Orthogonal wavelets on direct products of cyclic groups

We describe a method for constructing compactly supported orthogonal wavelets on a locally compact Abelian group G which is the weak direct product of a countable set of cyclic groups of pth order. For all integers p, n ≥ 2, we establish necessary and sufficient conditions under which the solutions of the corresponding scaling equations with p ⁿ numerical coefficients generate multiresolution analyses in L ²(G). It is noted that the coefficients of these scaling equations can be calculated from the given values of p ⁿ parameters using the discrete Vilenkin-Chrestenson transform. Besides, we obtain conditions under which a compactly supported solution of the scaling equation in L ²(G) is stable and has a linearly independent system of “integer” shifts. We present several examples illustrating these results.     

On the One Sided Translates of Powers of Continuous Functions on Locally Compact Groups

We prove that the one sided translates of powers of a real valued continuous (respectively compactly supported) function with a unique maximum (or minimum) span a dense subspace in C(G) and C(X) (respectively in L ^P (G) and L ^P (X)) for any locally compact group G and for any Riemannian symmetric space X.     

Group structure and the pointwise ergodic theorem for connected amenable groups

Let G be a connected amenable group (thus, an extension of a connected normal solvable subgroup R by a connected compact group $\text{K =}\text{G}\text{R}$). We show how to explicitly construct sequences {U_n} of compacta in G in terms of the structural features of G which have the following property: For any “reasonable” action G × L^p(X, μ) ↓ L^p(X, μ) on an L^p space, 1 <p < ∞, and any f ∈ L^p(X, μ), the averages

$\text{A}{}_{\mathrm{n}}\text{f=}\text{1}\text{|U}{}_{\mathrm{n}}\text{|}\text{∫}\text{U}{}_{\mathrm{n}}\text{T}{}_{\mathrm{g}}{}^{\mathrm{?1f}}\text{dg (|E|=}\text{left Haar measure in}\text{G)}$

converge in L^p norm, and pointwise μ-a.e. on X, to G-invariant functions $\text{f1}$ in L^p(X, μ). A single sequence {U_n} in G works for all L^p actions of G. This result applies to many nonunimodular groups, which are not handled by previous attempts to produce noncommutative generalizations of the pointwise ergodic theorem.     

Abelian groups admitting a Fréchet-Urysohn pseudocompact topological group topology

We show that every Abelian group G with r₀(G)=|G|=|G|^ω admits a pseudocompact Hausdorff topological group topology T such that the space (G,T) is Fréchet-Urysohn. We also show that a bounded torsion Abelian group G of exponent n admits a pseudocompact Hausdorff topological group topology making G a Fréchet-Urysohn space if for every prime divisor p of n and every integer k≥0, the Ulm-Kaplansky invariant f_p,k of G satisfies (f_p,k)^ω=f_p,k provided that f_p,k is infinite and f_p,k>f_p,i for each i>k.Our approach is based on an appropriate dense embedding of a group G into a Σ-product of circle groups or finite cyclic groups.     

Finite range decompositions of positive-definite functions

We give sufficient conditions for a positive-definite function to admit decomposition into a sum of positive-definite functions which are compactly supported within disks of increasing diameters Ln. More generally we consider positive-definite bilinear forms f→v(f,f) defined on . We say v has a finite range decomposition if v can be written as a sum v=∑Gn of positive-definite bilinear forms Gn such that Gn(f,g)=0 when the supports of the test functions f,g are separated by a distance greater or equal to Ln. We prove that such decompositions exist when v is dual to a bilinear form φ→∫²|Bφ| where B is a vector valued partial differential operator satisfying some regularity conditions.     

Group-valued continuous functions with the topology of pointwise convergence

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G^?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.     

Invariant subspaces of some function spaces on a locally compact group

Let G be a σ-compact and locally compact group. If f?L^∞(G) let U_f be the closed subspace of L^∞(G) generated by the left translations of f. Conditions are given which ensure that each function in U_f may be expanded in an essentially unique way as an absolutely convergent series of translations of f. In this case U_f contains subspaces which are isometrically isomorphic to l¹. If G is metrizable and nondiscrete there is a continuum Γ in L^∞(G) such that, for each f?Γ, U_f contains no non-zero continuous function, and for f, g?Γ with f ≠ g, U_f ∩ U_g = {0}. If G is non-compact, metrizable, and non-discrete there is a continuum Γ of bounded continuous functions on G such that, for each f?Γ, U_f contains no non-zero left uniformly continuous function, and for f, g?Γ with f ≠ g, U_f ∩ U_g = {0}. The subspaces U_f above are translation invariant but are not convolution invariant.     

The continuum limit of critical random graphs

We consider the Erd?s–Rényi random graph G(n, p) inside the critical window, that is when p?=?1/n?+?λn ^?4/3, for some fixed ${\lambda \in \mathbb{R}}$ . We prove that the sequence of connected components of G(n, p), considered as metric spaces using the graph distance rescaled by n ^?1/3, converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n ^?1/3 converges in distribution to an absolutely continuous random variable with finite mean.     

