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L. Klebanov proved the following theorem. Let ξ 1 , ⋯ , ξ n be independent random variables. Consider linear forms L 1 = a 1 ξ 1 + ⋯ + a n ξ n , L 2 = b 1 ξ 1 + ⋯ + b n ξ n , L 3 = c 1 ξ 1 + ⋯ + c n ξ n , L 4 = d 1 ξ 1 + ⋯ + d n ξ n , where the coefficients a j , b j , c j , d j are real numbers. If the random vectors (L 1 , L 2) and (L 3 , L 4) are identically distributed, then all ξ i for which a i d j - b i c j ≠ 0 for all j = 1 , n ¯ are Gaussian random variables. The present article is devoted to an analog of the Klebanov theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers. [ABSTRACT FROM AUTHOR]

We prove the following group analogue of the well-known Heyde theorem on a characterization of the Gaussian distribution on the real line. Let X be a second countable locally compact Abelian group containing no subgroups topologically isomorphic to the 2-dimensional torus. Let G be the subgroup of X generated by all elements of X of order 2 and let α be a topological automorphism of the group X such that Ker (I + α) = { 0 } . Let ξ 1 and ξ 2 be independent random variables with values in X and distributions μ 1 and μ 2 with nonvanishing characteristic functions. If the conditional distribution of the linear form L 2 = ξ 1 + α ξ 2 given L 1 = ξ 1 + ξ 2 is symmetric, then μ j are convolutions of Gaussian distributions on X and distributions supported in G. We also prove that this theorem is false if X is the 2-dimensional torus. [ABSTRACT FROM AUTHOR]

Let X be a locally compact Abelian group with the connected component of zero of dimension 1. Let ξ 1 and ξ 2 be independent random variables with values in X with nonvanishing characteristic functions. We prove that if a topological automorphism α of the group X satisfies the condition Ker (I + α) = { 0 } and the conditional distribution of the linear form L 2 = ξ 1 + α ξ 2 given L 1 = ξ 1 + ξ 2 is symmetric, then the distributions of ξ j are convolutions of Gaussian distributions on X and distributions supported in the subgroup { x ∈ X : 2 x = 0 } . This result can be viewed as a generalization of the well-known Heyde theorem on the characterization of the Gaussian distribution on the real line. [ABSTRACT FROM AUTHOR]

Using an intrinsic approach, we study some properties of random fields which appear as tail fields of regularly varying stationary random fields. The index set is allowed to be a general locally compact Hausdorff Abelian group G . The values are taken in a measurable cone, equipped with a pseudo norm. We first discuss some Palm formulas for the exceedance random measure ξ associated with a stationary (measurable) random field Y = (Y s) s ∈ G . It is important to allow the underlying stationary measure to be σ -finite. Then we proceed to a random field (defined on a probability space) which is spectrally decomposable, in a sense which is motivated by extreme value theory. We characterize mass-stationarity of the exceedance random measure in terms of a suitable version of the classical Mecke equation. We also show that the associated stationary measure is homogeneous, that is a tail measure. We then proceed with establishing and studying the spectral representation of stationary tail measures and with characterizing a moving shift representation. Finally we discuss anchoring maps and the candidate extremal index. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of shift preserving operators acting on shift-invariant spaces in L²(G), where G is a locally compact Abelian group. We generalize some results related to shift-preserving operator and its associated range operator from L²(ℝd) to L²(G). We investigate the matrix structure of range operator R(ξ) on range function J associated to shift-invariant space V, in the case of a locally compact Abelian group G. We also focus on some properties like as normal and unitary operator for range operator on L²(G). We show that shift preserving operator U is invertible if and only if fiber of corresponding range operator R is invertible and investigate the measurability of inverse R-1(ξ) of range operator on L²(G). [ABSTRACT FROM AUTHOR]

In this article, we construct a Riesz multiresolution analysis (MRA) on locally compact Abelian groups (LCA) starting from a suitably given scaling function. Subsequently, we investigate all the conditions under which a scaling function generates a Riesz MRA for . Besides, all the results are braced with illustrative examples. Towards the culmination, several exceptions are discussed regarding the nonexistence of dilative automorphism of . [ABSTRACT FROM AUTHOR]

