Abstract
A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2π], then the nth partial sum of its formally differentiated Fourier series divided by n converges to π-1[F(x+0)-F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of (nonperiodic) functions of bounded variation is also well known.
The aim of the present article is to extend these results to the (m, n)th rectangular partial sum of double Fourier or Fourier-Stieltjes series of a function F(x, y) of bounded variation over the closed square [0, 2π]×[0, 2π] in the sense of Hardy and Krause. As corollaries, we also obtain the following results:
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(i)
The terms of the Fourier or Fourier-Stieltjes series of F(x, y) determine the atoms of the (periodic) Borel measure induced by (an appropriate extension of) F.
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(ii)
In the case of periodic functions F(x, y) of bounded variation, the class of double Fourier-Stieltjes series coincides with the class of series that can be obtained from their Fourier series by a formal termwise differentiation with respect to both x and y.
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Communicated by Tom Körner
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Móricz, F. Extension of a theorem of Fejér to double Fourier-Stieltjes series. The Journal of Fourier Analysis and Applications 7, 601–614 (2001). https://doi.org/10.1007/BF02513078
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DOI: https://doi.org/10.1007/BF02513078