GABOR ANALYSIS OF THE SPACES M（p,q,w）（R~d） AND S（p,q,r,w,ω）（R~d）

Let g be a non-zero rapidly decreasing function and w be a weight function. In this article in analog to modulation space, we define the space M(p, q, w)(Rd) to be the subspace of tempered distributions f ∈ S′(Rd) such that the Gabor transform Vg(f) of f is in the weighted Lorentz space L(p, q, wdμ) (R2d). We endow this space with a suitable norm and show that it becomes a Banach space and invariant under time frequence shifts for 1 ≤ p, q ≤∞. We also investigate the embeddings between these spaces and the dual space of M(p, q, w)(Rd). Later we define the space S(p, q, r, w, ω)(Rd) for 1 < p < ∞, 1 ≤ q ≤∞. We endow it with a sum norm and show that it becomes a Banach convolution algebra. We also discuss some properties of S(p, q, r, w, ω)(Rd). At the end of this article, we characterize the multipliers of the spaces M(p, q, w)(Rd) and S(p, q, r, w, ω)(Rd).     

Resonance Problems for Hardy-Sobolev Operator

We consider the resonance problems for Hardy-Sobolev operator L w u := m j p u m w /(| x | p )| u | p m 2 u , 0 h w < (( N m p ) p ) p / at an arbitrary eigen value. Here j p is the usual p -Laplacean. We prove the existence of weak solutions assuming a standard Landesman-Lazer condition.     

Interpolation and best polynomial approximation in the domain ofp-adic integers

, , . , . Lip

---

---

The authors are indebted to Professor R. Bojanic for his valuable remarks and suggestions, especially for the simplification of the proof of Theorem 4.     

A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

---

---

This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

On the Space of Sequences of p-Bounded Variation and Related Matrix Mappings

The difference sequence spaces (), c(), and c ₀() were studied by Kzmaz. The main purpose of the present paper is to introduce the space bv _p consisting of all sequences whose differences are in the space _p , and to fill up the gap in the existing literature. Moreover, it is proved that the space bv _p is the BK-space including the space _p . We also show that the spaces bv _p and _p are linearly isomorphic for 1 p . Furthermore, the basis and the -, -, and -duals of the space bv _p are determined and some inclusion relations are given. The last section of the paper is devoted to theorems on the characterization of the matrix classes (bv _p : ), (bv : _p ), and (bv _p : ₁), and the characterizations of some other matrix classes are obtained by means of a suitable relation.     

Nonparametric Estimation in a Model with a Trend

Let Yt be a stochastic process taking values in R and defined by the model Yt=azt+(Xt)+ t where {zt} is a deterministic sequence, {Xt} is strictly stationary and strongly mixing, and {t} is i.i.d. We study asymptotic properties of nonparametric estimators of density and regression with rates of convergence, and their behavior on estimation when () is polynomial. It is shown that the estimator of the coefficients of () constructed from the nonparametric estimators of regression is consistent when the deterministic {zt} converges in Cesàro mean, and may be inconsistent when {zt} is periodic under some ordinary conditions.     

On weakly neighborly polyhedral maps of arbitrary genus

A weakly neighborly polyhedral map (w.n.p. map) is a 2-dimensional cell-complex which decomposes a closed 2-manifold without boundary, such that for every two vertices there is a 2-cell containing them. We lay the foundation for an investigation of the w.n.p. maps of arbitrary genus. In particular we find all the w.n.p. maps of genus 0, we prove that for everyg> the number of w.n.p. maps of genusg (orientable or not) is finite, and we find an upper bound for the number of vertices in a w.n.p. map of genusg>0. This upper bound grows as (4g(^2/3) wheng→∞.     

Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms

Summary We describe a large class of one-parameter families , {}, , of two-dimensional diffeomorphisms which arestable for <0, exhibit acycle for =0, and thereafter have a bifurcation set of positive but arbitrarily smallrelative measure for in small intervals [0, ]. A main assumption is that the basic sets involved in the cycle havelimit capacities that are not too large.The second author acknowledges hospitality and financial support from IMPA/CNPq during the period this paper was prepared     

Flat points of two-dimensional Brownian motion

Summary SupposeZ(·) is a two-dimensional Brownian motion. It is shown that a.s. there existt ₀ and >0 such thatZ(t ₀) is an extremal point of the convex hull of {Z(t)|t ₀–tt₀} and also an extremal point of the convex hull of {Z(t)|t ₀tt₀+} and, moreover, the tangent lines to the convex hulls atZ(t ₀) form a non-zero angle.The result is related to the following unsolved problem of S.J. Taylor. Do there exist a.s.t ₀ and >0 such that the intersection of the convex hulls of {Z(t)|t ₀–tt₀} and {Z(t)|t ₀tt₀+} contains onlyZ(t ₀)?This research was partially supported by Grant-in-Aid for Scientific Research (No. 400101540202), Ministry of Education, Science and Culture     

Morphisms of the Heat Equation

In this article we consider a certain class of mappings between Riemannian manifolds of the type M × R^{+ *}, which preserve two objects associated to the heat equation. First we show that maps which pull back local solutions of the heat equation to local solutions of the heat equation, called heat equation morphisms have to be the product of a homothetic submersion and an affine map of R^{+ *}. Then using the above characterisation, we study maps which pull back the heat kernel to the heat kernel. We call such maps heat kernel morphisms. We show in Theorem 3 that, in the case of compact manifolds, a map : M × M × R^{+ *} N × N × R^{+ *} of the form (x,y,t) = ( (x), (y),h(t)), with surjective, is a heat kernel morphism if and only if is a homothetic covering of N()-sheets and constant dilation such that N() = ^m (m = dim M) and h(t) = ²t. In particular, if is bijective then it must be an isometry and h(t) = t. A similar problem was considered from a different point of view in [6].     

