Abstract
We prove the following theorem. Assume f ∈ L ∞(R 2) with bounded support. If f is continuous at some point (x 1,x 2) ∈ R 2, then the double Fourier integral of f is strongly q-Cesàro summable at (x 1,x 2) to the function value f(x 1,x 2) for every 0 < q < ∞. Furthermore, if f is continuous on some open subset \(\mathcal{G}\) of R 2, then the strong q-Cesàro summability of the double Fourier integral of f is locally uniform on \(\mathcal{G}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Auscher and M. J. Carro, On relations between operators on R N, TN, and ZN, Studia Math., 101 (1992), 166–182.
G. H. Hardy, Divergent Series, Oxford Univ. Press (London, 1949).
F. Móricz, Strong Cesàro summability and statistical limit of Fourier integrals, Analysis, 25 (2005), 79–86.
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press (New Jersey, 1971).
A. Zygmund, Trigonometric Series, Cambridge Univ. Press (UK, 1959).
Author information
Authors and Affiliations
Additional information
Research partially supported by the Australian Research Council and NNSF of China under Grant # 1007-1007.
Research partially supported by the Australian Research Council and the Hungarian National Foundation for Scientific Research under Grant T 046 192.
Rights and permissions
About this article
Cite this article
Brown, G., Feng, D. & Móricz, F. Strong Cesàro summability of double Fourier integrals. Acta Math Hung 115, 1–12 (2007). https://doi.org/10.1007/s10474-007-4185-z
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10474-007-4185-z
Key words and phrases
- double Fourier transform and integral
- inversion formula
- partial (or Dirichlet) integral
- summability (C, 1)
- strong q-Cesàro summability