B. J. González

Mehler-Fock transforms of generalized functions via the method of adjoints

In this paper we analyse the Mehler-Fock transform of generalized functions via the method of adjoints. For a distribution of compact support, we prove that its Mehler-Fock transform agrees with its transform via the kernel method. A Paley-Wiener type theorem is established.

Gradient estimates for positive solutions of the Laplacian with drift

Let M be a complete Riemannian manifold of dimension n without boundary and with Ricci curvature bounded below by −K, where K ≥ 0. If b is a vector field such that ‖b‖ ≤ γ and ∇b ≤ K∗ on M, for some nonnegative constants γ and K∗, then we show that any positive C∞(M) solution of the equation ∆u(x) + (b(x)|∇u(x)) = 0 satisfies the estimate ‖∇u‖ u2 ≤ n(K + K∗) w + γ2 w(1− w) , on M , for all w ∈ (0, 1). In particular, for the case when K = K∗ = 0, this estimate is advantageous for small values of ‖b‖ and when b ≡ 0 it recovers the celebrated Liouville theorem of Yau (Comm. Pure Appl. Math. 28 (1975), 201–228).

A family of Wiener transforms associated with a pair of operators on Hilbert space

In this article, for each pair of operators A and B on the complexification H of a given Hilbert space H′, we consider a family of Wiener transforms given by where f belongs to the class 𝒫 of complex-valued polynomials on H′ and the integration on H′ is performed with respect to the isonormal distribution g c of the variance parameter c>0. We characterize the composition formula, the inversion formula, the Parseval relation and the unitary extension of the Wiener transform to L 2 (H′, dg c ).

New Lp-boundedness properties for the Kontorovich–Lebedev and Mehler–Fock transforms

ABSTRACT In this paper, the authors present a systematic study of several new -boundedness properties for the Kontorovich–Lebedev transform and the Mehler–Fock transform on the spaces and . Relevant connections with various earlier related results are also pointed out.

A variant of the Stieltjes transform on distributions of compact support

Abstract In this paper, by means of a generalized Lambert transform and the Mobius numbers, we obtain an inversion formula for a variant of the Stieltjes transform introduced by Goldberg [R.R. Goldberg, An inversion of the Stieltjes transform, Pacific J. Math. 8 (1958) 213–217.] and defined by G ( x ) = ∫ 0 ∞ f ( t ) t x 2 + t 2 d t , x > 0 on the space E ′ ( ( 0 , ∞ ) ) of distributions of compact support. We also study the analyticity of G.

The Second Quantization And Its General Integral Finite-Dimensional Representation

The purpose of this paper is to obtain an explicit integral representation for the Second Quantization acting over {\open C}^{n} . In concrete terms, we prove that the Segal duality transform maps \Gamma (U), U being a unitary matrix over {\open C}^{n} , in terms of a matrix Wiener transform over {\open R}^{n} .

Matrix Wiener transform

Abstract In this paper we analyse some properties of the matricial expression of the Fourier–Wiener transform, a matrix transform firstly treated by Cameron and Martin for analytic functions [3] , [4] . Here the referred properties are a composition formula, a Parseval relation and an inversion formula, which, according to Segal (1956) [13] extends an unitary explicit integral representation of the second quantization for one integral operator of the Wiener transform [12] . This work includes the unitary extension of the transform to L 2 ( R n , d μ c ) , where f belongs to the class of complex valued polynomials on R n , and dμ c being the Gaussian measure on R n as a unitary map [5] .

A distributional inversion formula for a generalization of the Stieltjes and Poisson transforms

In this paper, we obtain a distributional inversion formula for a generalization of the Stieltjes and Poisson transforms by means of a generalized Lambert transform and the Möbius numbers. Moreover, making use of the Post-Widder inversion operator of the distributional Stieltjes transform, we prove a distributional inversion formula for this transform.

Distributional representation of a Fourier inversion formula

In this article, we prove an inversion formula for the distributional Fourier transform on the space , k ∈ ℤ, k < 0. This result is applied to obtain a representation on 𝒮′ for any distribution of , as a limit of a sequence of ordinary functions.

A new class of Abelian theorems for the Mehler–Fock transforms

The main object of this paper is to derive several new Abelian theorems for the Mehler–Fock transforms. The results presented here are compared with those given earlier by R. S. Pathak and R. N. Pandey [Math. Soc. 3 (1987), 91–95]. Some applications and particular cases are also considered.