Sur la répartition des valeurs de la fonction d'Euler

Let (x) denote the number of those integers n with (n) x, where denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that (x)-Ax R(x), where A=(2)(3)/(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.     

The Solution Set to BVP for Some Functional Differential Inclusions

We consider a multivalued BVP x'(t) A(t)x((t))+ F(t,x((t))), Lx= . Under appropriate assumptions on A, L and F, we prove that for sufficiently small the set of solutions to this problem is a nonempty infinite dimensional AR-space (Theorem 4).     

Marcinkiewicz integral on weighted Hardy spaces

In this paper we give sufficient conditions to imply the $H^{1}_{w}-L^{1}_{w}$ boundedness of the Marcinkiewicz integral operator $\mu_\Omega$, where w is a Muckenhoupt weight. We also prove that, under the stronger condition $\Omega \in {\rm Lip}_\alpha$, the operator $\mu_\Omega$ is bounded from $H^{p}_{w}$ to $L^{p}_{w}$ for $\max\{n/(n+1/2), n/(n+\alpha)\}$ < p < 1.     

On Weighted Estimates of Solutions for Nonlinear Higher Order Elliptic Problems

In the domain D = z m F it is considered a degenerate nonlinear higher order elliptic equation such that the corresponding energetic space is W 1, q (D, w q ) 7 W m , p ( D , w p ), mp < q < n , and w q , w p are weighted functions from some Muckenhoupt classes. A behavior of a solution u ( x ) is studied for the equation under consideration with the boundary value condition , where and f ( x ) = 1 for x ] F . The pointwise estimate for u ( x ) is proved in terms of the weighted higher order capacity of the set F and the distance from the point x to the set F .     

The weakly neighborly polyhedral maps on the non-orientable 2-manifold with Euler characteristic −2

A weakly neighborly polyhedral map (w.n.p. map) is a 2-dimensional cell-complex which decomposes a closed 2-manifold without boundary, such that for every two vertices there is a 2-cell containing them. We prove that there are just eight non-orientable w.n.p. maps with Euler characteristic −2 and we describe them.     

Quasi-coherence of Hardy spaces in several complex variables

In this paper, we prove that the Hardy spaceH ^p (), 1p<, over a strictly pseudoconvex domain in ⁿ with smooth boundary is quasi-coherent. More precisely, we show that Toeplitz tuplesT with suitable symbols onH ^p () have property (). This proof is based on a well known exactness result for the tangential Cauchy-Riemann complex.     

Homogene mehrparametrische lineare Programmierung

Zusammenfassung Bei der Ausarbeitung von Verfahren zur Lösung von linearen Programmen mit mehreren Parametern in der rechten Seite oder in der Zielfunktion wurde bisher stillschweigend angenommen, daß der Vektorb0 oderc0 ist. In diesem Artikel wird der Fall erörtert, in dem die ursprüngliche rechte Seiteb=0 oder die ursprünglichen Zielfunktionskoeffizientenc=0 sind. Außerdem wird der Spezialfall behandelt, für den zwarb0, aber jede Komponenteb _i vonb mit jeweils einem Parameter _i 0 multipliziert wird (oder analog fürc _j).

---

Summary In working out methods for solving linear programming problems with a vector-parameter in the right hand side or in the objective function there has been always implicitely assumed thatb0 orc0. In this paper there is solved the multiparametric linear programming problem for the case when the originalb=0 or the originalc=0. Besides there is also treated a specific problem, namely,b0, but each componentb _i ofb is multiplied by a parameter _i 0 (or similarly forC _j).

     

A complete asymptotic expansion of the jacobi functions with error bounds

In this paper we give a complete asymptotic expansion of the Jacobi functions ^{(, )} (t) as + . The method we employed to get the complete expansion follows that of Olver in treating similar problems. By using a Gronwall-Bellman type inequality for an improper integral in which the integrand is an unbounded function and contains a parameter, we get an error bound of the asymptotic approximation which is different from that of Olver's.     

The Cohomology of SL(3, ?[1/2])

We compute the cohomology of SL(3, [1/2]) with coefficients in the prime fields and in the integers. On the way we obtain the cohomology of certain mod-2 congruence subgroups of SL(3, ) with coefficients in p for p>2. Finally we compute the cohomology of GL(3, [1/2]).     

The weakly neighborly polyhedral maps on the 2-manifold with euler characteristic −1

A weakly neighborly polyhedral map (w.n.p. map) is a two-dimensional cell-complex which decomposes a closed 2-manifold without boundary, such that for every two vertices there is a 2-cell containing them. We prove that there are just four w.n.p. maps with Euler characteristic –1 and we describe them.