1983; Zbl 0525.42009); \textit{K. I. Oskolkov}, ibid. 18, 515-526 (1975; Zbl 0339.42001); ibid. 25, 551-555 (1979; Zbl 0414.42002)].
scientific article; zbMATH DE number 3927586