A NOTE ON ANALYTIC MEASURES

Let G be a compact Abelian group with character group X. Let S be a subset of X such that, for some rcal-valued homomorphism on X, the set S N -1(]_, (X)]) is finite for all in X. Suppose that p is measure in M(G) such that vanishes off of S, then p is absolutely continuous with respect to the Haar meas,I,’c Oil G.

X, and let denote the adjoint homomorphism of .Thus is the continuous homomorphism from R into G such that the identity Xo(r) :exp((x)r) holds for all tin R, and all X in X.We denote by M(G) the linear spacc of all complex-valued regular Borel measures on G.In the terminology of de Leew and Glicksberg [1], a measure p in M(G) is called -analy$c if its Fourier transform vanishes on {X X: (X) < 0 }.Suppose that S is a nonvoid subset of X.Let Ms(G denote the closed linear subspace of M(G) consisting of the measures p with vanishing off of S. The set S will be called a B-set (B for Bochner) if there is a nonzero homomorphism from X into R such that the set S-(]-x),(X)]) is finite for all X in X.The homomorphism may depend on S, and may not be unique.For example, a sector with opening less than r in thc lattice plane 7xl is a B-set.The first orthant in :w (the weak direct product of countably many copies of l) is also a B-set.Once we have chosen a homomorphism , we will refer to S as a B-set with respect to the homomorphism .h theorem due to Bochner [2], on 1 , the two-dimensional torus, asserts that if p M(T) is such that ] vanishes off of a sector of opening less than r, then p is absolutely continuous.(The expression "absolutcly continuous" will always mean absolutely continuous with respect to the Haar measure on the group in consideration.)h generalization of this result is given in de Leew and Glicksberg [1], Theorem (3.4).
It is easy to construct B-sets in lxl that are contained in no sector with opening less than r.For example, consider the set S:{(x,y)lxl: y>log(l+lxl)}.Using results from [1], we will show that the conclusion of Bochner's theorem holds for B-sets.We have the following theorem.
(1.1) THEOREM.Let S be a B-set in X. Suppose that p is in Ms(G), then/ is absolutely continuous.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: