A Distributional Version of the Ferenc Lukács Theorem
DOI:
https://doi.org/10.5644/SJM.01.1.08Keywords:
Distributions, Fourier series, conjugate series, theorem of Ferenc LukácsAbstract
The theorem of F. Lukács determines the generalized jumps of a periodic, integrable function in terms of a logarithmic average of the partial sums of its conjugate Fourier series. Recently, F. Móricz gave a version of Lukács result for the Abel-Poisson means of the conjugate Fourier series, under an extended notion of jump. In this article we give a generalization that applies to periodic distributions under a much extended notion of jump, namely, that of distributional point values of Łojasiewicz. Our generalization is obtained by obtaining results on the local boundary behaviour of an analytic function with distributional boundary values near a point where the boundary generalized function has a jump.
2000 Mathematics Subject Classification. 42A50, 46F10, 42A16
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