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Norm Convergence of Riesz-Bochner Means For Radial Functions

Published online by Cambridge University Press:  20 November 2018

G. V. Welland
G. V. Welland*
Affiliation:
University of Missouri-St. Louis, St. Louis, Missouri
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It is well known now that certain spherical methods in k (≧2) dimensions are rather poor for reconstructing a function from its Fourier transform. Consider a function f in L1(Rk), k ≧ 2,

and

where both integrals are integrals in Rk, the first over the whole space the second over the ball of radius R; x • y is the usual Euclidean inner product of x and y in Rk and \z\2 = z • z.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 1 , 01 February 1975 , pp. 176 - 185
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Chandresekharan, K. and Minakshisundaran, S., Typical means (Oxford University Press, Oxford, 1952).Google Scholar
2. Copson, E. T., Theory of functions of a complex variable (Oxford University Press, Oxford 1935).Google Scholar
3. Fefferman, C. L., The multiplier problem for the ball, Ann. of Math. 94 (1971), 330–336.Google Scholar
4. Hardy, G. H., Littlew∞d, J. E., and Polya, G., Inequalities (Cambridge Univ. Press, Cambridge, 1934).Google Scholar
5. Herz, C. S., On the mean inversion of Fourier and Hankel transforms, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 996–999.Google Scholar
6. Hunt, R. A., On the convergence of Fourier series, Proc. Conf. Orthogonal Expansions and their Continuous Analogues (Edwardsville, 111., 1967), Southern Illinois University Press, Carbondale, 111., 1968, pp. 235–255.Google Scholar
7. Shapiro, V. L., Fourier series in several variables, Bull. Amer. Math. Soc. 70 (1964), 48–93.Google Scholar
8. Stein, E. M., Localization and summability of multiple Fourier series, Acta Math. 100 (1958), 93–147.Google Scholar
9. Stein, E. M., On certain exponential sums arising in multiple Fourier series, Ann. of Math. 73 (1961), 87–109.Google Scholar
10. Stein, E. M., Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–496.Google Scholar
11. Stein, E. M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Series (1971).Google Scholar
12. Watson, G. N., A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1922).Google Scholar
13. Whittaker, E. T. and Watson, G. N., A course of modern analysis (Cambridge University- Press, Cambridge, 1902).Google Scholar