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In this paper we use Nachbin's holomorphy types to generalize some recent results concerning hypercyclic convolution operators on Fr\'echet spaces of entire functions of bounded type of infinitely many complex variables.

We study the type set of singular measures of fractional type on the Heisenbrg group., arXiv admin note: text overlap with arXiv:1607.01071

Our goal is to find classes of convolution semigroups on Lie groups G that give rise to interesting processes in symmetric spaces G/K. The K–bi–invariant convolution semigroups are a well–studied example. An appealing direction for the next step is to generalise to right K–invariant convolution semigroups, but recent work of Liao has shown that these are in one–to–one correspondence with K–bi–invariant convolution semigroups. We investigate a weaker notion of right K–invariance, but show that this is, in fact, the same as the usual notion. Another possible approach is to use generalised notions of negative definite functions, but this also leads to nothing new. We finally find an interesting class of convolution semigroups that are obtained by making use of the Cartan decomposition of a semisimple Lie group, and the solution of certain stochastic differential equations. Examples suggest that these are well–suited for generating random motion along geodesics in symmetric spaces.

A discrete singular convolution method is presented for computation of the deflection analysis of beams resting on elastic foundation.. In the method of discrete singular convolution partial space derivatives of a function appearing in a differential equation are approximated by means of some kernels. Results are compared with existing solutions available from other analytical and numerical methods. The method presented gives accurate results and is computationally efficient.

We amend two proofs in Rend Circ Mat Palermo 63:301–309 (2014), with a slightly different argument based on the choice of a suitable convolution product.

Financial data such as asset returns, exchange rates, or option prices cannot be modeled effectively by classical distributions such as the Gaussian. These types of data have probability density functions that are thick-tailed and negatively skewed. To account for these features, we propose a new method of generating classes of distribution functions through convolution of smooth and non-smooth characteristic functions where the smoothing parameter is used to control the thickness of the density tails. To illustrate the advantages of using such class of distributions, we consider special cases in which the smooth characteristic functions are of those of the uniform, the normal and the compact supported cosine distributions and the non-smooth is the characteristic function of the Cauchy distribution. As a comparison criterion between distributions, we use the Stiltjes-Hamburger conditions for moments’ existence and show how the proposed distributions outperform the Student and Pearson IV distributions, which are commonly used by financial engineers to model stock returns.

Using explicit constructions of the Weierstrass mock modular form, we offer a closed formula for generating the values of shifted convolution $L$-values for certain elliptic curves that can be computed to arbitrary precision. These identities provide a surprising relation between weight $2$ newforms and shifted convolution $L$-values., Journal version

Sufficient conditions for the existence of at least one solution of a nonlinear integral equation with a general kernel are established. The existence result is proved in $C([0,T],E)$, where $E$ denotes an arbitrary Banach space. We use the Darbo-Sadovskii fixed point theorem and techniques of measure of noncompactness. We extend and generalize results obtained by other authors in the context of fractional differential equations. One example illustrates the theoretical results.

We develop a new type of convolution between two signals on the 2-sphere. This is the first type of convolution on the 2-sphere which is commutative. Two other advantages, in comparison with existing definitions in the literature, are that 1) the new convolution admits anisotropic filters and signals and 2) the domain of the output remains on the sphere. Therefore, the new convolution well emulates the conventional Euclidean convolution. In addition to providing the new definition of convolution and discussing its properties, we provide the spectral analysis of the convolution output. This convolutional framework can be useful in filtering applications for signals defined on the 2-sphere.

We study the asymptotic behavior near the boundary of u ( x, y ) = K y * μ ( x ), defined on the half-space + × N by the convolution of an approximate identity K y (·) ( y > 0) and a measure μ on N . The Poisson and the heat kernel are unified as special cases in our setting. We are mainly interested in the relationship between the rate of growth at boundary of u and the s -density of a singular measure μ. Then a boundary limit theorem of Fatou's type for singular measures is proved. Meanwhile, the asymptotic behavior of a quotient of K μ and Kν is also studied, then the corresponding Fatou-Doob's boundary relative limit is obtained. In particular, some results about the singular boundary behavior of harmonic and heat functions can be deduced simultaneously from ours. At the end, an application in fractal geometry is given.

