A generalization of \(H_{\mu}\)-spaces and Hankel transforms
We define test function spaces \(H_{\mu,M,a}\) and \(H_{\mu}^{\Omega,b}\). The space \(H_{\mu,M,a}\) is not a subspace of Zemanian's space \(H_{\mu}\). For \(M(x)=x^{1/\alpha}\), \(0<\alpha <1\), \(H_{\mu,M,a}\) is related to \textit{W. Y. K. Lee}'s space \(H_{\mu,\alpha}\) [SIAM J. Math. Anal. 5, 336-348 (1974; Zbl 0282.44002)]. The space \(H_{\mu}^{\Omega,b}\) is a generalization of \textit{A. H. Zemanian}'s space \({\mathcal Y}_{\mu,b}\) [SIAM J. Appl. Math. 14, 678-690 (1966; Zbl 0154.138)]. For arbitrary \(\Omega\), \({\mathcal Y}_{\mu,b}\subset H_{\mu}^{\Omega,b}\). Using adjoint methods, we define the generalized Hankel transform on dual spaces \(H'_{\mu,M,a}\), \(H'_{\mu,M}\), \((H_{\mu}^{\Omega,b})'\) and \((H^{\Omega}_{\mu})'\). Finally, we derive some operational formulae which are used to solve some generalized boundary value problems.
scientific article; zbMATH DE number 4024034