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Distributions and Their Fourier transforms

  • Vladimir B. Vasil’ev
Chapter

Abstract

Distributions are linear continuous functionals over the space of so-called basic functions. Space of basic functions we choose as the Schwartz space S(ℝ m ) (ℝ m is m — dimensional Euclidean space) of infinitely differentiable on ℝ m functions decreasing under |x| → ∞ more rapidly than any power of |x|−1, x = (x 1, ..., x m ), \(|x| = \sqrt {{x_1}^2 + \ldots + {x_m}^2} .\) We determine the counting number of norms in S(ℝ m ) by the formula
$$||\varphi |{|_p} = \sup {(1 + |x{|^2})^{p/1}}|{D^a}\varphi (x)|,\varphi \in S({\mathbb{R}^m}),p = 0,1 \ldots ,|a| \leqslant p$$
(1.1.1)
where \({D^a}\varphi = \frac{{{\partial ^{|\alpha |}}\varphi }}{{\partial {x_1}^{{\alpha _1}} \ldots \partial {x_m}^{{\alpha _m}}}}\), α is multi index, |α| = α 1 + ⋯ + α m ; with the help of these norms we define the convergence concept in S(ℝ m ). Namely we say the sequence φ 1, ..., φ k , ... of functions from S(ℝ m ) converges to function φS(ℝ m ) iff ∥φ k φ p → 0, k → ∞ for all ρ = 0, 1, ... The last statement, by virtue of (1.1.1) is equivalent to saying that x α D β φ k (x) uniformly tends to zero under k → ∞ for arbitrary multiindex \(\alpha ,\beta ,{x^\alpha } \equiv {x_1}^{{\alpha _1}} \ldots {x_m}^{{\alpha _m}}\).

Keywords

Fourier Transform Inverse Fourier Transform Dimensional Euclidean Space Regular Distribution Schwartz Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Vladimir B. Vasil’ev
    • 1
  1. 1.Department of Mathematical AnalysisNovgorod State UniversityNovgorodRussia

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