Fourier Stieltjes Research Articles

Using the nonstationary spectral method to analyze asymptotic macrodispersion in uniformly recharged heterogeneous aquifers

This paper describes an investigation of the influence of uniformly distributed groundwater recharge on asymptotic macrodispersion in two-dimensional heterogeneous media. This is performed using a nonstationary spectral approach [Li, S.-G., McLaughlin, D., 1991. A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis. Water Resour. Res. 27 (7), 1589–1605; Li, S.-G., McLaughlin, D., 1995. Using the nonstationary spectral method to analyze flow through heterogeneous trending media. Water Resour. Res. 31 (3), 541–551] based on Fourier–Stieltjes representations for the perturbed quantities. To solve the problem analytically, focus is placed on the case where the local longitudinal dispersivity α L is much smaller than the integral scale of log transmissivity λ (i.e., α L / λ ≪ 1). The closed-form expressions are obtained for describing the spectrum of flow velocity, the variability of flow velocity and asymptotic macrodispersion, in terms of the statistical properties and the integral scale of log transmissivity, local transport parameters and a parameter β [Rubin, Y., Bellin, A., 1994. The effects of recharge on flow nonuniformity and macrodispersion. Water Resour. Res. 30 (4), 939–948] used to characterize the degree of flow nonuniformity due to the groundwater recharge. The impact of β on these results is examined.

The spine of a Fourier-Stieltjes algebra

We define the spine A *(G) of the Fourier–Stieltjes algebra B (G) of a locally compact group G. This algebra encodes information about much of the fine structure of B (G), particularly information about certain homomorphisms and idempotents. We show that A *(G) is graded over a certain semi-lattice, that of non-quotient locally precompact topologies on G. We compute the spine's spectrum G*, which admits a semi-group structure. We discuss homomorphisms from A *(G) to B (H) where H is another locally compact group; and we show that A *(H) contains the image of every completely bounded homomorphism from the Fourier algebra A (H) of any amenable group G. We also show that A *(G) contains all of the idempotents in B (G). Finally, we compute examples for vector groups, abelian lattices, minimally almost periodic groups and the (ax + b)-group; and we explore the complexity of A *(G) for the discrete rational numbers and free groups.

$\varepsilon$-Kronecker and $I_{0}$ sets in abelian groups, I: arithmetic properties of $\varepsilon$-Kronecker sets

A subset $E$ of the locally compact abelian group $\Gamma$ is “$\varepsilon$-Kronecker” if every continuous function from $E$ to the unit circle can be uniformly approximated on $E$ by a character with error less than $\varepsilon$. The set $E\subset \Gamma$ is $I_0$ if every bounded function on $E$ can be interpolated by the Fourier–Stieltjes transform of a discrete measure on the dual group.We show that if $\varepsilon\,{<}\,\sqrt2$ then an $\varepsilon$-Kronecker set is $I_0$, but this is not true for at least one $\sqrt 2$-Kronecker set. $\varepsilon$-Kronecker sets in ${\mathbb Z}$ need not be finite unions of Hadamard sets. As with Sidon sets, $\varepsilon$-Kronecker sets with $\varepsilon\,{<}\,2$ do not contain arbitrarily long arithmetic progressions or large squares. When $\varepsilon\,{<}\,\sqrt 2$ they can contain only a bounded number of pairs with common differences and their step length tends to infinity. Related results and examples are given to show the sharpness of these results.

$\varepsilon$-Kronecker and $I_{0}$ sets in abelian groups, II: sparseness of products of $\varepsilon$-Kronecker sets

A subset $E$ of the locally compact abelian group $\Gamma$ is “$\varepsilon$-Kronecker” if every continuous function from $E$ to the unit circle can be uniformly approximated on $E$ by a character with error less than $\varepsilon$. The set $E\subset \Gamma$ is $I_0$ if every bounded function on $E$ can be interpolated by the Fourier Stieltjes transform of a discrete measure on the dual group.We show that products (sums) of $\varepsilon$-Kronecker sets can be all of the group if the number of terms is sufficiently large, but are shown to be $U_0$ sets (sets of uniqueness in the weak sense) if the number is small. Results about cluster points of products are extended from Hadamard to $\varepsilon$-Kronecker sets. One consequence of that is that finite unions of translates of a fixed $\varepsilon$-Kronecker set are $I_0$.

