On problems of approximation in \(L_ 2\) spaces (Q1333075)

The aim of the present paper is to answer some questions raised by \textit{P. Goyaliya} [Publ. Math. 26, 263-266 (1979; Zbl 0471.42003)]. P. Goyaliya (loc. cit.) proved that: \[ (1) \quad E_n (f^{(s)}) \leq 2^{- 1/2} \omega \bigl( f^{(s)}, \pi/(n + 1) \bigr), \quad \text{and} \qquad (2) \quad E_n (f) \leq Mn^{-s} E_n (f^{(s)}), \] for every function \(f\), such that \(f^{(s)} \in L^2_{2 \pi}\), \(s \in \{0\} \cup N\). For \(g \in L^2_{2 \pi}\) one puts \(E_n (g) = |g - S_n (g) |_{L^2_{2 \pi}}\), where \(S_n (g)\) is the partial sum of the Fourier series of \(g\). In the same paper, P. Goyaliya asked if these results hold for the case of \((C,1)\)-summability or for other orthogonal expansions, viz. Legendre series, Bessel series etc. The author proves that (1) can be extended to (C,1)-summability but (2) cannot. The case of Legendre series is also considered.