Approximation of \(\tilde{f}\), conjugate function of \(f\) belonging to a subclass of \(L^p \)-space, by product means of conjugate Fourier series. (English) Zbl 1439.42007
The authors prove results for the conjugate Fourier series comparable to results proved for Fourier series by W. Łenski and B. Szal [Acta Comment. Univ. Tartu. Math. 13, 11–24 (2009; Zbl 1193.42032)]. They introduce summability methods using infinite matrices and prove a convergence result in \(L^p\), for \(p >1\), with a separate result for \(p = 1\) that does not require Hölder. For \(f\) in a \(L^p\) class defined by the behavior of its \(L^p\) modulus of continuity, the degree of approximation of the summed conjugate series to the conjugate function is expressed (for each \(x\)) in terms of the coefficients of the matrices and the modulus of continuity of the function defining the conjugate Fourier series. The results are rather technical; details are in the paper for those interested.
Reviewer: Raymond Johnson (Columbia)
MSC:
| 42A50 | Conjugate functions, conjugate series, singular integrals |
| 42A24 | Summability and absolute summability of Fourier and trigonometric series |
| 42B35 | Function spaces arising in harmonic analysis |
| 40A25 | Approximation to limiting values (summation of series, etc.) |
Keywords:
\(L^p\tilde{(\omega)}_{\beta} \)-class; product means; conjugate Fourier series; degree of approximationCitations:
Zbl 1193.42032References:
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