It is already known that the convolution of a bounded density with itself can be estimated at the root- n rate using the two asymptotically equivalent kernel estimators: (i) Frees estimator ( Frees, 1994 ) and (ii) Saavedra and Cao estimator ( Saavedra and Cao, 2000 ). In this work, we investigate the efficiency of these estimators of the convolution of a bounded density. The efficiency criterion used in this work is that of a least dispersed regular estimator described in Begun et al. (1983) . This concept is based on the Hajek–Le Cam convolution theorem for locally asymptotically normal (LAN) families.

X. L. Shi and W. Wang [12] studied the question on determination of jumps for functions via conjugate convolution operators. In the present paper a different way to calculate jumps is discussed. We show that derivative convolution operators can be used to determine jumps for functions. Furthurmore, we point out that the assumption “ϕ is even” in Shi–Wang’s theorem can be removed.

We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution \mu one can define convolution powers with respect to free additive convolution and with respect to Boolean convolution, where the exponent considered in the power is a suitably chosen linear map \eta from B to B, instead of being a non-negative real number. More precisely, the Boolean convolution power is defined whenever \eta is completely positive, while the free additive convolution power is defined whenever \eta - 1 is completely positive (where 1 stands for the identity map on B). In connection to these convolution powers we define an evolution semigroup related to the Boolean Bercovici-Pata bijection. We prove several properties of this semigroup, including its connection to the B-valued free Brownian motion. We also obtain two results on the operator-valued analytic function theory related to the free additive convolution powers with exponent \eta. One of the results concerns analytic subordination for B-valued Cauchy-Stieltjes transforms. The other gives a B-valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a B-valued free Brownian motion., Comment: 33 pages, no figures

In this paper, we solve a long-standing problem on Bernoulli convolutions. In particular, we show that the Bernoulli convolution μ ρ with contraction rate ρ ∈ ( 0 , 1 ) admits a spectrum if and only if ρ is the reciprocal of an even integer.

The linear canonical transform (LCT), which is a generalization of the Fourier transform (FT), has many applications in several areas, including signal processing and optics. Many properties for this transform are already known, but an extension of convolution theorem of FT is still not having a widely accepted closed form expression. In the literature of recent past different authors have tried to formulate convolution theorem for LCT, but none have received acclamation because their definition do not generalize very nicely the classical result for the FT. Moreover, those definitions exhibit only partial invariance properties which prevent their actual use in many applications of signal processing. The purpose of this paper is to introduce a new convolution structure for the LCT that preserves the translation invariance property. Indeed, an effective translation invariance is obtained by slightly modifying the former definitions and by introducing linear canonical translation operators.

Dual integral equations involving convolution transforms, investigated by Y. Tanno, are extended to Zemanian's generalized function space L′ c, d by interpreting convergence in the weak distributional sense. For this purpose, certain properties of the classical convolution transform are also obtained under appropriate conditions on the sequences defining the convolution kernel.

The paper deals with the notions of weak stability and weak generalized convolution with respect to a generalized convolution, introduced by Kucharczak and Urbanik. We study properties of such objects and give examples of weakly stable measures with respect to the Kendall convolution. Moreover, we show that in the context of non-commutative probability, two operations: the q-convolution and the (q,1)-convolution satisfy the Urbanik’s conditions for a generalized convolution, interpreted on the set of moment sequences. The weak stability reveals the relation between two operations.

The fractional Fourier transform FRFT-a generalization of the well-known Fourier transform FT-is a comparatively new and powerful mathematical tool for signal processing. Many results in Fourier analysis have currently been extended to the FRFT, including the ordinary convolution theorem. However, the extension of the ordinary convolution theorem associated with the FRFT has been developed differently and is still not having a widely accepted closed-form expression. In this paper, a generalized convolution theorem for the FRFT is proposed, and the dual of it is also presented. The ordinary convolution theorem and some of its existing extensions related to the FRFT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented. Copyright © 2012 John Wiley & Sons, Ltd.

The linear canonical transform (LCT), which is a generalized form of the classical Fourier transform (FT), the fractional Fourier transform (FRFT), and other transforms, has been shown to be a powerful tool in optics and signal processing. Many results of this transform are already known, including its convolution theorem. However, the formulation of the convolution theorem for the LCT has been developed differently and is still not having a widely accepted closed-form expression. In this paper, we first propose a generalized convolution theorem for the LCT and then derive a corresponding product theorem associated with the LCT. The ordinary convolution theorem for the FT, the fractional convolution theorem for the FRFT, and some existing convolution theorems for the LCT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented.

