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1: 1.16 Distributions
§1.16(v) Tempered Distributions
For a detailed discussion of tempered distributions see Lighthill (1958). …
§1.16(vii) Fourier Transforms of Tempered Distributions
The Fourier transform ( u ) of a tempered distribution is again a tempered distribution, and …
2: 2.6 Distributional Methods
To each function in this equation, we shall assign a tempered distribution (i. …, a continuous linear functional) on the space 𝒯 of rapidly decreasing functions on . …
2.6.11 f , ϕ = 0 f ( t ) ϕ ( t ) d t , ϕ 𝒯 .
3: Bibliography K
  • D. A. Kofke (2004) Comment on “The incomplete beta function law for parallel tempering sampling of classical canonical systems” [J. Chem. Phys. 120, 4119 (2004)]. J. Chem. Phys. 121 (2), pp. 1167.
  • 4: Errata
  • Subsection 1.16(vii)

    Several changes have been made to

    1. (i)

      make consistent use of the Fourier transform notations ( f ) , ( ϕ ) and ( u ) where f is a function of one real variable, ϕ is a test function of n variables associated with tempered distributions, and u is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

    2. (ii)

      introduce the partial differential operator D in (1.16.30);

    3. (iii)

      clarify the definition (1.16.32) of the partial differential operator P ( D ) ; and

    4. (iv)

      clarify the use of P ( D ) and P ( x ) in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

  • 5: 1.14 Integral Transforms
    1.14.1 ( f ) ( x ) = f ( x ) = 1 2 π - f ( t ) e i x t d t .