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Some continuous analogs of the expansion in Jacobi polynomials and vector-valued orthogonal bases. (English. Russian original) Zbl 1115.33008

Funct. Anal. Appl. 39, No. 2, 106-119 (2005); translation from Funkts. Anal. Prilozh. 39, No. 2, 31-46 (2005).
Denote \[ R(s)=\left(\begin{matrix} \Gamma(\frac{1}{2}-\alpha-is)\Gamma(\frac{1}{2}-\alpha+is)&\Gamma(\frac{1}{2}-i\beta-is)\Gamma(\frac{1}{2}-i\beta+is)\\ \Gamma(\frac{1}{2}+i\beta-is)\Gamma(\frac{1}{2}+i\beta+is)&\Gamma(\frac{1}{2}+\alpha-is)\Gamma(\frac{1}{2}+\alpha+is)\end{matrix}\right). \] For the Hilbert space \(H_{\alpha,\beta},0\leq\alpha\leq 1/2,\beta\in\mathbb R,\alpha+i\beta\neq 0,\) with the inner product \[ \langle(\varphi_1,\varphi_2),(\psi_1,\psi_2)\rangle=\frac{1}{2\pi}\int_0^{\infty}(\varphi_1(s),\varphi_2(s))R(s)\left(\frac{\overline{\psi_1(s)}}{\psi_2(s)}\right)\frac{ds}{| \Gamma(2is)| ^2} \] the author finds an analogue of the index hypergeometric transform (it is called the double hypergeometric transform, \(f\to (\varphi_1,\varphi_2),\varphi_j(s)= \int_{-\infty}^{\infty}f(x)\overline{Q_j(\alpha,\beta;x,s)}dx)\) and proves that it is a unitary bijective operator from \(L^2(\mathbb R)\) to \(H_{\alpha,\beta}.\) The inversion formula is given by \[ f(x)=\langle(Q_1,Q_2),(\overline{\varphi_1},\overline{\varphi_2})\rangle. \] A generalization to \(\alpha>1/2\) is given (with the space \(H_{\alpha,\beta}\oplus W_{\alpha,\beta},\) where \(W_{\alpha,\beta}\) is the finite-dimensional linear space with the inner product \[ \langle c,c^{\prime}\rangle=\frac{1}{2\pi}\sum_{k=0}^n\frac{2\alpha-2k-1}{\Gamma(2\alpha-k)k!}c_k\overline{c_k}'). \]
Also, an \({}_3F_2\)-orthogonal basis in a space of functions ranging in \(\mathbb C^2\) is constructed.The basis lies in the analytic continuation of continuous Hahn polynomials with respect to the index \(n\) of a polynomial.
Proofs use the spectral decomposition of the hypergeometric differential operator on the contour \(\text{Re}\, z=1/2.\) Applications to tensor products of unitary representations of \(SL_2(\mathbb R)\) are discussed.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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