For a second countable locally compact abelian group G, we study a system of translates generated by f ∈ L 2 (G) . We find some equivalent conditions of this family to have some fundamental frame properties. More precisely, let Γ be a uniform lattice in G (a closed subgroup which is cocompact and discrete) and Γ ∗ be the annihilator of Γ in G ^ . For f ∈ L 2 (G) , the Γ ∗ -periodic function Φ f is defined as Φ f (ξ) = ∑ γ ∈ Γ ∗ | f ^ (ξ + γ) | 2 on Γ ^ (the dual group of Γ ) and some of its properties are investigated. In particular, it is shown that if Φ f is continuous, then the family { f (. + γ) } γ ∈ Γ cannot be a redundant frame. Among other things, it is shown that there is an isometry from L 2 (G) into L 2 (Γ ^) in such a way that the system of translates in L 2 (Γ ^) is transferred to a nice Fourier-like system in L 2 (Γ ^) . Also, the canonical and oblique duals of the frames of translates are investigated. [ABSTRACT FROM AUTHOR]

We introduce the notion of intrinsic semilattice entropy h ̃ in the category ℒ qm of generalized quasimetric semilattices and contractive homomorphisms. By using appropriate categories and functors F : → ℒ qm , we find specific known entropies h ̃ on as intrinsic functorial entropies, that is, as h ̃ = h ̃ ∘ F. These entropies are the intrinsic algebraic entropy, the algebraic and the topological entropies for locally linearly compact vector spaces, the topological entropy for totally disconnected locally compact groups and the algebraic entropy for compactly covered locally compact abelian groups. [ABSTRACT FROM AUTHOR]

We introduce positive operator-valued measure (POVM) generated by the projective unitary representation of a direct product of locally compact Abelian group G with its dual G ^ . The method is based upon the Pontryagin duality allowing to establish an isometrical isomorphism between the space of Hilbert–Schmidt operators in L 2 (G) and the Hilbert space L 2 (G ^ × G) . Any such a measure determines a pair of hybrid (containing classical and quantum parts) quantum channels consisting of the measurement channel and the channel transmitting an initial quantum state to the ensemble of quantum states on the group. It is shown that the second channel can be called a complementary channel to the measurement channel. [ABSTRACT FROM AUTHOR]

In this note we present that the patch counting entropy can be obtained as a limit and investigate which sequences of compact sets are suitable to define this quantity. We furthermore present a geometric definition of patch counting entropy for Delone sets of infinite local complexity and that the patch counting entropy of a Delone set equals the topological entropy of the corresponding Delone dynamical system. We present our results in the context of (non‐compact) locally compact abelian groups that contain Meyer sets. [ABSTRACT FROM AUTHOR]

According to the well-known Heyde theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. In the article we study analogues of this theorem for some locally compact Abelian groups. We consider linear forms of two independent random variables with values in a locally compact Abelian group X, whose characteristic functions do not vanish. Unlike most previous works, we do not impose any restrictions on coefficients of the linear forms. They are arbitrary topological automorphisms of X. [ABSTRACT FROM AUTHOR]

Let be a locally compact Abelian group with dual group , let be a power bounded measure on , and let be a strongly regular matrix. We show that the sequence converges in the -norm for every if and only if is clopen in , where is the Fourier–Stieltjes transform of . If is a probability measure, then is clopen in if and only if the closed subgroup generated by the support of is compact. [ABSTRACT FROM AUTHOR]

The aim of this article is to present a novel method for constructing wavelet frames on locally compact Abelian (LCA) groups by employing the frame multiresolution analysis (FMRA) structure of L 2 (G) . The objective is accomplished in two steps: firstly, we show that there always exist a pair of functions ψ 1 and ψ 2 in the detailed subspace W 0 of an FMRA such that their translates constitute a frame for W 0 , and in general, { D j T λ ψ ℓ : j ∈ Z , λ ∈ Λ , ℓ = 1 , 2 } forms a frame for L 2 (G) . Secondly, we establish sufficient conditions for an FMRA on LCA groups to acknowledge that a single function ψ ∈ W 0 can generate a frame for W 0 . Nevertheless, the underlying procedure is demonstrated via some illustrative examples. [ABSTRACT FROM AUTHOR]

Let G be a locally compact abelian group, ϕ be a topological isomorphism on G, and L be a uniform lattice in G. We provide a development of the L 1 (G / ϕ (L)) function-valued product on L p (G) called (ϕ (L) , p) -bracket product, where 1 < p < ∞ . Among other things, we study ϕ (L) -factorable operators and we prove Riesz representation type theorem for L p (G) . [ABSTRACT FROM AUTHOR]

In this paper, we define the wavelet multiplier and Landau–Pollak–Slepian (L.P.S) operators on the Hilbert space L 2 (G 2 , H) , where G is a locally compact abelian topological group, and H is the quaternion algebra; Also, we will investigate some of their properties. In particular, we show that they are bounded linear operators, as well in Schatten p-class spaces, 1 ≤ p ≤ ∞ , and we determine their trace class. [ABSTRACT FROM AUTHOR]

