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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 β are missing. Also the authors claim to get an estimate for |T ˜ n,p -f ˜| in L p -norm which is really never used in the proof, where 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 T ˜ n,p (x)-f ˜(x) for all x by assuming conditions, which seem to be in non-existence.

Surprisingly they assume fW(L p ,ψ 1 (t)) but they mention on page 40, line 7, that ψ(t)W(L p ,ψ 1 (t)), where ψ(t)=f(x+t)-f(-t). Unfortunately whatever they have proved is also not correct since on page 40, lines 5-6 from bottom

0 π/n t -(2+β)q dtt 1-(2+β)q 1-(2+β)q 1 π/n ·

Also the paper contains a number of misprints.

MSC:
42A10Trigonometric approximation
41A25Rate of convergence, degree of approximation
42A50Conjugate functions, conjugate series, singular integrals, one variable