Generalized Jacobi transform

Let X be a random variable with characteristic function ϕ. In the case where X is integer-valued and n is a positive integer, a formula (in terms of ϕ) for the probability that n divides X is presented. The derivation of this formula is quite simple and uses only the basic properties of expectation and complex numbers. The formula easily generalizes to one for the distribution of X mod n. Computational simplifications and the relation to the inversion formula are also discussed; the latter topic includes a new inversion formula when the range of X is finite.When X may take on a more general distribution, limiting considerations of the previous formulas suggest others for the distribution, density, and moments of the fractional part X — [X]. These are easily derived using basic properties of Fourier series. These formulas also yield an alternative inversion formula for ϕ when the range of X is bounded.Applications to renewal theory and random walks are suggested. A by-product of the approach is a probabilistic method for the evaluation of infinite series.

The Inverse Segal–Bargmann Transform for Compact Lie Groups

This book is a comprehensive study of the Radon transform, which operates on a function by integrating it over hyperplanes. The book begins with an elementary and graphical introduction to the Radon transform, tomography and CT scanners, followed by a rigorous development of the basic properties of the Radon transform. Next the author introduces Grassmann manifolds in the study of the k-plane transform (a version of the Radon transform) which integrates over k-dimensional planes rather than hyperplanes. The remaining chapters are concerned with more advanced topics, such as the attenuated Radon transform and generalized Radon transforms defined by duality of homogeneous spaces and double fibrations. Questions of invertibility and the range of the Radon transform are dealt with and inversion formulas are developed with particular attention to functions on L2 spaces and some discussion of the case of Lp spaces.

The ordinary Weierstrass transformation is extended to a class of generalized functions of n independent variables as follows. A testing function space $\eta _\mu $ is constructed, which is a countably normed space and contains as a member the Weierstrass kernel, $K(x - \tau ,1)$, considered as a function of $\tau $. Then, the generalized Weierstrass transform $F(s)$ of any member of the dual space $\eta '_\mu $ is obtained by applying f to the kernel function : $F(s) = \langle {f(\tau ),K(x - \tau ,1)} \rangle $. Next, a theorem is given concerning the convergence behavior of the one-dimensional inversion formulas of P. G. Rooney for the ordinary transformation. On the basis of this result we are able to extend these inversion formulas to the one-dimensional generalized transformation and then construct an inversion formula for the n-dimensional case. Finally, an application to the heat equation for an n-dimensional medium is given.

Radon transforms on Grassmann manifolds

The fundamental space ζ is defined as the set of entire analytic functions [test functions φ(z)], which are rapidly decreasing on the real axis. The variable z corresponds to the complex energy plane. The conjugate or dual space ζ′ is the set of continuous linear functionals (distributions) on ζ. Among those distributions are the propagators, determined by the poles implied by the equations of motion and the contour of integration implied by the boundary conditions. All propagators can be represented as linear combinations of elementary (one pole) functionals. The algebra of convolution products is also determined. The Fourier transformed space ζ̃ contains test functions φ̃(x). These functions are extra-rapidly decreasing, so that the exponentially increasing solutions of higher-order equations are distributions on ζ̃.

Let [Formula: see text] and [Formula: see text] be two bounded linear operators on a Banach space [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] and [Formula: see text], then [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] have some common spectral properties. Drazin invertibility and polaroidness of these operators are also discussed. Cline’s formula for Drazin inverse in a ring with identity is also studied under the assumption that [Formula: see text] for some positive integer [Formula: see text].

Gel'fand and Shilov in [3; pp. 151-54] have extended the Hilbert transform to generalized functions of the space Φ' and proved the inversion formula where the elements of the testing function space Φ belong to the dual space of a testing function space Ψ equipped with the topology generated by a countable set of norms. They claimed that a locally integrable function f(χ) satisfying the asymptotic order and 0<e<1, is Hilbert transformable according to their theory. As it stands their this claim is incorrect: in this paper we have modified their technique and have extended the Hilbert transform and its inversion formula to the space of tempered distributions on . We have given very simple examples to illustrate the applications of our results in solving some singular differential and integro-differential equations. AMS Subject Classification (1980) Primary 46F12 Secondary 44A15

Inversion formula for the growth function of a cancellative monoid

In this paper, we study an integral transform where the kernel is a solution of the nth differential equation in the complex domain y (n)+λ n y=0, n being an arbitrary positive integer. The case n=2 is reduced to the classical Fourier transform. For the case of a real positive argument, an inversion formula is established.

