Multiplication product formula for associated homogeneous distributions on R

The set of associated homogeneous distributions (AHDs) on R, ??^′(R), consists of the distributional analogues of power‐log functions with domain in R. This set contains the majority of the (one‐dimensional) distributions one typically encounters in physics applications. The recent work done by the author showed that the set ??^′(R) admits a closed convolution structure (??^′(R), *). By combining this structure with the generalized convolution theorem, a distributional multiplication product was defined, resulting in also a closed multiplication structure (??^′(R), .). In this paper, the general multiplication product formula for this structure is derived. Multiplication of AHDs on R is associative, except for critical triple products. These critical products are shown to be non‐associative in a simple and interesting way. The non‐associativity is necessary and sufficient to circumvent Schwartz's impossibility theorem on the multiplication of distributions. Copyright © 2011 John Wiley & Sons, Ltd.     

The convolution and multiplication of one‐dimensional associated homogeneous distributions

The set of associated homogeneous distributions (AHDs) with support in R is an important subset of the tempered distributions because it contains the majority of the (one‐dimensional) distributions typically encountered in physics applications (including the δ distribution). In a previous work of the author, a convolution and multiplication product for AHDs on R was defined and fully investigated. The aim of this paper is to give an easy introduction to these new distributional products. The constructed algebras are internal to Schwartz’ theory of distributions and, when one restricts to AHDs, provide a simple alternative for any of the larger generalized function algebras, currently used in non‐linear models. Our approach belongs to the same class as certain methods of renormalization, used in quantum field theory, and are known in the distributional literature as multi‐valued methods. Products of AHDs on R, based on this definition, are generally multi‐valued only at critical degrees of homogeneity. Unlike other definitions proposed in this class, the multi‐valuedness of our products is canonical in the sense that it involves at most one arbitrary constant. A selection of results of (one‐dimensional) distributional convolution and multiplication products are given, with some of them justifying certain distributional products used in quantum field theory. Copyright © 2012 John Wiley & Sons, Ltd.     

Structure theorems for associated homogeneous distributions based on the line

Associated homogeneous distributions (AHDs) with support in the line R are the distributional generalizations of one‐dimensional power‐log functions. In this paper, we derive a number of practical structure theorems for AHDs based on R and being complex analytic with respect to their degree of homogeneity in some region of the complex plane. Each theorem gives a representation that is designed to have a distinct advantage for calculating either convolution products, multiplication products, generalized derivatives and primitives, Fourier transforms or Hilbert transforms of AHDs. Copyright © 2008 John Wiley & Sons, Ltd.     

The convolution of associated homogeneous distributions on R–Part II

This is the second in a series of two papers in which we construct a convolution product for the set ?′ (R) of associated homogeneous distributions (AHDs) with support in R. In Part I we showed that if f ^a and g ^b are AHDs with degrees of homogeneity a ? 1 and b ? 1, the convolution f ^a * g ^b exists as an AHD, if the resulting degree of homogeneity a + b?1 ? N. In this article, we develop a functional extension process, based on the Hahn–Banach theorem, to give a meaning to the convolution product of two AHDs of degrees a ? 1 and b ? 1, in the critical case that a + b ? 1 ∈ N. With respect to this construction, the structure (?′(R), *) is shown to be closed.     

Convolution theorems associated with some integral operators and convolutions

In this article, various convolution theorems involving certain weight functions and convolution products are derived. The convolution theorems we obtain are more general, convenient, and efficient than the complicated convolution theorem of the Hartley transform. Further results involving new variants of generalizations of Fourier and Hartley transforms are also discussed.     

Convolution structure associated with the Jacobi-Dunkl operator on IR

In this paper, a product formula for the eigenfunction of the Jacobi-Dunkl differential-difference operator is derived. It leads to a uniformly bounded convolution of point measures and a signed hypergroup on IR. 2000 Mathematics Subject Classification Primary—34K99, 44A15, 44A35, 43A15     

几何随机变量的卷积

使用组合数学与概率论的方法研究了几何随机变量的卷积.     

A product formula for spherical representations of a group of automorphisms of a homogeneous tree, I

Let be the group of automorphisms of a homogeneous tree , and let be a lattice subgroup of . Let be the tensor product of two spherical irreducible unitary representations of . We give an explicit decomposition of the restriction of to . We also describe the spherical component of  explicitly, and this decomposition is interpreted as a multiplication formula for associated orthogonal polynomials.

     

Convolution on distribution spaces characterized by regularization

Locally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space of test functions into a given locally convex lattice of measures. The convolutor spaces are endowed with the topology of uniform convergence on bounded sets. Their locally convex structure is characterized via regularization and function-valued seminorms under mild structural assumptions on the space of measures. Many recent generalizations of classical distribution spaces turn out to be special cases of the general convolutor spaces introduced here. Recent topological characterizations of convolutor spaces via regularization are extended and improved. A valuable property of the convolutor spaces in applications is that convolution of distributions inherits continuity properties from those of bilinear convolution mappings between the locally convex lattices of measures.     

Convolution products of nonidentical distributions on a topological semigroup

In this paper we present a number of sufficient conditions on a sequence of probability measuresµ _n on a locally compact (second countable) Hausdorff topological semigroupS that guarantee the weak convergence of the sequence of convolution productsµ _k,n µ _{k + 1} *···*µ _n (k), asn, for allk0.     

Optimal quantization for dyadic homogeneous Cantor distributions

For a large class of dyadic homogeneous Cantor distributions in ?, which are not necessarily self‐similar, we determine the optimal quantizers, give a characterization for the existence of the quantization dimension, and show the non‐existence of the quantization coefficient. The class contains all self‐similar dyadic Cantor distributions, with contraction factor less than or equal to 1/3. For these distributions we calculate the quantization errors explicitly. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linearization formula for the product of associated q-ultraspherical polynomials

We obtain an explicit formula for the linearization coefficient of the product of two associated q-ultraspherical polynomials in terms of a multiple of a balanced, terminating very-well-poised ₁₀φ₉ series. We also discuss the nonnegativity properties of the coefficients as well as some special cases. 2000 Mathematics Subject Classification Primary—33D45; Secondary—33D8 This work was supported in part by an NSERC grant A6197.     

独立指数分布卷积的矩的计算

推导了∑from i=1 to s(λir)/(π j=1 j≠i to s (λi-λj))求和公式,从而解决了独立指数分布卷积的矩的计算.     

Explicit formula of holomorphic automorphism group on complex homogeneous bounded domains

We known that the maximal connected holomorphic automorphism group Aut (D)(0) is a semi-direct product of the triangle group T(D) and the maximal connected isotropic subgroup Iso(D)(0) of a fixed point in the complex homogeneous bounded domain D and any complex homogeneous bounded domain is holomorphic isomorphic to a normal Siegel domain D(VN,F). In this paper, we give the explicit formula of any holomorphic automorphism in T(D(VN, F)) and Iso(D(VN,F))(0), where G(0) is the unit connected component of the Lie group G.     

Convolution properties for certain subclasses of analytic functions

The authors introduce two new subclasses of analytic functions. The object of the present paper is to investigate some convolution properties of functions in these subclasses     

A product formula for the TASEP on a ring

For a random permutation sampled from the stationary distribution of the TASEP on a ring, we show that, conditioned on the event that the first entries are strictly larger than the last entries, the order of the first entries is independent of the order of the last entries. The proof uses multi‐line queues as defined by Ferrari and Martin, and the theorem has an enumerative combinatorial interpretation in that setting. Finally, we present a conjecture for the case where the small and large entries are not separated. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 247–259, 2016