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


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
$$\phi ^{\left( {\text{k}} \right)} \left( {\text{t}} \right) = \frac{{\phi _{\text{k}}^0 \left( {\text{t}} \right)\ln ^{\ell _{\text{k}} } \left( {{\text{t}} - {\text{a}}} \right)\ln ^{{\text{q}}_{\text{k}} } \left( {{\text{b}} - {\text{t}}} \right)}} {{\left( {{\text{t}} - {\text{a}}} \right)^{{\text{m}} + \alpha _{\text{k}} + {\text{k}}} \left( {{\text{b}} - {\text{t}}} \right)^{{\text{r}} + {{\beta }}_{\text{k}} + {\text{k}}} }},{\text{k}} = 0,1,2, \ldots ,$$
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].


Boundary Condition Fourier Transform Generalize Function Basic Function Functional Analysis 
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    G. Bremermann, “Distributions, Complex Variables, and Fourier Transforms”, Mir, Moscow, (1968), (in Russian).MATHGoogle Scholar
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    R. Edwards, “Functional Analysis, Theory and Applications”, Mir, Moscow, (1969), (in Russian).MATHGoogle Scholar
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    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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