Substitution theorems for integral transforms with symmetric kernels
The authors deduce a number of general theorems for a class of integral transforms. If we use the notation \[ h(p)=T\{f(x);p\}=\int^{\infty}_{0}K(p,x)f(x)dx \] and \(h(p;(b,c))=T\{f(x)[U(x-b)-U(x-c)];p\}=\int^{c}_{b}K(p,x)f(x)dx\) then one of the theorems proved by the authors is: If \(h_ 1(p)=T_ 1\{k(x)h_ 2[g(x)];p\}\) and \(h_ 2(p^{\sigma})=T_ 2\{f(x);p\}\) then \(h_ 1(p)=\sigma \int^{\infty}_{0}\phi (p,u)f(u)du\) where \[ \phi (p,u)=T_ 2\{K_ 1[ph(y^{\sigma})K[h(y^{\sigma})]y^{\sigma - 1}h'(y^{\sigma});u\}. \] Suitable special cases are mentioned.
scientific article; zbMATH DE number 4063766