On the near differentiability property of Banach spaces

Let μ be a scalar measure of bounded variation on a compact metrizable abelian group G. Suppose that μ has the property that for any measure σ whose Fourier–Stieltjes transform σ ˆ vanishes at ∞, the measure μ * σ has Radon–Nikodým derivative with respect to λ, the Haar measure on G. Then L. Pigno and S. Saeki showed that μ itself has Radon–Nikodým derivative. Such property is not shared by vector measures in general. We say that a Banach space X has the near differentiability property if every X-valued measure of bounded variation shares the above property. We prove that Banach spaces with the Radon–Nikodým property have the near differentiability property, while Banach spaces with the near differentiability property enjoy the near Radon–Nikodým property. We also show that the Banach spaces L 1 [ 0 , 1 ] and L 1 / H 0 1 have the near differentiability property. Lastly, we show that Banach spaces with the near differentiability property have type II- Λ-Radon–Nikodým property, whenever Λ is a Riesz subset of type 0 of G ˆ .

Representations of locally compact groups on QSLp-spaces and a p-analog of the Fourier–Stieltjes algebra

For a locally compact group G and p ∈ (1, oo), we define Bp(G) to be the space of all coefficient functions of isometric representations of G on quotients of subspaces of L p spaces. For p = 2, this is the usual Fourier-Stieltjes algebra. We show that Bp(G) is a commutative Banach algebra that contractively (isometrically, if G is amenable) contains the Figa-Talamanca-Herz algebra Ap(G). If 2 ≤ q ≤ p or p ≤ q ≤ 2, we have a contractive inclusion B q (G) ⊂ Bp(G). We also show that Bp(G) embeds contractively into the multiplier algebra of Ap(G) and is a dual space. For amenable G, this multiplier algebra and Bp(G) are isometrically isomorphic.

Banach algebras on compact right topological groups

It is shown how the basic constructs of harmonic analysis, such as convolution, algebras of measures and functions (including Fourier–Stieltjes algebras) can be developed for compact Hausdorff right topological groups. In particular, the properties and structure of these new objects are compared with their classical analogues in the topological group case.

Amenability and weak amenability of the Fourier algebra

Let G be a locally compact group. We show that its Fourier algebra A(G) is amenable if and only if G has an abelian subgroup of finite index, and that its Fourier–Stieltjes algebra B(G) is amenable if and only if G has a compact, abelian subgroup of finite index. We then show that A(G) is weakly amenable if the component of the identity of G is abelian, and we prove some partial results towards the converse.

Completely bounded homomorphisms of the Fourier algebras

For locally compact groups G and H let A ( G ) denote the Fourier algebra of G and B ( H ) the Fourier–Stieltjes algebra of H. Any continuous piecewise affine map α : Y ⊂ H → G (where Y is an element of the open coset ring) induces a completely bounded homomorphism Φ α : A ( G ) → B ( H ) by setting Φ α u = u ∘ α on Y and Φ α u = 0 off of Y. We show that if G is amenable then any completely bounded homomorphism Φ : A ( G ) → B ( H ) is of this form; and this theorem fails if G contains a discrete nonabelian free group. Our result generalises results of Cohen (Amer. J. Math. 82 (1960) 213–226), Host (Bull. Soc. Math. France (1986) 114) and of the first author (J. Funct. Anal. (2004) 213). We also obtain a description of all the idempotents in the Fourier–Stieltjes algebras which are contractive or positive definite.

A characterization of the closed unital ideals of the Fourier–Stieltjes algebra B( G) of a locally compact amenable group G

Let G be a locally compact amenable group, B( G) its Fourier–Stieltjes algebra and I be a closed ideal of it. In this paper we prove the following result: The ideal I has a unit element iff it is principal. This is the noncommutative version of the Glicksberg–Host–Parreau Theorem. The paper also contains an abstract version of this theorem.

