Almost everywhere convergence of the spherical partial sums for radial functions. (English) Zbl 0639.42010
Let \(f\in L^ p(R^ n)\), \(n\geq 2\), be a radial function and let \(S_ Rf\) be the spherical partial sums operator. We prove that if \(2n/n+1<p<2n/n-1\) then \(S_ Rf(x)\to f(x)\) a.e. as \(R\to \infty\). The result is false for \(p<2n/n+1\) and \(p>2n/n-1\).
Reviewer: E.Prestini
MSC:
| 42B05 | Fourier series and coefficients in several variables |
Keywords:
radial functionReferences:
| [1] | Herz, C.: On the mean inversion of the Fourier and Hankel transforms. Proc. Nat. Acad. Sci. U.S.A.40, 996–999 (1954). · Zbl 0059.09901 · doi:10.1073/pnas.40.10.996 |
| [2] | Kenig, C., Tomas, P.: The weak behavior of the spherical means. Proc. Amer. Math. Soc.78, 48–50 (1980). · Zbl 0432.42014 · doi:10.1090/S0002-9939-1980-0548082-8 |
| [3] | Chanillo, S.: The multiplier for the ball and radial functions. J. Funct. Anal.55, 18–24 (1984). · Zbl 0531.42004 · doi:10.1016/0022-1236(84)90015-6 |
| [4] | Fefferman, C.: The multiplier problem for the ball. Ann. Math.94, 330–336 (1971). · Zbl 0234.42009 · doi:10.2307/1970864 |
| [5] | Hunt, R., Young, W. S.: A weighted norm inequality for Fourier series. Bull. Amer. Math. Soc.80, 274–277 (1974). · Zbl 0283.42004 · doi:10.1090/S0002-9904-1974-13458-0 |
| [6] | Stein, E. M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Univ. Press. 1971. · Zbl 0232.42007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
