Integral Representations and Isomorphism of Isotropy Classes

  • S. M. Nikol’skii
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 205)


The Fourier-transform of the function (1 + |x|2)r/2, for sufficiently large r > 0, may be obtained effectively, since it is a function of |x|, and the well-known formula1 $$\matrix{\overline{(1+\mid x\mid^2)^{-r/2}}={1\over (2\pi)^{n/2}}\int {e^{iu\xi}d\xi\over (1+\mid \xi \mid^2)^{r/2}}\cr ={1\over{\mid u\mid^{{n-r}\over 2}}}{\mathop\int\limits_0^\infty}{\varrho^{n/2}\over (1+\varrho^2)^{r/2}}I_{{n-2}\over 2}(\mid u\mid\varrho)d\varrho,}$$ where Iμ is the Bessel Function of order μ, is applicable to it.


Integral Representation Entire Function Regular Function Trigonometric Polynomial Exponential Type 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1975

Authors and Affiliations

  • S. M. Nikol’skii
    • 1
  1. 1.Sergei Mihailovič Nikol’skii Steklov MathematicalInstitute Academy of SciencesMoscowUSA

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