E. R. Negrin

Poisson transform on Boehmians

Convolution theorem for Poisson transform on compactly supported distributions is obtained. Applying the convolution theorem, Poisson transform is extended as a linear continuous map from a suitable Boehmian space into another Boehmian space satisfying the convolution theorem.

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} .

Wick products of the CAR algebra

The purpose of this paper is to put in a precise mathematical (algebraic) form the Wick products of the CAR algebra. We state in detail the reduction of the ordinary product of Fermi fields in terms of a finite sum of monomials in the creation and annihilation operators in which all creation operators occur to the left of all annihilation operators (Wick-ordered) and the Fock (vacuum) state of the former.

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.