# 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].

Convolution

## Preview

### References

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