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The outputs of linear shift-invariant systems are usually defined in terms of the convolution of input signals with the impulse response functions characterizing the systems. In many areas, as for instance, electrical engineering, digital signal and image processing, statistics, physic, optics, etc., convolution systems defined on finite groups are used. Such systems can be modeled and represented by convolution matrices. The problem is that due to the complexity of systems, dealing with large matrices is required. In this paper, we discuss representation of convolution systems on finite groups by Heterogeneous decision diagrams (HDDs). Such representations permit compact representations of convolution systems, and thanks to that, efficient manipulations and computations related to investigation of features and applications of such systems.

Gaussian convolution is of fundamental importance in linear scale-space theory and in numerous applications. We introduce iterated extended box filtering as an efficient and highly accurate way to compute Gaussian convolution. Extended box filtering approximates a continuous box filter of arbitrary non-integer standard deviation. It provides a much better approximation to Gaussian convolution than conventional iterated box filtering. Moreover, it retains the efficiency benefits of iterated box filtering where the runtime is a linear function of the image size and does not depend on the standard deviation of the Gaussian. In a detailed mathematical analysis, we establish the fundamental properties of our approach and deduce its error bounds. An experimental evaluation shows the advantages of our method over classical implementations of Gaussian convolution in the spatial and the Fourier domain.

This paper discusses relationships among convolution matrices and fixed-polarity matrices for polynomial expressions of discrete functions on finite groups. Switching and multiple-valued functions are considered as particular examples of discrete functions on finite groups. It is shown that if the negative literals for variables are defined in terms of the shift operators on domain groups, then there is a relationship between the polarity matrices and convolution matrices. Therefore, the recursive structure of polarity matrices follows from the recursive structure of convolution matrices. This structure is determined by the assumed decomposition of the domain groups for the considered functions.

We study a subspace of the Fock space, called Boolean Fock space, and its associated non-communicative processes obtained by combinations of annihilators and creators. These processes include the Boolean Brownian and Poisson processes obtained by replacing the classical convolution by its Boolean counterpart, and a family of Bernoulli processes. Using a quantum stochastic calculus constructed by time changes, we complete the existing non-commutative relations between basic probability laws. In particular the uniform distribution has the role played by the exponential law in the classical setting of tensor independence.

Let K be a linear operator mapping a linear set X ⊂ D’(ℝ d ) into D’(ℝ d ). We denote X and K(X) by dom K and im K (respectively). The operator K is called local if does not increase the support, i.e. $$T\; \in \;dom\;k\; \Rightarrow \;\operatorname{supp} \;k\left( T \right)\; \subset \;\operatorname{supp} \;T $$ (Loc) In other words, for any open O ⊂ℝd the distribution K (T)|O depends only on the restriction of T onto the same set O: $$\${T_{1,}}{T_2}\; \in \;dom\;K,\;{T_1}|0 = {T_2}|0 \Rightarrow \;K\left( {{T_1}} \right)|0 = K\left( {{T_2}} \right)|0$$ (1) .

In this part, we present a survey of mean-periodicity phenomena which arise in connection with classical questions in complex analysis, partial differential equations, and more generally, convolution equations. A common feature of the problem we shall consider is the fact that their solutions depend on techniques and ideas from complex analysis. One finds in this way a remarkable and fruitful interplay between mean-periodicity and complex analysis. This is exactly what this part will try to explore.

As the high-dimensional generalization of the boundedness of singular integrals on Lipschitz curves, the \(L^{p}(\Sigma )\)-boundedness of the Cauchy-type integral operators on the Lipschitz surfaces \(\Sigma \) is a meaningful question. The increase of the dimensions means that we need to apply a new method to solve the above question. In 1994, C. Li, A. McIntosh and S. Semmes embedded \(\mathbb {R}^{n+1}\) into Clifford algebra \(\mathbb {R}_{(n)}\) and considered the class of holomorphic functions on the sectors \(S_{w,\pm }\), see [1]. They proved that if the function \(\phi \) belongs to \(K(S_{w,\pm })\), then the singular integral operator \(T_{\phi }\) with the kernel \(\phi \) on Lipschitz surface is bounded on \(L^{p}(\Sigma )\).

