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The Problem of the Jump and the Sokhotski Formulas in the Space of Generalized Functions on a Segment of the Real Axis

  • L. V. Kartashova
  • V. S. Rogozhin

Abstract

Let Sm,n (m,n are fixed, m ≧ 0, n ≧ 0) denote the linear countable normed space of smooth functions that can be represented with their derivatives in the form
$$\phi ^{\left( {\text{k}} \right)} \left( {\text{t}} \right) = \frac{{\phi _{\text{k}}^0 \left( {\text{t}} \right)\ln ^{\ell _{\text{k}} } \left( {{\text{t}} - {\text{a}}} \right)\ln ^{{\text{q}}_{\text{k}} } \left( {{\text{b}} - {\text{t}}} \right)}} {{\left( {{\text{t}} - {\text{a}}} \right)^{{\text{m}} + \alpha _{\text{k}} + {\text{k}}} \left( {{\text{b}} - {\text{t}}} \right)^{{\text{r}} + {{\beta }}_{\text{k}} + {\text{k}}} }},{\text{k}} = 0,1,2, \ldots ,$$
(1)
where 0 ≦ αk < 1, 0 ≦ βk < l, ℓk,qk ≧ 0, \(\phi _{\text{k}}^0 \left( {\text{t}} \right)\) (k = 0,1,2,…)are smooth functions on (a,b) and H-continuous on [a,b]; a function ψ is an H-function or Hölder’s function, from Hλ, λ > 0, if there is a constant A so that \(\left| {\psi \left( {{\text{t}}_{\text{1}} } \right) - {{\psi }}\left( {{\text{t}}_{\text{2}} } \right)} \right| < {\text{A}}\left| {{\text{t}}_{\text{1}} - {\text{t}}_2 } \right|^\lambda\) for all t1, t2 ∈ [a, b].

Keywords

Boundary Condition Fourier Transform Generalize Function Basic Function Functional Analysis 
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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References

  1. 1.
    G. Bremermann, “Distributions, Complex Variables, and Fourier Transforms”, Mir, Moscow, (1968), (in Russian).MATHGoogle Scholar
  2. 2.
    R. Edwards, “Functional Analysis, Theory and Applications”, Mir, Moscow, (1969), (in Russian).MATHGoogle Scholar
  3. 3.
    D. Mitrović, A singular convolution equation in the space of distributions, Publ. Inst. Math. Beograd, 21(35), 151–163, (1977).MathSciNetGoogle Scholar

Copyright information

© Plenum Press, New York 1988

Authors and Affiliations

  • L. V. Kartashova
    • 1
  • V. S. Rogozhin
    • 1
  1. 1.Rostov State UniversityRostov on DonUSSR

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