The offset linear canonical transform (OLCT), which is a time-shifted and frequency-modulated version of the linear canonical transform, has been shown to be a powerful tool for signal processing and optics. However, some basic results for this transform, such as convolution and correlation theorems, remain unknown. In this paper, based on a new convolution operation, we formulate convolution and correlation theorems for the OLCT. Moreover, we use the convolution theorem to investigate the sampling theorem for the band-limited signal in the OLCT domain. The formulas of uniform sampling and low-pass reconstruction related to the OLCT are obtained. We also discuss the design method of the multiplicative filter in the OLCT domain. Based on the model of the multiplicative filter in the OLCT domain, a practical method to achieve multiplicative filtering through convolution in the time domain is proposed.

We establish a solvability criterion for convolution equations in the Beurling classes of ultradifferentiable functions of mean type on an interval. Under study is also the question of degeneration of convolution equations into infinite-order equations with constant coefficients.

We suggest a constructive method for solving a nonsingular convolution equation with two kernels whose kernel functions are integrable on the entire line. The right-hand side is assumed to be integrable with power p ⩾ 1.

The author's weighted inequality for two functions is used to estimate a natural connection between solutions and kernels of the first-kind convolution Volterra equations and related convolution integral equations. The solution–kernel estimates and some functionally parameterized solutions lead to various integro-differential, hypergeometric, convolution, exponential, and other inequalities. Also, the obtained estimates allow us to establish certain non-trivial properties of the unknown and complicated solutions. Several examples and applications are discussed.

We prove that a semigroup generated by finitely many truncated convolution operators on Lp[0, 1] with 1 ≤ p < ∞ is non-supercyclic. On the other hand, there is a truncated convolution operator, which possesses irregular vectors.

Conditions are discovered which ensure that the space of entire functions can be represented as the sum of an ideal in the space of entire functions and the kernel of a convolution operator. In this way conditions for the multipoint de la Vallee-Poussin problem to have a solution are found. Bibliography: 14 titles.

A characterization of the convolution of Gaussian and Poisson laws in the set of infinitely divisible distributions is provided.

We study a mean value of the shifted convolution problem over the Hecke eigenvalues of a fixed non-holomorphic cusp form. We attain a result also for a weighted case. Furthermore, we point out that the proof yields analogous upper bounds for the shifted convolution problem over the Fourier coefficients of a fixed holomorphic cusp form in mean., Comment: 21 pages. arXiv admin note: text overlap with arXiv:1110.3950 .

In this paper, numerical algorithms based on fast convolution for the fractional integral and fractional derivative are proposed. Two examples are also included which show the efficiency of the derived method.

Let M=(M)n∈ℕ be a discrete-time normal martingale satisfying some mild requirements. In this paper we show that through the full Wiener integral introduced by Wang et al. (‘An alternative approach to Privault’s discrete-time chaotic calculus’, J. Math. Anal. Appl.373 (2011), 643–654), one can define a multiplication-type operation on square integrable functionals of M, which we call the convolution. We examine algebraic and analytical properties of the convolution and, in particular, we prove that the convolution can be used to represent a certain family of conditional expectation operators associated with M. We also present an example of a discrete-time normal martingale to show that the corresponding convolution has an integral representation.

We derive some characteristic properties of the convolution operator acting on white noise functions and prove that the convolution product of white noise distributions coincides with their Wick product. Moreover, we show that the S-transform and the Laplace transform coincide on Fock space.

This paper describes an application of Rota and collaborator’s ideas, about the foundation on combinatorial theory, to the computing of solutions of some linear functional partial differential equations. We give a dynamical interpretation of the convolution families of polynomials. Concretely, we interpret them as entries in the matrix representation of the exponentials of certain contractive linear operators in the ring of formal power series. This is the starting point to get symbolic solutions for some functional–partial differential equations. We introduce the bivariate convolution product of convolution families to obtain symbolic solutions for natural extensions of functional-evolution equations related to delta-operators. We put some examples to show how these symbolic methods allow us to get closed formulas for solutions of genuine partial differential equations. We create an adequate framework to base theoretically some of the performed constructions and to get some existence and uniqueness results.

Motivated by the convolution product of white noise functionals, we introduce a new notion of convolution products of white noise operators. Then we study several interesting relations between the convolution products and the quantum generalized Fourier-Mehler transforms, and study a quantum-classical correspondence.

