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$ W\sp p$-spaces and Fourier transform

Authors: R. S. Pathak and S. K. Upadhyay
Journal: Proc. Amer. Math. Soc. 121 (1994), 733-738
MSC: Primary 46F05; Secondary 42A38, 46E10
MathSciNet review: 1185272
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Abstract: The spaces $ W_M^p,W_{M,a}^p,{W^{\Omega ,p}},{W^{\Omega ,b,p}},W_M^{\Omega ,p},W_{M,a}^{\Omega ,b,p}$ generalizing the spaces of type W due to Gurevich (also given by Friedman, and Gelfand and Shilov) are investigated. Here M, $ \Omega $ are certain continuous increasing convex functions, a, b are positive constants and $ 1 \leq p < \infty $. The Fourier transformation F is shown to be a continuous linear mapping as follows: $ F:W_{M,a}^p \to {W^{\Omega ,1/a,r}},F:{W^{\Omega ,b,p}} \to W_{M,1/b}^r,F:W_{M,a}^{\Omega ,b,p} \to W_{M,1/b}^{\Omega ,1/a,r}$. These results will be used in investigating uniqueness classes of certain Cauchy problems in future work.

References [Enhancements On Off] (What's this?)

  • [1] Avner Friedman, Generalized functions and partial differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0165388
  • [2] I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 3: Theory of differential equations, Translated from the Russian by Meinhard E. Mayer, Academic Press, New York-London, 1967. MR 0217416
  • [3] B. L. Gurevich, New types of test function spaces and spaces of generalized functions and the Cauchy problem for operator equations, dissertation, Kharkov, 1956. (Russian)
  • [4] S. K. Upadhyay, On certain weighted $ {L^p}$-spaces and Fourier and Hankel transforms of distributions, Ph.D. thesis, Banaras Hindu University, Varanesi, 1993.

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