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A class of functions and their degree of approximation. (English) Zbl 0856.42005
The reviewer observes that the statement of the given theorem is not complete since the conditions on $p$, $q$ and $\beta$ are missing. Also the authors claim to get an estimate for $|\widetilde T_{n, p}- \widetilde f|$ in $L_p$-norm which is really never used in the proof, where $\widetilde T_{n, p}(x)$ are almost Nörlund means of the Fourier series of $f$ at $x$. In fact, they have tried to get an estimate for $\widetilde T_{n, p}(x)- \widetilde f(x)$ for all $x$ by assuming conditions, which seem to be in non-existence. Surprisingly they assume $f\in W(L^p, \psi_1(t))$ but they mention on page 40, line 7, that $\psi(t)\in W(L^p, \psi_1(t))$, where $\psi(t)= f(x+ t)- f(- t)$. Unfortunately whatever they have proved is also not correct since on page 40, lines 5-6 from bottom $$\int^{\pi/n}_0 t^{- (2+ \beta)q} dt\ne \Biggl[{t^{1- (2+ \beta)q}\over 1- (2+ \beta)q} \Biggr]^{\pi/n}_1.$$ Also the paper contains a number of misprints.
42A10Trigonometric approximation
41A25Rate of convergence, degree of approximation
42A50Conjugate functions, conjugate series, singular integrals, one variable