Let g be a continuous, compactly supported function on R such that the integer translates of g constitute a partition of unity. We show that the Gabor system (g, a, b), with window g and time-shift and frequency-shift parameters a, b > 0 has no lower frame bound larger than 0 if b = 2, 3, . . . and a > 0. In particular, (g, a, b) is not a Gabor frame if g is a continuous, compactly supported wa...

Let $G:\mathbb R\rightarrow \mathbb R$ be a continuous function. Under some assumptions on $G$, $s,\alpha ,p$ and $q$ we prove that $$\{G(f):f\in A_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha })\}\subset })$$ implie

Throughout the paper L1[a,b] denotes the set of integrable functions on [a,b], Lloc(a,b] the set of functions x : (a,b]→ R which are integrable on [a− ε,b] for arbitrary small ε > 0, AC[a,b] is the set of absolutely continuous function on [a,b], and ACloc(a,b] is the set of functions x : (a,b]→ R which are absolutely continuous on [a− ε,b] for arbitrary small ε > 0. Let T be a positive number. ...

ing the above, for a (not necessarily countable) family . . . φ2 // B1 φ1 // Bo of Banach spaces with continuous linear transition maps as indicated, not recessarily requiring the continuous linear maps to be injective (or surjective), a (projective) limit limiBi is a topological vector space with continuous linear maps limiBi → Bj such that, for every compatible family of continuous linear map...

In this article we originate a new class of Grill Set, namely GSβ-Open which is parallel to the β Open Set in Topological Space (X, θ, G). addition, entitle GSβ-continuous and GSβ-open functions by applying review some its important properties. Many examples are given explain concept lucidly. The properties GSβ open sets investigated studied. theorems based on arbitrary union finite intersectio...

Given two metric spaces S1, S2 and a measurable function f : S1 → S2, sup­ pose S1 is equipped with some probability measure P. This induces a proba­ bility measure on S2 which is denoted by Pf−1 and is defined by Pf−1(A) = P(f−1(A)) for every measurable set A ⊂ S2. Then for any random variable X : S2 → R, its expectation EPf−1 [X] is equal to EP[X(f)]. (Convince your­ self that this is the cas...

Let Z be a Banach lattice endowed with positive cone C and an order-continuous norm j.j . Let G be a left-amenable semigroup of positive linear endomorphisms of Z . Then the positive fixed points Co of Z under G form a lattice cone, and their linear span Z0 is a Banach lattice under an order-continuous norm ||.||0 which agrees with |.| on Co. A counterexample shows that under the given conditio...

where p > , φp(x) = |x|p–x for x =  and φp() = , c ∈ Cn(R,R) and c(t + T) ≡ c(t), f is a continuous function defined in R and periodic in t with f (t, ·) = f (t + T , ·) and f (t, ) = , g(t,x) = g(x) + g(t,x), where g : R × (, +∞) → R is an L-Carathéodory function, g(t, ·) = g(t + T , ·), g ∈ C((,∞);R) has a singularity at x = , e : R→ R is a continuous periodic function with ...

Contents 1. Introduction 597 2. Braid groups in positive genus 601 3. Relative completion of mapping class groups 603 4. Mixed Hodge structures on Torelli groups 608 5. Review of continuous cohomology 613 6. Remarks on the representations of sp g 616 7. Continuous cohomology of Torelli groups 618 8. The lower central series quotients of a surface group 621 9. The action of t 1 g on p g 623 10. ...

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K compact, subgroup. We use decay properties of the spherical functions to show that convolution product any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ hence absolutely measure with respect Haar measure. The number r approximately rank . For special case meas...