Stochastic wave field solution of the 2D elastic wave equation based on the random Fourier–Stieltjes increments

An analytical solution of the stochastic wave equation is presented to model 2D heterogeneous geological environments. In the formulation, a plane-harmonic seismic wave propagates in a medium having random elastic properties in the horizontal and vertical directions. The 2D random field representation is introduced in the stiffness properties of the medium by assuming it has log-normal probability density functions. The constitutive stress and displacement laws with the momentum balance equation for total stress yield a partial differential equation, which is developed using a perturbation approach by assuming a 2D random geological medium having material heterogeneity randomly distributed in the horizontal ( x) and vertical ( z) directions. The method yields a double integral representation of the displacement wave vector based on the Green's tensor and the Fourier–Stieltjes increments. The double integral is reduced to one integral representation by removing the singularities. The final form of the integral is used to construct the stochastic wave field displacement components expressed in terms of a single integral that is appropriate for calculations. This paper also describes a numerical approach that predicts the stochastic wave field used to test the applicability of the theory by simulating a 2D randomly heterogeneous geological medium. Synthetic vertical and horizontal component seismograms based on this random medium indicate a decrease in wave amplitude and wave broadening effects at different depths of the random velocity images. The results suggest that the attenuation and dispersion of waves traveling between two wells are caused by the presence of scatterers observed in the 2D random velocity distribution. Large scatterers produce strong reflections that are observed in the random horizontal wave field seismograms. In general, the random wave field seismograms show characteristic seismic signatures that are associated with the structural distribution of the heterogeneities. In particular, random wave field signatures associated with heterogeneous low velocity zones are observed in the simulations. This kind of signature has been observed in crosswell data recorded at the Gypsy test site in Oklahoma.

Particle Description of Zero-Energy Vacuum I: Virtual Particles

First the “frame problem” is sketched: The motion of an isolated particle obeys a simple law in Galilean frames, but how does the Galilean character of the frame manifest itself at the place of the particle? A description of vacuum as a system of virtual particles will help to answer this question. For future application to such a description, the notion of global particle is defined and studied. To this end, a systematic use of the Fourier transformation on the Poincare group is needed. The state of a system of n free particles is represented by a statistical operator W, which defines an operator-valued measure on ^Pn (^P is the dual of the Poincare group). The inverse Fourier–Stieltjes transform of that measure is called the characteristic function of the system; it is a function on Pn. The main notion is that of global characteristic function: it is the restriction of the characteristic function to the diagonal subgroup of Pn; it represents the state of the system, considered as a single particle. The main properties of characteristic functions, and particularly of global characteristic functions, are studied. A mathematical Appendix defines two functional spaces involved.

A stochastic wavefield solution of the acoustic wave equation based on the random Fourier–Stieltjes increments

In this paper we investigate the problem of compressional wave seismic propagation in random media. This problem is important because almost all geologic media is spatially heterogeneous by nature, consisting of a random agglomerate of many-sized rocks, soil and strata. In our formulation, a plane-harmonic seismic wave propagates in a medium having random material properties in the vertical direction. The random field representation is introduced through the intrinsic rock physical properties of the elastic medium. Each of these intrinsic properties is assumed to have a log-normal probability density function, and the random field representation is expressed in terms of these log-normal probability density functions. The constitutive law, the mass balance, and the moment balance equations are written in the Fourier–Stieltjes representation using random Lamé coefficients and random mass density. The stochastic wave equation is developed by introducing a perturbation approach based on an infinite series expansion of both random coefficients and the displacement solution in terms of σ-parameters (standard deviations of the random material properties). The method yields an integral representation of the displacement wavefield based on the Green's function. This representation is expressed in terms of the random rock physical properties of the medium. The key feature of this paper is that we have expressed the solution as a function of statistical parameters of 1D random medium, including the second order moments. Contrary to most previous derivations, the solutions can also simulate the coda and can be easily extended to simulate waves propagating in 2D and 3D random media. To test the displacement wave solution, synthetic seismograms and dispersion due to scattering effects were calculated for stiffness and density fluctuations of the random medium. This paper is the underlying foundation for the development of the effective propagation vector of acoustic waves in randomly heterogeneous media. This development is presented in a companion paper. In this case, an analytical expression is obtained using a second order perturbation solution. The solution is obtained in terms of the standard deviations of the density and the Young's modulus, respectively, as well as the cross-correlation coefficient and an integral that includes the spectral density and a kernel. In addition, this paper introduces practical expressions for the calculation of the effective attenuation and phase velocity of waves in randomly heterogeneous media. In this companion paper the solution is applied to interpret phase velocity curves that were obtained from interwell acoustic data recorded at Buckhorn test site, Illinois. The objective in this case is to be able to simulate the effect of scattering and intrinsic attenuation associated with acoustic waves in randomly heterogeneous media.

