scholarly journals On random almost periodic trigonometric polynomials and applications to ergodic theory

2006 ◽  
Vol 34 (1) ◽  
pp. 39-79
Author(s):  
Guy Cohen ◽  
Christophe Cuny
Keyword(s):  
Ergodic Theory ◽  
Trigonometric Polynomials ◽  
Almost Periodic

scholarly journals The convolution-induced topology onL∞(G)and linearly dependent translates inL1(G)

1982 ◽  
Vol 5 (1) ◽  
pp. 11-20 ◽  
Author(s):  
G. Crombez ◽  
W. Govaerts
Keyword(s):  
Weak Topology ◽  
Trigonometric Polynomials ◽  
Major Result ◽  
Almost Periodic ◽  
Convolution Operators ◽  
Norm Topology ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Hausdorff Group ◽  
Induced Topology

Given a locally compact Hausdorff groupG, we consider onL∞(G)theτc-topology, i.e. the weak topology under all convolution operators induced by functions inL1(G). As a major result we characterize the trigonometric polynomials on a compact group as those functions inL1(G)whose left translates are contained in a finite-dimensional set. From this, we deduce thatτcis different from thew∗-topology onL∞(G)wheneverGis infinite. As another result, we show thatτccoincides with the norm-topology if and only ifGis discrete. The properties ofτcare then studied further and we pay attention to theτc-almost periodic elements ofL∞(G).

On random almost periodic series and random ergodic theory

2006 ◽  
Vol 26 (03) ◽  
pp. 683 ◽  
Author(s):  
GUY COHEN ◽  
CHRISTOPHE CUNY
Keyword(s):  
Ergodic Theory ◽  
Almost Periodic

scholarly journals The research of some classes of almost periodic at infinity functions

2021 ◽  
Vol 21 (1) ◽  
pp. 4-14
Author(s):  
Irina A. Vysotskaya ◽  
◽  
Irina I. Strukova ◽  
Keyword(s):  
Fourier Coefficients ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Trigonometric Polynomials ◽  
Factor Space ◽  
Almost Periodic ◽  
Bounded Solutions ◽  
Banach Modules ◽  
Definition Of ◽  
Necessary And Sufficient

The article under consideration is devoted to continuous almost periodic at infinity functions defined on the whole real axis and with their values in a complex Banach space. We consider different subspaces of functions vanishing at infinity, not necessarily tending to zero at infinity. We introduce the notions of slowly varying and almost periodic at infinity functions with respect to these subspaces. For almost periodic at infinity functions (with respect to a subspace) we give four different definitions. The first definition (approximating) is based on the approximation theorem. In the classical version, for almost periodic functions, they are represented as uniform closures of trigonometric polynomials. In our case, the Fourier coefficients are slowly varying at infinity functions. The second definition, which is an analogue of G. Bohr’s definition of an almost periodic function, is based on the concept of an ε-period. The third definition meets S. Bochner’s criterion for the almost periodicity of functions. The fourth definition is given in terms of factor space. With the help of the results of the theory of almost periodic vectors in Banach modules those four definitions are proved to be equivalent. In addition, it was proved that the introduced spaces of slowly varying and almost periodic at infinity functions with respect to different subspaces of functions vanishing at infinity coincide with the spaces of ordinary slowly varying and almost periodic at infinity functions, respectively. The feasibility of consideration of these functions is due to the fact that the solutions of some important classes of differential and difference equations are almost periodic at infinity. We consider differential equations whose right-hand side is a function vanishing at infinity and obtain necessary and sufficient conditions for their bounded solutions to be almost periodic at infinity functions. We also study an asymptotic representation of the solutions.

scholarly journals Finite dimensional H-invariant spaces

1997 ◽  
Vol 56 (3) ◽  
pp. 353-361
Author(s):  
K.E. Hare ◽  
J.A. Ward
Keyword(s):  
Compact Group ◽  
Closed Subspace ◽  
Sufficient Conditions ◽  
Trigonometric Polynomials ◽  
Left Translation ◽  
Almost Periodic ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Invariant Spaces

A subset V of M(G) is left H-invariant if it is invariant under left translation by the elements of H, a subset of a locally compact group G. We establish necessary and sufficient conditions on H which ensure that finite dimensional subspaces of M(G) when G is compact, or of L∞(G) when G is locally compact Abelian, which are invariant in this weaker sense, contain only trigonometric polynomials. This generalises known results for finite dimensional G-invariant subspaces. We show that if H is a subgroup of finite index in a compact group G, and the span of the H-translates of μ is a weak*-closed subspace of L∞(G) or M(G) (or is closed in Lp(G)for 1 ≤ p < ∞), then μ is a trigonometric polynomial.We also obtain some results concerning functions that possess the analogous weaker almost periodic condition relative to H.

scholarly journals Distribution of Small Values of Bohr Almost Periodic Functions with Bounded Spectrum

2019 ◽  
pp. 571-578
Author(s):  
Wayne M. Lawton
Keyword(s):  
Periodic Function ◽  
Riemann Hypothesis ◽  
Mahler Measure ◽  
Trigonometric Polynomials ◽  
Almost Periodic Function ◽  
Almost Periodic ◽  
Periodic Functions ◽  
The Mean ◽  
Relationship Of ◽  
The Relationship

For f a nonzero Bohr almost periodic function on R with a bounded spectrum we proved there exist Cf > 0 and integer n > 0 such that for every u > 0 the mean measure of the set f x : jf(x)j < u g is less than Cf u1=n: For trigonometric polynomials with n + 1 frequencies we showed that Cf can be chosen to depend only on n and the modulus of the largest coefficient of f: We showed this bound implies that the Mahler measure M(h); of the lift h of f to a compactification G of R; is positive and discussed the relationship of Mahler measure to the Riemann Hypothesis

Ergodic Theory

1983 ◽  
Author(s):  
Karl E. Petersen
Keyword(s):  
Ergodic Theory

Approximation of Unbounded Functions by Trigonometric Polynomials

2017 ◽  
Vol 13 (4) ◽  
pp. 106-116
Author(s):  
Alaa A. Auad ◽  
◽  
Mousa M. Khrajan
Keyword(s):  
Trigonometric Polynomials

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

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