Convergence parameter associated with a Markov chain and a family of functions

The proposed definition of convergence parameter R(W) corresponding to a Markov chain X with a measurable state space (E,?) and any nonempty setW of bounded below measurable functions f: E → ? is wider than the well-known definition of convergence parameter R in the sense of Tweedie or Nummelin. Very often, R(W) < ∞, and there exists a set playing the role of the absorbing set inNummelin’s definition ofR. Special attention is paid to the case in whichE is locally compact, X is a Feller chain on E, and W coincides with the family ? ₀ ⁺ of all compactly supported continuous functions f ≥ 0 (f ? 0). In particular, certain conditions for R(? ₀ ⁺ )^?1 to coincide with the norm of an appropriate modification of the chain transition operator are found.     

On Fourier frame of absolutely continuous measures

Let μ be a compactly supported absolutely continuous probability measure on Rn, we show that L²(K,dμ) admits a Fourier frame if and only if its Radon-Nikodym derivative is bounded above and below almost everywhere on the support K. As a consequence, we prove that if μ is an equal weight absolutely continuous self-similar measure on R¹ and L²(K,dμ) admits a Fourier frame, then the density of μ must be a characteristic function of self-similar tile. In particular, this shows for almost everywhere 1/2<λ<1, the L² space of the λ-Bernoulli convolutions cannot admit a Fourier frame.     

Compactly supported positive definite radial functions

We provide criteria for positive definiteness of radial functions with compact support. Based on these criteria we will produce a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest ones arecut-off polynomials, which consist of a single polynomial piece on [0, 1] and vanish on [1, ∞). More precisely, for any given dimensionn and prescribedC ^k smoothness, there is a function inC ^k (? ⁿ ), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean distance. Another example is derived from odd-degreeB-splines.     

On covariant functions and distributions under the action of a compact group

Let G be a compact subgroup of GLn(R) acting linearly on a finite dimensional complex vector space E. B. Malgrange has shown that the space C^∞G(Rn,E) of C^∞ and G-covariant functions is a finite module over the ring C^∞G(Rn) of C^∞ and G-invariant functions. First, we generalize this result for the Schwartz space SG(Rn,E) of G-covariant functions. Secondly, we prove that any G-covariant distribution can be decomposed into a sum of G-invariant distributions multiplied with a fixed family of G-covariant polynomials. This gives a generalization of an Oksak result proved in [4].     

Real Paley-Wiener Theorems for the Hankel Transform

We first characterize the image of the compactly supported smooth even functions under the Hankel transform as a subspace of the Schwartz space. We then describe the space of smooth L^p-functions whose Hankel transform has compact support as a subspace of the space of L^p-functions.     

The rate of approximation of Gaussian radial basis neural networks in continuous function space

There have been many studies on the dense theorem of approximation by radial basis feedforword neural networks, and some approximation problems by Gaussian radial basis feedforward neural networks(GRBFNs)in some special function space have also been investigated. This paper considers the approximation by the GRBFNs in continuous function space. It is proved that the rate of approximation by GRNFNs with n~d neurons to any continuous function f defined on a compact subset K(R~d)can be controlled by ω(f, n~(-1/2)), where ω(f, t)is the modulus of continuity of the function f .     

Density of systems of trigonometric polynomials in L p , 0 < p < 1

Suppose that μ is a nonnegative finite regular Borel measure on an infinite compact abelian group Γ. Let M be a finite subset of the dual group G of Γ. It is shown that if p ∈ (0, 1), then the linear span of GobM is dense in L ^p (μ). The result is extended to L ^p -spaces of operator-valued functions which are p-integrable with respect to certain operator-valued measures. An application to a theorem by Koosis is given.     

Continuity of Operators Intertwining with Convolution Operators

Let G be a locally compact abelian group, let μ be a bounded complex-valued Borel measure on G, and let T_μ be the corresponding convolution operator on L¹(G). Let X be a Banach space and let S be a continuous linear operator on X. Then we show that every linear operator Φ: X→L¹(G) such that ΦS=T_μΦ is continuous if and only if the pair (S,T_μ) has no critical eigenvalue.     

Hardy-type inequalities on planar and spatial open sets

Several Hardy-type inequalities with explicit constants are proved for compactly supported smooth functions on open sets in the Euclidean space ? ⁿ .     

Amenability of the Second Dual of Hypergroup Algebras

The amenability of the Banach algebra L ¹(G), the measure algebra M(G) and their second duals of a locally compact group have been considered by a number of authors. During these investigations it has been shown that L ¹(G)^** is amenable if and only if G is finite. If LUC (G)^*, the dual of the space of left uniformly continuous functions on G, is amenable, then G is compact and M(G) is amenable. Finally, if M(G)^** is amenable, then G is finite. The aim of this paper is to generalize all of the above results to the locally compact hypergroups.     

On the solvability of the jump problem in clifford analysis

Let Ω be a bounded open and oriented connected subset of ? ⁿ which has a compact topological boundary Γ, let C be the Dirac operator in ? ⁿ , and let ?_0,n be the Clifford algebra constructed over the quadratic space ? ⁿ . An ?_0,n -valued smooth function f : Ω → ?_0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F ⁺ ? F ^? = f on Γ, where the components F ^± are extendable to monogenic functions in Ω^± with Ω⁺ := Ω, and Ω^? := ? ⁿ \ (Ω ? Γ), respectively.