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

Let S_m,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 1 $$\phi ^{\left( {\text{k}} \right)} \left( {\text{t}} \right) = \frac{{\phi _{\text{k}}^0 \left( {\text{t}} \right)\ln ^{\ell _{\text{k}} } \left( {{\text{t}} - {\text{a}}} \right)\ln ^{{\text{q}}_{\text{k}} } \left( {{\text{b}} - {\text{t}}} \right)}} {{\left( {{\text{t}} - {\text{a}}} \right)^{{\text{m}} + \alpha _{\text{k}} + {\text{k}}} \left( {{\text{b}} - {\text{t}}} \right)^{{\text{r}} + {{\beta }}_{\text{k}} + {\text{k}}} }},{\text{k}} = 0,1,2, \ldots ,$$ where 0 ≦ α_k < 1, 0 ≦ β_k < l, ℓ_k,q_k ≧ 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 H_λ, 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].