summary:In this note we show that the two definitions of generalized index ${ }_{2}F_{1}$-transform given in the previous works [1] and [2] agree for distributions of compact support

González, Benito J.

Negrin, Emilio R.

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeBenito J. González; Emilio R. NegrinThe link between the kernel method and the method of adjoints for thegeneralized index 2F1-transformCommentationes Mathematicae Universitatis Carolinae, Vol. 37 (1996), No. 4, 691--694Persistent URL: http://dml.cz/dmlcz/118878Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1996Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 37,4 (1996)691–694 691The link between the kernel method and the method ofadjoints for the generalized index 2F1−transformBenito J. González, Emilio R. NegrinAbstract. In this note we show that the two definitions of generalized index 2F1-transformgiven in the previous works [1] and [2] agree for distributions of compact support.Keywords: index transform, hypergeometric function, distributions of compact supportClassification: 44A15, 46F12The generalized index 2F1-transform of f ∈ E′(I), I = (0,∞), was defined bythe kernel method in [1] as:(1) F (τ) = 〈f(x),F(µ, α, τ, x)〉, τ ∈ I,whereF(µ, α, τ, x) = 2F1(µ+12+ iτ, µ+12− iτ ;µ+ 1;−x)xαand 2F1(µ+12 + iτ, µ+12 − iτ ;µ+ 1;−x) is the Gauss hypergeometric functiondefined for |x| < 1 as the sum of the Gauss series, while for |x| ≥ 1, is defined asits analytic continuation [4, p. 431]. Here µ and α are complex parameters.In [2] the generalized index 2F1-transform was analyzed by the method ofadjoints. It was required to introduce two Fréchet spaces V and W satisfyingthat the operator L :W −→ V given by(2) (Lψ) (x) =∫ ∞0ψ(τ)F(µ, α, τ, x) dτ, x > 0,was continuous whenever Re µ > −12 . To be more precise, the spaces V and Wwere defined byV = {φ ∈ C∞(I) : γk(φ) <∞, k ∈ N ∪ {0}}whereγk(φ) = supx∈I∣∣∣x−α(x+ 1)µ+32Akxφ(x)∣∣∣ ,692 B.J.González, E.R.NegrinAx being the differential operator defined byxα−µ(x+ 1)−µDxxµ+1(x+ 1)µ+1Dxx−α,andW = {ψ ∈ C∞(R), even : ̺r(ψ) <∞, r ∈ N ∪ {0}} ,where̺r(ψ) = supx∈I∣∣∣∣∣∣x32 exF[(µ+ 12)2+ τ2]rψ(τ)τΓ(µ+ 12 + iτ)Γ(µ+ 12 − iτ) (x)∣∣∣∣∣∣,and F denotes Fourier transform with respect to the τ -variable.The generalized index 2F1-transform was defined as the adjoint operator of L,namely,(3) 〈L′f, ψ〉 = 〈f,Lψ〉, f ∈ V ′, ψ ∈ W.In this note we establish the link between these two definitions. Here, it willbe proved that both definitions agree for distributions of compact support. Thecorresponding result for the Kontorovich-Lebedev transform was established in[3, Proposition 2.4].Theorem. Let f ∈ E ′(I) and α, µ being complex parameters with Re α > 0,Re µ > 0, 18 < Re (µ−α) <14 and Re (µ− 2α) < −1. Then, for any ψ ∈ W , onehas(4) 〈L′f, ψ〉 = 〈T〈f(x),F(µ,α,τ,x)〉, ψ(τ)〉,where T〈f(x),F(µ,α,τ,x)〉 is the member of W′ given by(5) 〈T〈f(x),F(µ,α,τ,x)〉, ψ(τ)〉 =∫ ∞0〈f(x),F(µ, α, τ, x)〉ψ(τ) dτ.Proof: First, observe that D(I) ⊂ V ⊂ E(I) and the topology of V is strongerthan the one induced on it by E(I). Thus, E ′(I) is a subspace of V ′.Also, from [1, Theorem 3.2, p. 662] the function F (τ) given by (1) is an entirefunction such that(i) F (τ) = O(1) as τ → 0+;(ii) there exists a r ∈ N ∪ {0} such that F (τ) = O(τ2r−Re µ−12)as τ → ∞.From this and having into account that, for any ψ ∈ W one has(i) ψ(τ) = O(τ2) as τ → 0+;(ii) for all p ∈ N, ψ(τ) = O (τ−p), as τ → ∞;The link between the kernel method and the method of adjoints . . . 693(cf. [2, Proposition 2.2 and Proposition 2.4]) it follows that the integral of theright hand side of (5) has sense.Clearly, if f ∈ D(I), the result of this Theorem holds. In fact, assume thatf has its support contained in the closed interval [a, b] ⊂ I, then, using Fubinitheorem, one has〈L′f, ψ〉 = 〈f,Lψ〉=∫ baf(x)(Lψ)(x)dx =∫ baf(x)∫ ∞0F(µ, α, τ, x)ψ(τ) dτ dx=∫ ∞0ψ(τ)∫ baf(x)F(µ, α, τ, x) dx dτ =∫ ∞0F (τ)ψ(τ) dτ.So, the result holds for f ∈ D(I).Now, for f ∈ E ′(I) and using [5, Theorem 28.2(i), p. 301], there exists a se-quence of functions {fn}n∈N belonging to D(I) which converges to f in E′(I) andso, in V ′. Furthermore, from [2, Proposition 2.3] it follows that L′ is a continuousmapping from V ′ into W ′. Thus{L′fn}n∈N converges to L′f in W ′ as n→ ∞.Since, for all ψ ∈ W ,〈L′fn, ψ〉 =∫ ∞0Fn(τ)ψ(τ) dτwhereFn(τ) = 〈fn(x),F(µ, α, τ, x)〉,it follows that(6) 〈L′f, ψ〉 = limn→∞∫ ∞0Fn(τ)ψ(τ) dτ.Clearly, from the asymptotic growth of Fn and ψ, the limit in (6) goes into theintegral. Also, noting thatlimn→∞Fn(τ) = limn→∞〈fn(x),F(µ, α, τ, x)〉= 〈f(x),F(µ, α, τ, x)〉 = F (τ),the result holds. References[1] Hayek N., González B.J., The index 2F1-transform of generalized functions, Comment.Math. Univ. Carolinae 34.4 (1993), 657–671.[2] Hayek N., González B.J., On the distributional index 2F1-transform, Math. Nachr. 165(1994), 15–24.[3] Lisena B., On the Generalized Kontorovich-Lebedev Transform, Rend. Math. Appl. (7) 9(1989), 87–101.694 B.J.González, E.R.Negrin[4] Prudnikov A.P., Brychkov Y.A., Marichev O.I., Integrals and Series, vol. 3, Gordon andBreach Science Publishers, New York, 1990.[5] Treves F., Topological Vector Spaces, Distributions and Kernels, Academic Press, NewYork, 1967.Departamento de Análisis Matemático, Universidad de La Laguna, 38271 CanaryIslands, SpainE-mail : bgonzalez@ull.esenegrin@ull.es(Received January 16, 1996)

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The link between the kernel method and the method of adjoints for the generalized index $_2F_1$-transform

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The link between the kernel method and the method of adjoints for the generalized index $_2F_1$ -transform

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The link between the kernel method and the method of adjoints for the generalized index 2F1_2F_12​F1​-transform

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The link between the kernel method and the method of adjoints for the generalized index $_2F_1$ -transform