Classification and geometric aspects of vector valued Fourier transforms

It is shown that for any locally compact abelian group ?? and 1 ≤ p ≤ 2, the Fourier type p norm with respect to ?? of a bounded linear operator T between Banach spaces, denoted by ‖T |???^??_p‖, satisfies ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the direct product of ?₂, ?₃, ?₄, … It is also shown that if ?? is not of bounded order then Cⁿ_p ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the circle group, n is a onnegative integer and C^p = . From these inequalities, for any locally compact abelian group ?? ‖T |???^??₂‖ ≤ ‖T |???^??₂‖, and moreover if ?? is not of bounded order then ‖T |???^??₂‖ = ‖T |???^??₂‖. The Hilbertian property and B‐convexity are discussed in the framework of Fourier type p norms. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Note on Classes of BANACH Spaces Related to Stable Measures

The problem under consideration is the following: Let S: E′ → L_q, T: E′ → L_p, 0 < q ≦ 2, 0 < p ≦ 2, be operators, ‖Sa‖ ≦ ‖Ta‖, such that, T generates a stable measure on E, i.e., exp (-‖Ta‖ ^p), a ? E′, is the characteristic function of a RADON measure on E. Does this imply, that exp (-‖Sa‖ q), a ? E′, is the characteristic function of a RADON measure, too? In general this is not true provided q or p less than 2. A BANACH space is said to be of (q,p)-cotype if the answer to the above question is “yes”. We establish several properties of this classification and obtain as an application the well-known classes due to MOUCHTARI, TIEN, WERON and MANDREKAR, WERON, Finally we apply our results to so-called S-spaces.     

A connection between Schur multiplication and Fourier interpolation. II

Given m × n matrices A = [a_jk ] and B = [b_jk ], their Schur product is the m × n matrix A ○ B = [a_jkb_jk ]. For any matrix T, define ‖T‖ _S = max_{X ≠O} ‖T ○ X ‖/‖X ‖ (where ‖·‖ denotes the usual matrix norm). For any complex (2n – 1)‐tuple μ = (μ _{–n +1}, μ _{–n +2}, …, μ _{n –1}), let T_μ be the Hankel matrix [μ –_{n +j +k –1}]_j,k and define ??_μ = {f ∈ L ₁[–π, π] : f? (2j ) = μ_j for –n + 1 ≤ j ≤ n – 1} . It is known that ‖T_μ‖ _S ≤ infequation/tex2gif-inf-18.gif ‖f ‖₁. When equality holds, we say T_μ is distinguished. Suppose now that μ _j ∈ ? for all j and hence that T_μ is hermitian. Then there is a real n × n hermitian unitary X and a real unit vector y such that 〈(T_μ ○ X )y, y 〉 = ‖T_μ ‖_S . We call such a pair a norming pair for T_μ . In this paper, we study norming pairs for real Hankel matrices. Specifically, we characterize the pairs that norm some distinguished Schur multiplier T_μ . We do this by giving necessary and suf.cient conditions for (X, y ) to be a norming pair in the n ‐dimensional case. We then consider the 2‐ and 3‐dimensional cases and obtain further results. These include a new and simpler proof that all real 2 × 2 Hankel matrices are distinguished, and the identi.cation of new classes of 3 × 3 distinguished matrices. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On measure-preserving functions over ?3

This paper is devoted to (discrete) p-adic dynamical systems, an important domain of algebraic and arithmetic dynamics [31]?C[41], [5]?C[8]. In this note we study properties of measurepreserving dynamical systems in the case p = 3. This case differs crucially from the case p = 2. The latter was studied in the very detail in [43]. We state results on all compatible functions which preserve measure on the space of 3-adic integers, using previous work of A. Khrennikov and author of present paper, see [24]. To illustrate one of the obtained theorems we describe conditions for the 3-adic generalized polynomial to be measure-preserving on ?₃. The generalized polynomials with integral coefficients were studied in [17, 33] and represent an important class of T-functions. In turn, it is well known that T-functions are well-used to create secure and efficient stream ciphers, pseudorandom number generators.     

NILPOTENT GROUPS AND ABELIANIZATION

We study the question of what properties of nilpotent groups are shared by their abelianizations. We identify two such properties—that of being a π-torsion group, where π is a family of primes, and that of having q^th roots, for some prime q. We use these properties to provide simplified proofs of the following theorems in the localization of nilpotent groups.

Let H, K be subgroups of the nilpotent group N and let P be a family of primes. Then [H, K] P = [H_P, K_p]

Let the group G act on the nilpotent group N. Then G acts compatibly on N_p and (Γ^G _i N)_P = Γ^G _i(N_p).

The second theorem above is then applied to the study of the localization of relative groups, in the sense of [4].     

