On \(K_ \mu\) and \(I_ \mu\) transforms of generalized functions
Let \(L_{a,b}^ \mu\), \(L_ a^ \mu\) be Fréchet spaces consisting of \(\varphi(t)\in C^ \infty(I)\), \(I=(0,\infty)\), and \(Y_{a,b}\), \(Y_ a^ \mu\) Fréchet spaces consisting of complex functions \(\Phi(s)\). The authors regard here the \(K_ \mu\)-transformation \(K_ \mu\{\varphi\}(s)=\int_ 0^ \infty \sqrt{st} K_ \mu(st)\varphi(t)dt\) as an isomorphism from \(L_{a,b}^ \mu\) onto \(Y_{a,b}\), and the \(I_ \mu\)-transformation \(I_ \mu\{\varphi\}(s)=\int_ 0^ \infty \sqrt{st}I_ \mu(st)\varphi(t)dt\) as an isomorphism from \(L_ a^ \mu\) onto \(Y_ a^ \mu\). Here \(K_ \mu\) (resp. \(I_ \mu\)) is the modified Bessel function of the third (resp. first) kind. The mixed Parseval equation \[ \int_{c-i\infty}^{c+i\infty} \Phi_ 1(s)\Phi_ 2(s)ds=\pi i\int_ 0^ \infty \varphi_ 1(t)\varphi_ 2(t)dt \] holds for suitable \(\varphi_ 1\), \(\varphi_ 2\) and certain \(c\in R\). Here \(\Phi_ 1=K_ \mu\{\varphi_ 1\}\) and \(\Phi_ 2=I_ \mu\{\varphi_ 2\}\). They define the generalized \(K_ \mu\)-transformation on \((L_ a^ \mu)'\) by using the generalized Parseval equation \[ \langle K_ \mu' f,I_ \mu\varphi\rangle=\pi i\langle f,\varphi\rangle \qquad\text{for}\qquad \forall\varphi\in L_ a^ \mu, \] and the generalized \(I_ \mu\)-transformation on \((L_{a,b}^ \mu)'\) by \[ \langle I_ \mu 'f,K_ \mu\varphi\rangle=\pi i\langle f,\varphi\rangle\qquad\text{for}\qquad \forall\varphi\in L_{a,b}^ \mu. \] Their \(K_ \mu'\) is an extension of the one by \textit{A. H. Zemanian} [SIAM J. Appl. Math. 14, 1350-1365 (1966; Zbl 0154.139)]. They also give an inversion formula for the \(I_ \mu\)-transformation. Next let \(P(s)=\gamma_ 0(s-\gamma_ 1^ 2)^{k_ 1}\dots(s- \gamma_ q^ 2)^{k_ q}\) and \(S_ \mu=t^{-\mu- (1/2)}(d/dt)t^{2\mu+1}(d/dt)t^{-\mu-(1/2)}\). Since the relation \(K_ \mu '\{P(S_ \mu)f\}(s)=P(s^ 2)K_ \mu '\{f\}(s)\) holds for \(f\in(L_ a^ \mu)'\) with \(\mu\geq 0\), \(a>0\), they solve the distribution equation \(P(S_ \mu)u=g\) by applying their generalized transformations.
scientific article; zbMATH DE number 62061