The aim of this paper is to obtain a solution of a certain multidimensional convolution integral equation of Fredholm type whose kernel involves a generalized polyno- mial set. A number of results follow as special cases from the main theorem by specifying the parameters of the generalized polynomial set.

We consider the problem of azimuthally symmetric convolution of signals defined on the 2-Sphere. Applications of such convolution include but are not limited to: geodesy, astronomical data (such as the famous Wilkinson Microwave Anisotropy Probe data), and 3D beamforming/sensing. We review various definitions of convolution from the literature and show a nontrivial equivalence between different definitions. Some convolution formulations based on SO(3) are shown not to be well formed for applications and we demonstrate a simpler framework to understand, use and generalize azimuthally symmetric convolution.

In this paper, we drive several interesting subordination results of analytic functions defined by convolution.

We specify the smallest mode of the ordinary multinomials leading to the expression of the maximal probability of convolution powers of the discrete uniform distribution. The generating function for an extension of the maximal probability is given.

This article contains results of Watson convolution and translation. For this, a space of multipliers on the space T(λ, μ) is developed. A continuity result for Watson translation is obtained. Also, a structure theorem is developed to study the continuity result for convolution of two distributions.

We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a sub-family of Askey–Wimp–Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case. At the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution, the others being the Cauchy distribution and the free 1/2-stable distribution.

In this paper we characterize those bounded linear transformations Tf carrying L1(ℝ1) into the space of bounded continuous functions on ℝ1, for which the convolution identity T(f * g) = Tf · Tg holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.

Various functional spaces take place in Resurgence theory : multiplicative spaces of formal series expansions that one would like to sum; convolutive spaces of analytic functions, the elements of which coming from the former ones by formal Borel transformations; multiplicative spaces of analytic functions deduced from the previous ones by Laplace transformations, thus giving the Borel-Laplace transforms whose asymptotics give back the formal objects one started with. This thesis is devoted to the construction of convolution algebras. Our aim is to provide an original and self-contained proof of the stability under convolution products of the space of endlessly continuable functions, in a way understandable by any youngsearcher in the field. The second part of the thesis concentrates on the convolution space of endlessly continuable functions with simple singularities. We show how the use of the alien derivations bring deep knowlege on the singular structure. We end our work with some problems, still open according to us but of great importance in practice and for which we think that our methods could be applied as well.

We introduce an averaging framework, where the solution of a time-varying equation with a small amplitude is approximated by the solution of a slowly-varying auxiliary system, generated by convolving the original equation with a kernel function. The effect of the convolution is smoothing of the equation, thus, making it more amenable to numerical computations. We present tight results on the approximation error for general classes of vector fields and kernels.

In this paper we determine the limiting distributional behavior for sums of infinitesimal conditionally free random variables. We show that the weak convergence of classical convolution and that of conditionally free convolution are equivalent for measures in an infinitesimal triangular array, where the measures may have unbounded support. Moreover, we use these limit theorems to study the conditionally free infinite divisibility. These results are obtained by complex analytic methods without reference to the combinatorics of c-free convolution. Received by the editors May 14, 2008; revised March 2, 2009. Published electronically September 30, 2010. The author was supported in part by a Coleman Postdoctoral Fellowship at Queen’s University. AMS subject classification: 46L53, 60F05.

An error analysis is given for convolution quadratures based on strongly A-stable Runge–Kutta methods, for the non-sectorial case of a convolution kernel with a Laplace transform that is polynomially bounded in a half-plane. The order of approximation depends on the classical order and stage order of the Runge–Kutta method and on the growth exponent of the Laplace transform. Numerical experiments with convolution quadratures based on the Radau IIA methods are given on an example of a time-domain boundary integral operator.

We study the distributioneαx(♢+m2)kδform≥0, where(♢+m2)kis the diamond Klein-Gordon operator iteratedktimes,δis the Dirac delta distribution,x=(x1,x2,…,xn)is a variable inℝn, andα=(α1,α2,…,αn)is a constant. In particular, we study the application ofeαx(♢+m2)kδfor solving the solution of some convolution equation. We find that the types of solution of such convolution equation, such as the ordinary function and the singular distribution, depend on the relationship betweenkandM.

We consider a Banach algebra generated by multiplicative discrete convolution operators. We construct a symbolic calculus for this algebra and in terms of this calculus we describe criteria for the Noetherian property of operators and obtain a formula for their index.