In this paper, we consider the eigenvalue problem for a tensor of arbitrary even rank. In this connection, we state definitions and theorems related to the tensors of moduli ℂ2p(Ω) and ℝ2p(Ω), where p is an arbitrary natural number and Ω is a domain of the n-dimensional Riemannian space ℝn. We introduce the notions of minor tensors and extended minor tensors of rank (2ps) and order s, the corresponding notions of cofactor tensors and extended cofactor tensors of rank (2ps) and order (N−s), and also the cofactor tensors and extended cofactor tensors of rank 2p(N−s) and order s for rank-(2p) tensor. We present formulas for calculation of these tensors through their components and prove the Laplace theorem on the expansion of the determinant of a rank-(2p) tensor by using the minor and cofactor tensors. We also obtain formulas for the classical invariants of a rank-(2p) tensor through minor and cofactor tensors and through first invariants of degrees of a rank-(2p) tensor and the inverse formulas. A complete orthonormal system of eigentensors for a rank-(2p) tensor is constructed. Canonical representations for the specific strain energy and determining relations are obtained. A classification of anisotropic linear micropolar media with a symmetry center is proposed. Eigenvalues and eigentensors for tensors of elastic moduli for micropolar isotropic and orthotropic materials are calculated.

This paper is devoted to an investigation of a topological ring of analytic functions. Specifically, this ring, denoted by R, is the set of functions analytic on the unit disc with the usual addition and scalar multiplication, the Hadamard product for its ring multiplication, and the compact-open topology. The ring R is identified algebrai cally with a subring RA of the ring of continuous functions on the non-negative integers X. The operations in fT are the usual pointwise operations, ana the structure of R is determined by considering its isomorph iT. In Chapter I we are concerned with the problems of identifying the maximal ideal space of R and describing the maximal ideals intrinsically. We first show, using theorems on general rings of continuous functions, that the maximal ideals are in one-to-one correspondence with the points of the Stone-Cech compactification px of X. We next give an intrinsic description of the maximal ideals, using the properties of the power series expansions for analytic functions. Using this description we strengthen the prev ious theorem appreciably and show that the maximal ideal space with the hul1-kernel topology is homeomorphic to pX. Finally, the Hadamard product is used to give a simple iv characterization of the dual space of the topological linear space of analytic functions on the unit disc. This dual space is isomorphic to the set of functions in R whose radius of convergence exceeds one, which is exactly the intersection of the maximal ideals corresponding to points of pX -X (the dense maximal ideals of R) . In Chapter II we continue the investigation of the maximal ideals by studying the structure of their associated residue class rings. The complex number field 0 is isomorphically embedded in R/M, where M is a maximal ideal of R. If M corresponds to a point of X, then R/M and the isomorph 0* of 0 are identical; whereas, if M corresponds to a point of ^X -X, then R/M is a transcendental extension of 0* having transcendence degree c, the cardinality of the continuum. Moreover, we show, in the second case, that R/M is algebraically closed. Using theorems on transcendental extensions and algebraically closed fields, we show that, in either case, R/M and 0 are isomorphic fields. The two classes of maximal ideals are distinguished by the fact that their residue class rings admit radically different types of complex-valued isomorphisms. In Chapter III we are concerned primarily with the structure of the closed ideals of R. The basic tool used is the rotational completeness theorem for analytic v functions, which we proved using the methods and results of harmonic analysis. We show that the closure of every principal ideal is principal, give a necessary and suffi cient condition that a principal ideal be closed, and show that every closed ideal is a principal ideal generated by an idempotent element of R. Using these theorems we indicate connections with the general theory of dual rings, of which R is an example, and raise several questions for further i investigation in the direction of releasing some of the restrictions with which most of the results so far have been obtained.

The multiplicative formal group X + Y + XY is linearly reductive in characteristic p>0

Central hypergroups have been studied by Ross [21 ] (in the commutative case) and by Hauenschild et al. [11 ]. The first purpose of this paper is to complete and improve some of the results in [11 ]. Secondly, we are concerned with spectral synthesis in central hypergroups. The basic development of harmonic analysis for hypergroups can be found in [3, 7, 16, 22, 23]. In Sect. 1 we prove that the dimension function ~--*d o from the dual space /~ofa central hypergroup Kinto N is continuous and that /~ is a locally compact Hausdorff space. We use this result to define a Plancherel meausre on/~. This measure leads to a simple formulation of the Plancherel theorem and the Inversion formula for central hypergroups [11 ] (and hence also for central groups [8, 9]). Section 2 contains some results on the convolution algebra LI(K). After establishing that Kis polynomially growing we show that L ~ (K) is symmetric. Using Dixmier's functional calculus, we prove that LI(K) is completely regular. Some results on spectral synthesis are given in Sect. 3. As the main result we prove a projection theorem for central hypergroups. Let r be the canonical mapping f rom/s onto 2. Then a closed subset F of 2 is spectral if and only if its inverse image r a (F) is a spectral set. Finally, we show that the L~-kernel ker 0 contains bounded approximate units for all Q of/~.

In this paper we prove that if E is the strict inductive limit of a sequence of Mackey spaces {En} such that for every positive integer n, the topological dual space of En, E′n, provided with the Mackey topology μ(E′n,En), is ultrabornological, then the topological dual space E′ of E, provided with the Mackey topology μ(E′,E), is ultrabornological. We also prove that if E is a strict (LF)-space and G a closed subspace of E′ [μ(E′,E)] such that E′[μ(E′,E)] /G is sequentially complete, then E′[μ(E′,E)]/G is complete.