According to the well-known Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We study analogues of this theorem for some locally compact Abelian groups X containing an element of order 2. We prove that if X contains an element of order 2, this leads to the fact that a wide class of non-Gaussian distributions on X is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. While coefficients of linear forms are topological automorphisms of a group. [ABSTRACT FROM AUTHOR]

Introduction An orthonormal wavelet is a square-integrable function whose translates and dilates form an orthonormal basis for the Hilbert space . That is, given the unitary operators of translation for and dilation , we call an orthonormal wavelet if the set is an orthonormal basis for . This definition was later generalized to higher dimensions and to allow for other dilation and translation sets; let Hilbert space and an n × n expansive matrix A (i.e. a matrix with eigenvalues bigger than 1) with integer entries, then dilation operator is given by and the translation operator is given by for . A finite set is called multiwavelet if the set , is an orthogonal basis for . The concept of a multiresolution analysis, abbreviated as MRA is Central to the theory of wavelets. There is much overlaps between wavelet analysis and Fourier analysis. Indeed, wavelets can be thought of as non-trigonometric Fourier series. Thus, Fourier analysis is used as a tool to investigate properties of wavelets. Another concept is wavelet set. The term wavelet set was coined by Dai and Larson in the late 90s to describe a set W such that , the characteristic function of W, is the Fourier transform of an orthonormal wavelet on . At about the same time as the Dai and Larson paper, Fang and Wang first used the term MSF wavelet (minimally supported frequency wavelet) to describe wavelets whose Fourier transforms are supported on sets of the smallest possible measure. The importance of MSF wavelets as a source of examples and counterexamples has continued throughout wavelet history. A famous example due to Journe first showed that not all wavelets have an associated structure multiresolution analysis (MRA). The discovery of a non-MRA wavelet gave an important push to the development of more general structures such as frame multiresolution analyses (FMRAs) and generalized multiresolution analysis(GMRAs). In this paper we generalize wavelets and wavelet sets on locally compact Abelian group G with uniform lattice. Material and methods In this paper, we investigate wavelet sets on locally compact abelian groups with uniform lattice, where a uniform lattice H in LCA group G is a discrete subgroup of G such that the quotient group G/H is compact. So we review some basic facts from the theory of LCA groups and harmonic analysis. Then we define wavelet sets on these groups and characterize them by using of Fourier transform and multiresolution analysis. Results and discussion We extend theory of wavelet sets on locally compact abelian groups with uniform lattice. This is a generalization of wavelet sets on Euclidean space. We characterize wavelet sets by using of Fourier transform and multiresolution analysis. Also, we define generalized scaling sets and dimension functions on locally compact abelian groups and verify its relations with wavelet sets. Dimension functions for MSF wavelets are described by generalized scaling sets. In the setting of LCA groups, we define translation congruent and show wavelet sets are translation congruent, so we can define a map on G such that it is measurable, measure preserving and bijection. Conclusion The following conclusions were drawn from this research. Wavelet sets on locally compact groups by uniform lattice can be defined. This is a generalization of wavelet sets on Euclidean space. Characterization of wavelet sets on LCA groups can be done in different ways. A method is to use Fourier transform and translation congruent. Another way is to generaliz scaling set and dimension function. As an example, Cantor dyadic group is a non-trivial example that satisfies in the theory of wavelet sets on locally compact groups by uniform lattice. We find wavelet set and generalized scaling set for this group and show related wavelet is MRA wavelet../files/site1/files/63/7.pdf

Let G be a locally compact abelian group with Haar measure, Φ be a Young function and ω be a weight. We investigate the closed convex left invariant subsets of L Φ (G , ω) and apply the results to the study of the multipliers for L Φ (G , ω) . [ABSTRACT FROM AUTHOR]

Let G be a locally compact group. This work aims to study some important properties of the frame of translates for the space L 2 (G). Also, we showed that the frame of translates can be a frame for a proper subspace of L 2 (G) , in the case that G has a contractive automorphism. Furthermore, we characterize the frame properties for such systems and find an expression for the canonical dual frame. [ABSTRACT FROM AUTHOR]

The topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action. This relation holds both for the entropy relative to a net and for the receptive entropy of finitely generated monoid actions.