Fourier–Stieltjes Algebras of Locally Compact Groupoids

For locally compact groups, Fourier algebras and Fourier–Stieltjes algebras have proven to be useful dual objects. They encode the representation theory of the group via the positive definite functions on the group: positive definite functions correspond to cyclic representations and span these algebras as linear spaces. They encode information about the algebra of the group in the geometry of the Banach space structure, and the group appears as a topological subspace of the maximal ideal space of the algebra. Because groupoids and their representations appear in studying operator algebras, ergodic theory, geometry, and the representation theory of groups, it would be useful to have a duality theory for them. This paper gives a first step toward extending the theory of Fourier–Stieltjes algebras from groups to groupoids. IfGis a locally compact (second countable) groupoid, we show that B(G), the linear span of the Borel positive definite functions onG, is a Banach algebra when represented as an algebra of completely bounded maps on aC*-algebra associated withG. This necessarily involves identifying equivalent elements of B(G). An example shows that the linear span of the continuous positive definite functions need not be complete. For groups, B(G) is isometric to the Banach space dual ofC*(G). For groupoids, the best analog of that fact is to be found in a representation of B(G) as a Banach space of completely bounded maps from aC*-algebra associated withGto aC*-algebra associated with the equivalence relation induced byG. This paper adds weight to the clues in earlier work of M. E. Walter on Fourier–Stieltjes algebras that there is a much more general kind of duality for Banach algebras waiting to be explored.

Abel–Tauber Theorems for Fourier–Stieltjes Coefficients

We prove Abel–Tauber theorems which link the asymptotics of a function and its Fourier–Stieltjes coefficients under a weak Tauberian condition. Both cosine and sine coefficients are studied. The results in the cosine case can be applied to stationary time series with long-time memory. The analogues for Fourier–Stieltjes transforms are also given.

Asymptotics of Matrix Coefficients and Closures of Fourier–Stieltjes Algebras

The vanishing theorems of R. Howe and C. C. Moore are interpreted in a topological way. This yields a generalization of their results to connected locally compact totally minimal groups. The first section gives the complete classification of these groups and shows a lot of examples. It is proven that, among the connected groups, the totally minimal groups are exactly those groupsGwhose Fourier–Stieltjes algebras are uniformly dense in the C*-algebra of all weakly almost periodic functions WAP(G). This solves a problem which was intensely discussed by C. Chou. Furthermore it is shown that WAP(G) is generated as a C*-algebra by the continuous functions onG/Nwhich vanish at infinity, whereNranges over all closed normal subgroups.

Strictly Positive Definite Functions

We give a complete characterization of the strictly positive definite functions on the real line. By Bochner's theorem, this is equivalent to proving that if the separated sequence of real numbers {an} describes the points of discontinuity of a distribution function, there exists an almost periodic polynomial with the zeros {an}. We prove a useful necessary condition that every strictly normalized, positive definite functionfsatisfies |f(x)|<1 for allx≠0. It is a sufficient condition for strictly positive definiteness that if the carrier of a nonzero finite Borel measure on R is not a discrete set, then the Fourier–Stieltjes transformμofμis strictly positive definite.

Some Geometric Properties on the Fourier and Fourier Stieltjes Algebras of Locally Compact Groups, Arens Regularity and Related Problems

Some Geometric Properties on the Fourier and Fourier Stieltjes Algebras of Locally Compact Groups, Arens Regularity and Related Problems

The C*-Algebra Generated by Operators with Compact Support on a Locally Compact Group

Let G be a locally compact group and VN( G) be the von Neumann algebra generated by the left regular representation of G. Let UCB( Ĝ) denote the C*-subalgebra generated by operators in VN( G) with compact support. When G is abelian, UCB( Ĝ) corresponds to the space of bounded uniformly continuous functions on the dual group Ĝ of G. In this paper we prove among other things that for a large class of locally compact groups which include the Heisenberg group, the " ax + b" group and the motion group, the centre of the Banach algebra UCB( Ĝ)* is the Fourier Stieltjes algebra B( G).

Approximation by fourier stieltjes series

In this paper certain estimates of the rate of convergence of triangular matrix means of the Fourier Stieltjes series and its conjugate series are obtained.