We consider a stochastic dynamic system in discrete time with an observable component Xn ∈X⊂R v (signal). The signal is a Markov chain and, at any step, the conditional density of the observation given the signal is fixed in the exponential class. We face the problem of finding transition kernels for Xn for which the system admits filters of dimension v; we reduce this problem to an analytical procedure in which the first step can be solved in terms of convolution semigroups of probability distributions.

S. Ruscheweyh, using Hadamard convolution, introduced new definitions for some classes of regular functions. The definitions are used here to prove several properties of these classes.

Suppose that f(z) is an entire function and μ is a measure in ℂ n with compact support. We define the convolution operator \(\hat \mu \)(f) by $$\hat \mu (f)\, = \,g\,= f \star \mu \, = \,\mathop \smallint \limits_{{\mathbb{C}_n}} \,f(z + w)d\mu(w)$$ (9,1) It is a simple consequence of Cauchy’s Theorem in the polydisc, for instance, that this includes all finite order differential operators with constant coefficients, and if we choose \(\mu \, = \,\sum\limits_{v = 1}^s {{\lambda _v}\delta ({z^{(v)}})} \), δ the Dirac measure, then we obtain the finite difference operator \(\hat \mu (f)\,=\,\sum\limits_{v = 1}^s {{\lambda _v}f(z - {z^{(v)}})}\).

Kloss’ “general principle of convergence”, until now available for first-countable groups, is established for a wider class including e.g. all locally compact commutative groups. The proof combines Csiszar’s method with a Fourier analytic technique.

In the preceding chapter the problem of embedding an infinitely divisible probability measure in a continuous convolution semigroup was discussed in great detail. The next step towards a solution of the central limit theorem is the canonical representation of all continuous convolution semigroups in the sense of a Levy-Khintchine formula. As in the classical theory this formula will enable us in subsequent chapters to characterize special classes of infinitely divisible measures and to illuminate the role of the Gauss measures. A systematic presentation of the theory of canonical representations of continuous convolution semigroups of probability measures on a locally compact group G requires some knowledge of the theory of positive semigroups and their infinitesimal generators or generating functionals.

Let (µt)t>0 be a convolution semigroup on the LCA-group G, and let (P t )t>0 denote the induced Feller semigroup. It is often useful, in the study of (µt)t>0, to replace the potential operator \(N = \smallint _o^\infty {\text{ }}{{\text{P}}_s}ds{\text{ for (}}{{\text{P}}_t}{{\text{)}}_{t >0}}\) with the measure (if it exists) $$\kappa = \int\limits_0^\infty {\mu _t dt.} $$

This chapter is of quite general character: in it we make almost no direct reference to the Littlewood-Paley, Marcinkiewicz, and M. Riesz theorems, which are our main goals. The material developed here is, however, central to the proofs of all three types of result, and can be used to prove a great deal more besides.

Various additive formulae for supports and singular supports of convolutions are studied in terms of the Fourier transform in complex domain. Part of the material presented below was announced in our note [4]. The authors with to thank Professor L. Hormander for his most hepful comments.

In constructing models of certain linear control processes, one encounters the problem of approximation by exponential sums. A function fN(t) is called an exponential sum of order N+1 if it satisfies a linear differential equation of the form $${u^{(N + 1)}}(t)\; + \;{a_0}{u^{(N)}}(t)\; + \ldots + \;{a_N}\;u(t)\,\; = \;\,0$$ (1.1) for some constant, real coefficients a0,a1,a2,... aN. The problem of approximation by exponential sums, then, is to find a differential equation of the form (1.1) such that the solution is close, in some sense, to a given function f(t).

We shall be concerned with the theorem stated below. Its proof depends on the two propositions indicated afterwards.

The L-transformation permits the transformation of the convolution, a complicated integral representation, into a simple algebraic product. This facility can be utilized to produce simple proofs of integral relations which are otherwise difficult to verify.