We study translation preserving operators, that is operators commuting with translations by a closed subgroup of a locally compact abelian group. We show that there is a one-to-one correspondence between these operators and range operators. Furthermore, we obtain a necessary condition for a translation preserving operator to be Hilbert–Schmidt or of finite trace in terms of its range operator. The last result is proved in the case that the subgroup is discrete or compact. [ABSTRACT FROM AUTHOR]

According to the well-knows Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We study analogues of this theorem for some locally compact Abelian groups that contain an element of order 2. While coefficients of linear forms are topological automorphisms of a group. [ABSTRACT FROM AUTHOR]

Let G be a locally compact abelian group and Λ be a closed subgroup of the dual group G ^ . In this paper we investigate modulation preserving operators with respect to Λ and give a characterization of them in terms of modulation-range operators. [ABSTRACT FROM AUTHOR]

A.M. Kagan introduced a class of distributions D m , k in R m and proved that if the joint distribution of m linear forms of n independent random variables belongs to the class D m , m - 1 , then the random variables are Gaussian. A.M. Kagan's theorem implies, in particular, the well-known Skitovich–Darmois theorem, where the Gaussian distribution on the real line is characterized by independence of two linear forms of n independent random variables. In the note we describe a wide class of locally compact Abelian groups where A.M. Kagan's theorem is valid. [ABSTRACT FROM AUTHOR]

Heyde proved that a Gaussian distribution on a real line is characterized by the symmetry of the conditional distribution of one linear form given another. The present article is devoted to an analogue of the Heyde theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers. [ABSTRACT FROM AUTHOR]

According to the classical Skitovich–Darmois theorem the Gaussian measure on the real line is characterized by the independence of two linear forms of n independent random variables. A similar result was proved by Heyde, where the independence of linear forms is replaced by the symmetry of the conditional distribution of one of them given the other. We prove an analogue of Heyde’s theorem for linear forms of two independent random variables taking values in a connected locally compact Abelian group of dimension 1 containing no elements of order 2. Coefficients of the linear forms are topological automorphisms of the group. The proof is based on the description of all solutions of a functional equation in the class of characteristic functions (Fourier transforms) of probability measures. In contrast to the previous investigations we do not impose any restrictions on coefficients of the linear forms. We also investigate the role of elements of order 2 in Heyde’s theorem for the mentioned class of groups. [ABSTRACT FROM AUTHOR]

Let α be a topological automorphism on a locally compact abelian group G, satisfies α (p 2) = p for all p ∈ G . This paper deals with defining Fourier–Wigner, Wigner and Weyl transforms with respect to α , and among other things, it shows that Weyl transform has the same effect as Hilbert–Schmidt operators and the product of two wavelet multipliers. [ABSTRACT FROM AUTHOR]

We present characterizations of democratic property for systems of translates on a general locally compact abelian group, along a lattice in that group.. That way we generalize the results from Hernández et al. (J Approx Theory 171:105–127, 2013) on systems of integer translates. Furthermore, we investigate the possibilities of more operative characterizations for lattices with torsion group structure, mainly through examples and counterexamples. [ABSTRACT FROM AUTHOR]

Let G be a locally compact abelian group and X be a nonempty compact set of G. Given a positive definite kernel K : G × G → C whose real part is continuous on the diagonal of G × G , we provide a necessary and sufficient condition for the linear span of the set x ∈ X ↦ K (x , y) : y ∈ X to be dense in C(X) in the topology of uniform convergence. [ABSTRACT FROM AUTHOR]

In this paper we define a metric on the collection of all translation invariant spaces on a locally compact abelian group, and we study some topological properties of the metric space. [ABSTRACT FROM AUTHOR]

In this paper, we define the wavelet multiplier and Landau–Pollak–Slepian (L.P.S) operators on the Hilbert space L 2 (G) , where G is a locally compact abelian topological group and investigate some of their properties. In particular, we show that they are bounded linear operators, and are in Schatten p-class spaces, 1 ≤ p ≤ ∞ , and we determine their trace class. [ABSTRACT FROM AUTHOR]

Let G be a locally compact abelian group and let M (G) be the measure algebra of G. Assume that μ ∈ M (G) is power bounded, that is, sup n ≥ 0 ⁡ ‖ μ n ‖ 1 < ∞. This paper is concerned mainly with finding necessary and sufficient conditions for strong convergence of iterates of the convolution operators T μ f : = μ ⁎ f in L p (G) (1 ≤ p < ∞) spaces. Some related problems are also discussed. [ABSTRACT FROM AUTHOR]

Abstract Let G be a second countable locally compact abelian group. The aim of this paper is to characterize the left translates on the Heisenberg group H (G) to be frames and Riesz bases in terms of the group Fourier transform. [ABSTRACT FROM AUTHOR]

