Cesàro summability of the conjugate series and the double Hilbert transform
HTML articles powered by AMS MathViewer
- by John O. Basinger PDF
- Proc. Amer. Math. Soc. 56 (1976), 177-182 Request permission
Abstract:
If $f(x,y)$, a $2\pi$ periodic function in each variable, has a modulus of continuity ${w_f}(\delta ) = o(1/\log (1/\delta ))$ then \[ {\tilde \sigma _n}(x,y,f) - \int _{1/n}^\pi {\int _{1/n}^\pi {\frac {{[f(x + u,y + v) - f(x - u,y + v) - f(x + u,y - v) + f(x - u,y - v)]}}{{4\tan (u/2)\tan (v/2)}}} } dudv \to 0\quad {\text {uniformly}}\;{\text {in}}\;(x,y)\] where ${\tilde \sigma _n}(x,y,f)$ is the first arithmetic mean of the conjugate series. This theorem is best possible in that $o(1/\log (1/\delta ))$ cannot be replaced by $O(1/\log (1/\delta ))$.References
- Victor L. Shapiro, Fourier series in several variables, Bull. Amer. Math. Soc. 70 (1964), 48–93. MR 158222, DOI 10.1090/S0002-9904-1964-11026-0
- Victor L. Shapiro, Singular integrals and spherical convergence, Studia Math. 44 (1972), 253–262. MR 313711, DOI 10.4064/sm-44-3-253-262
- K. Sokół-Sokołowski, On trigonometric series conjugate to Fourier series of two variables, Fund. Math. 34 (1947), 166–182. MR 21619, DOI 10.4064/fm-34-1-166-182
- A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 56 (1976), 177-182
- MSC: Primary 42A40
- DOI: https://doi.org/10.1090/S0002-9939-1976-0410231-9
- MathSciNet review: 0410231