On the stability of the J transformation

The stability of a large class of nonlinear sequence transformations is analyzed. Considered are variants of the J transformation [17]. Suitable variants of this transformation belong to the most successful extrapolation algorithms that are known [20]. Similar to recent results of Sidi, it is proved that the _p {J} transformations, the Weniger S transformation, the Levin transformation and a special case of the generalized Richardson extrapolation process of Sidi are S-stable. An efficient algorithm for the calculation of stability indices is presented. A numerical example demonstrates the validity of the approach. This revised version was published online in June 2006 with corrections to the Cover Date.     

Divergenzverhalten mehrdimensionaler Shannonscher Abtastreihen

The behaviour of multidimensional Shannon sampling series for continuous functions is examined. A continuous functiong ₁ εC ₀[0,1]² with support in the rectangle [0,1]×[0,1/2] is indicated in the paper for which the two dimensional Shannon sampling series diverge almost everywhere in the rectangle [0,1]×[1/2,1]. This shows that the localization principle for Shannon sampling series cannot hold in two dimensions and in higher dimensions. The result solves a problem formulated by P.L. Butzer.     

On subsequences of the Haar system inL p [0, 1], (1<p<∞)

We show that ifX is the closed linear span inL _p [0,1] of a subsequence of the Haar system, thenX is isomorphic either tol _p or toL _p [0,1], [1<p<∞]. We give criteria to determine which of these cases holds; for a given subsequence, this is independent ofp. This is part of the second author's Ph.D. dissertation, written at the University of Alberta under the supervision of J. L. B. Galmen. The first author's research was partially supported by NRC A7552.     

PROBABILISTIC OSTROWSKI TYPE INEQUALITIES

New very general univariate and multivariate probabilistic Ostrowski type inequalities are established, involving ‖·‖_∞ and ‖·‖ _p , p≥1 norms of probability density functions. Some of these inequalities provide pointwise estimates to the error of probability distribution function from the expectation of some simple function of the engaged random variable. Other inequalities give upper bounds for the expectation and variance of a random variable. All are done over finite domains. At the end are given applications, especially for the Beta random variable.     

Divergenzverhalten mehrdimensionaler Shannonscher Abtastreihen

The behaviour of multidimensional Shannon sampling series for continuous functions is examined. A continuous function g ₁∈C ₀ [0,1]² with support in the rectangle [0,1] × [0,?] is indicated in the paper for which the two dimensional Shannon sampling series diverge almost everywhere in the rectangle [0,1] × [?,1]. This shows that the localization principle for Shannon sampling series cannot hold in two dimensions and in higher dimensions. The result solves a problem formulated by P.L. Butzer. Received: 21 December 1995 / Revised version: 5 October 1996     

Representations of nilpotent Lie algebras

Let (L,[p]) a finite dimensional nilpotent restricted Lie algebra of characteristic p 3 3, c ? L^*p \geq 3, \chi \in L^* a linear form. In this paper we study the representation theory of the reduced universal enveloping algebra u(L,c)u(L,\chi ). It is shown that u(L,c)u(L,\chi ) does not admit blocks of tame representation type. As an application, we prove that the nonregular AR-components of u(L,c)u(L,\chi ) are of types \Bbb Z [A_￥ ]\Bbb Z [A_\infty ] or \Bbb Z [A_n]/(t)\Bbb Z [A_n]/(\tau ).     

Local convergence in a generalized Fermat-Weber problem

In the Fermat-Weber problem, the location of a source point in ^N is sought which minimizes the sum of weighted Euclidean distances to a set of destinations. A classical iterative algorithm known as the Weiszfeld procedure is used to find the optimal location. Kuhn proves global convergence except for a denumerable set of starting points, while Katz provides local convergence results for this algorithm. In this paper, we consider a generalized version of the Fermat-Weber problem, where distances are measured by anl _p norm and the parameterp takes on a value in the closed interval [1, 2]. This permits the choice of a continuum of distance measures from rectangular (p=1) to Euclidean (p=2). An extended version of the Weiszfeld procedure is presented and local convergence results obtained for the generalized problem. Linear asymptotic convergence rates are typically observed. However, in special cases where the optimal solution occurs at a singular point of the iteration functions, this rate can vary from sublinear to quadratic. It is also shown that for sufficiently large values ofp exceeding 2, convergence of the Weiszfeld algorithm will not occur in general.     

Minimum permanents of doubly stochastic matrices with One fixed entry

For n  3 and sufficiently small a  0, the minimum value of the permanent function restricted on n × n doubly stochastic matrices with at least one entry equal to a is obtained. For n = 3, the explicit form of the function p is derived where p(a) = min{per(C):C = (c_ij )?Ω₃ c ₁₁ = a}a? [0, 1].     