Let X be a locally compact Abelian group, αj,βj be topological automorphisms of X. Let ξ1,ξ2 be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. It is known that if X contains no subgroup topologically isomorphic to the circle group T, then the independence of the linear forms L1=α1ξ1+α2ξ2 and L2=β1ξ1+β2ξ2 implies that μj are Gaussian distributions. We prove that if X contains no subgroup topologically isomorphic to T2, then the independence of L1 and L2 implies that μj are either Gaussian distributions or convolutions of Gaussian distributions and signed measures supported in a subgroup of X generated by an element of order 2. The proof is based on solving the Skitovich-Darmois functional equation on some locally compact Abelian groups. [ABSTRACT FROM AUTHOR]

Let , , be independent random variables with nonvanishing characteristic functions, and , be real numbers such that for . Let , . By C.R. Rao's theorem the distribution of the random vector determines the distributions of the random variables up to a change of location. We prove an analogue of this theorem for independent random variables with values in a locally compact Abelian group. We also prove an analogue of a similar C.R. Rao's theorem for independent random variables with values in an -adic solenoid. In so doing coefficients of linear forms are continuous endomorphisms of the group. [ABSTRACT FROM AUTHOR]

By Heyde's theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of n independent random variables with the other fixed. When n = 2 we prove analogues of this theorem in the case when independent random variables take values in a locally compact Abelian group X and coefficients of the linear forms are topological automorphisms of X. [ABSTRACT FROM AUTHOR]

We obtain a characterization for probability measures on a locally compact Abelian group X based on linear forms of Q-independent random elements taking values in X generalizing the earlier work of the author in [12]. [ABSTRACT FROM AUTHOR]

Let G be a locally compact abelian group and B be a commutative Banach algebra. Let L1(G,B) be the Banach algebra of B-valued Bochner integrable functions on G. In this paper we provide a complete description of continuous disjointness preserving maps on L1(G,B)-algebras based on a scarcely used tool: the vector-valued Fourier transform. We also present necessary and sufficient conditions for these operators to be compact. [ABSTRACT FROM AUTHOR]

In this paper, we extend to the context of locally compact abelian groups, the notion of Weyl transform. We also define and characterize Weyl multipliers for certain function spaces. A dual space characterization is provided for the space of Weyl multipliers on Lp spaces. Finally we also study the twisted shift-invariant subspaces. [ABSTRACT FROM AUTHOR]

In this article, we provide necessary and sufficient conditions for the orthogonality of two Bessel families when such families have the form of generalized translation invariant (GTI) systems over a second countable locally compact abelian (LCA) group G . The work is motivated by the utility of a recent notion given by Jakobsen and Lemvig on GTI systems in L 2 ( G ) , and the concept of the orthogonality (or strongly disjointness) of a pair of frames studied by Balan, Han, and Larson. Consequently, we deduce similar results for several function systems including the case of TI systems, and GTI systems on compact abelian groups. We apply our results to the Bessel families having wave-packet structure (combination of wavelet as well as Gabor structure), and hence a characterization for pairwise orthogonal wave-packet frame systems over LCA groups is obtained. In addition, we relate the well-established theory from literature with our results by observing several deductions in the context of wavelet and Gabor systems over LCA groups with G = R d , Z d , etc. [ABSTRACT FROM AUTHOR]

Suppose that f is the characteristic function of a probability measure on the real line $$\mathbb R$$ . In this paper, we deal with the following problem posed by N.G. Ushakov: Is it true that f is never determined by its imaginary part $$\mathfrak {I}f$$ ? In other words, is it true that for any characteristic function f there exists a characteristic function g such that $$\mathfrak {I}f\equiv \mathfrak {I}g$$ but $$ f\not \equiv g$$ ? We study this question in the more general case of the characteristic function defined on an arbitrary locally compact abelian group. A characterization of what characteristic functions are uniquely determined by their imaginary parts are given. As a consequence of this characterization, we obtain that several frequently used characteristic functions on the classical locally compact abelian groups are uniquely determined by their imaginary parts. [ABSTRACT FROM AUTHOR]

Let X be a locally compact Abelian group, Y be its character group. Following A. Kagan and G. Székely we introduce a notion of Q-independence for random variables with values in X. We prove group analogues of the Cramér, Kac-Bernstein, Skitovich-Darmois and Heyde theorems for Q-independent random variables with values in X. The proofs of these theorems are reduced to solving some functional equations on the group Y. [ABSTRACT FROM AUTHOR]

Let $pounds$ be the category of all locally compact abelian (LCA) groups. In this paper, the groups $G$ in $pounds$ are determined such that every extension $0to Xto Yto Gto 0$ with divisible, $sigma-$compact $X$ in $pounds$ splits. We also determine the discrete or compactly generated LCA groups $H$ such that every pure extension $0to Hto Yto Xto 0$ splits for each divisible group $X$ in $pounds$.