Distance Functions and Almost Global Solutions of Eikonal Equations

Consider a function u defined on  ⁿ , except, perhaps, on a closed set of potential singularities . Suppose that u solves the eikonal equation ‖Du‖ = 1 in the pointwise sense on  ⁿ \, where Du denotes the gradient of u and ‖·‖ is a norm on  ⁿ with the dual norm ‖·‖_?. For a class of norms which includes the standard p-norms on  ⁿ , 1 < p < ∞, we show that if  has Hausdorff 1-measure zero and n ≥ 2, then u is either affine or a “cone function,” that is, a function of the form u(x) = a ± ‖x ? z‖_?.     

All-norm approximation algorithms

A major drawback in optimization problems and in particular in scheduling problems is that for every measure there may be a different optimal solution. In many cases the various measures are different ℓ_p norms. We address this problem by introducing the concept of an all-norm ρ-approximation algorithm, which supplies one solution that guarantees ρ-approximation to all ℓ_p norms simultaneously. Specifically, we consider the problem of scheduling in the restricted assignment model, where there are m machines and n jobs, each job is associated with a subset of the machines and should be assigned to one of them. Previous work considered approximation algorithms for each norm separately. Lenstra et al. [Math. Program. 46 (1990) 259–271] showed a 2-approximation algorithm for the problem with respect to the ℓ_∞ norm. For any fixed ℓ_p norm the previously known approximation algorithm has a performance of θ(p). We provide an all-norm 2-approximation polynomial algorithm for the restricted assignment problem. On the other hand, we show that for any given ℓ_p norm (p>1) there is no PTAS unless P=NP by showing an APX-hardness result. We also show for any given ℓ_p norm a FPTAS for any fixed number of machines.     

Carleson's counterexample and a scale of Lorentz-BMO spaces on the bitorus

We introduce a full scale of Lorentz-BMO spaces BMO _L ^p,q on the bidisk, and show that these spaces do not coincide for different values ofp andq. Our main tool is a detailed analysis of Carleson's construction in [C]. The first author gratefully acknowledges support by the LMS and Proyecto BFM2002-04013. The second author gratefully acknowledges support by EPSRC and by the Nuffield Foundation.     

Partially Ordered Generalized Patterns and <Emphasis Type="Italic">k</Emphasis>-ary Words

Recently, Kitaev [9] introduced partially ordered generalized patterns (POGPs) in the symmetric group, which further generalize the generalized permutation patterns introduced by Babson and Steingrímsson [1]. A POGP p is a GP some of whose letters are incomparable. In this paper, we study the generating functions (g.f.) for the number of k-ary words avoiding some POGPs. We give analogues, extend and generalize several known results, as well as get some new results. In particular, we give the g.f. for the entire distribution of the maximum number of non-overlapping occurrences of a pattern p with no dashes (which is allowed to have repetition of letters), provided we know the g.f. for the number of k-ary words that avoid p.AMS Subject Classification: 05A05, 05A15.     

Sobolev's inequality for Riesz potentials with variable exponent satisfying a log-Hölder condition at infinity

Our aim in this paper is to deal with the boundedness of maximal functions in generalized Lebesgue spaces Lp(⋅) when p(⋅) satisfies a log-Hölder condition at infinity that is weaker than that of Cruz-Uribe, Fiorenza and Neugebauer [D. Cruz-Uribe, A. Fiorenza, C.J. Neugebauer, The maximal function on variable Lp spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003) 223-238; 29 (2004) 247-249]. Our result extends the recent work of Diening [L. Diening, Maximal functions on generalized Lp(⋅) spaces, Math. Inequal. Appl. 7 (2004) 245-254] and the authors Futamura and Mizuta [T. Futamura, Y. Mizuta, Sobolev embeddings for Riesz potential space of variable exponent, preprint]. As an application of the boundedness of maximal functions, we show Sobolev's inequality for Riesz potentials with variable exponent.     

Partial cotilting modules and the lattices induced by them

We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct summand of a cotilting module C such that the left Ext-orthogonal class ⊥P coincides with ⊥C. As an application, we characterize all cotilting torsion-free classes. Each partial cotilting module P defines a lattice L = [Cogen P₁P] of torsion-free classes. Similarly, each partial tilting module P′ defines a lattice L′ = [[Gen P′,P′⊥]] of torsion classes. Generalizing a result of Assem and Kerner, we show that the elements of L are determined by their Rej_p-torsion parts, and the elements of L′ by their Tr_p-torsion-free parts.     

On the p—Local Rank of Finite Groups

In this paper,we shall mainly study the p-solvable finite group in terms of p-local rank,and a group theoretic characterization will be given of finite p-solvabel groups with p-local rank two.Theorem A Let G be a finite p-solvable group with p-local rank plr(G)=2 and Op(G)=1.If P is a Sylow p-subgrounp of G,then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two.Theorem B Let G be a finite p-solvable group with Op(G)=1.Then the p-length lp(G)≤plr(G);if in addition plr(G)=lp (G) and p≥5 is odd,then plr(G)=0 or 1.