Extension of multilinear maps defined on subspaces

We study the problem of whether every multilinear form defined on the product of n closed subspaces has an extension defined on the product of the entire Banach spaces. We prove that the property derived from this condition (the Multilinear Extension Property) is local. We use this to prove that, for a wide variety of Banach spaces, there exist a product of closed subspaces and a multilinear form defined on it, which has no extension to the product of the entire spaces. We show that the ℓ _p spaces, with 1 ≤p ≤ ∞ and p ≠ 2, are among them and, more generally, every Banach space which fails to have type p for some p < 2 or cotype q for some q > 2.     

New models for locating a moving service facility

In this paper we analyze a new location problem which is a generalization of the well-known single facility location model. This extension consists of introducing a general objective function and replacing fixed locations by trajectories. We prove that the problem is well-stated and solvable. A Weiszfeld type algorithm is proposed to solve this generalized dynamic single facility location problem on L ^p spaces of functions, with p ∈(1,2]. We prove global convergence of our algorithm once we have assumed that the set of demand functions and the initial step function belong to a subspace of L ^p called Sobolev space. Finally, examples are included illustrating the application of the model to generalized regression analysis and the convergence of the proposed algorithm. The examples also show that the pointwise extension of the algorithm does not have to converge to an optimal solution of the considered problem while the proposed algorithm does.     

Refinable Functions with Exponential Decay: An?Approach via Cascade Algorithms

Refinable functions with exponential decay arise from applications such as the Butterworth filters in signal processing. Refinable functions with exponential decay also play an important role in the study of Riesz bases of wavelets generated from multiresolution analysis. A fundamental problem is whether the standard solution of a refinement equation with an exponentially decaying mask has exponential decay. We investigate this fundamental problem by considering cascade algorithms in weighted L _p spaces (1≤p≤∞). We give some sufficient conditions for the cascade algorithm associated with an exponentially decaying mask to converge in weighted L _p spaces. Consequently, we prove that the refinable functions associated with the Butterworth filters are continuous functions with exponential decay. By analyzing spectral properties of the transition operator associated with an exponentially decaying mask, we find a characterization for the corresponding refinable function to lie in weighted L ₂ spaces. The general theory is applied to an interesting example of bivariate refinable functions with exponential decay, which can be viewed as an extension of the Butterworth filters.     

The Fermat—Weber location problem revisited

The Fermat—Weber location problem requires finding a point in ^N that minimizes the sum of weighted Euclidean distances tom given points. A one-point iterative method was first introduced by Weiszfeld in 1937 to solve this problem. Since then several research articles have been published on the method and generalizations thereof. Global convergence of Weiszfeld's algorithm was proven in a seminal paper by Kuhn in 1973. However, since them given points are singular points of the iteration functions, convergence is conditional on none of the iterates coinciding with one of the given points. In addressing this problem, Kuhn concluded that whenever them given points are not collinear, Weiszfeld's algorithm will converge to the unique optimal solution except for a denumerable set of starting points. As late as 1989, Chandrasekaran and Tamir demonstrated with counter-examples that convergence may not occur for continuous sets of starting points when the given points are contained in an affine subspace of ^N . We resolve this open question by proving that Weiszfeld's algorithm converges to the unique optimal solution for all but a denumerable set of starting points if, and only if, the convex hull of the given points is of dimensionN.     

Fourier multipliers and spectral measures in Banach function spaces

It is classical that amongst all spaces L^p (G), 1 ≤ p ≤ ∞, for , or say, only L² (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L² (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L² (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L² (G); this fails for every p ≠ 2. We show that this special status of L² (G) amongst the spaces L^p (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure).      

The Geodesic Problem in Quasimetric Spaces

In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality d(x,y)≤σ(d(x,z)+d(z,y)) for some constant σ≥1, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzelà theorem) still hold in quasimetric spaces. Moreover, we explore conditions under which a quasimetric will induce an intrinsic metric. As an example, we introduce a family of quasimetrics on the space of atomic probability measures. The associated intrinsic metrics induced by these quasimetrics coincide with the d _α metric studied early in the study of branching structures arisen in ramified optimal transportation. An optimal transport path between two atomic probability measures typically has a “tree shaped” branching structure. Here, we show that these optimal transport paths turn out to be geodesics in these intrinsic metric spaces. This work is supported by an NSF grant DMS-0710714.     

Convergence analysis of the projective scaling algorithm based on a long-step homogeneous affine scaling algorithm

In this paper we give a new convergence analysis of a projective scaling algorithm. We consider a long-step affine scaling algorithm applied to a homogeneous linear programming problem obtained from the original linear programming problem. This algorithm takes a fixed fraction λ≤2/3 of the way towards the boundary of the nonnegative orthant at each iteration. The iteration sequence for the original problem is obtained by pulling back the homogeneous iterates onto the original feasible region with a conical projection, which generates the same search direction as the original projective scaling algorithm at each iterate. The recent convergence results for the long-step affine scaling algorithm by the authors are applied to this algorithm to obtain some convergence results on the projective scaling algorithm. Specifically, we will show (i) polynomiality of the algorithm with complexities of O(nL) and O(n ² L) iterations for λ<2/3 and λ=2/3, respectively; (ii) global covnergence of the algorithm when the optimal face is unbounded; (iii) convergence of the primal iterates to a relative interior point of the optimal face; (iv) convergence of the dual estimates to the analytic center of the dual optimal face; and (v) convergence of the reduction rate of the objective function value to 1−λ.     

Geometric Optimization Problems Likely Not Contained in $$\mathbb{A}\mathbb{P}\mathbb{X}$$

   Abstract. Maximizing geometric functionals such as the classical l _p -norms over polytopes plays an important role in many applications, hence it is desirable to know how efficiently the solutions can be computed or at least approximated. While the maximum of the l _∞ -norm over polytopes can be computed in polynomial time, for 2≤ p < ∞ the l _p -norm-maxima cannot be computed in polynomial time within a factor of 1.090 , unless P=NP. This result holds even if the polytopes are centrally symmetric parallelotopes. Quadratic Programming is a problem closely related to norm-maximization, in that in addition to a polytope P ⊂ R ⁿ , numbers c _ij , 1≤ i≤ j≤ n , are given and the goal is to maximize Σ _{1≤ i≤ j≤ n} c _ij x _i x _j over P . It is known that Quadratic Programming does not admit polynomial-time approximation within a constant factor, unless P=NP. Using the observation that the latter result remains true even if the existence of an integral optimal point is guaranteed, in this paper it is proved that analogous inapproximability results hold for computing the l _p -norm-maximum and various l _p -radii of centrally symmetric polytopes for 2≤ p < ∞.     

COEFFICIENT MULTIPLIERS ON MIXED NORM SPACES

§1　IntroductionSuppose thatf is analytic in the open unit disc D in the complex plane.We defineMp(r,f) =12π∫2π0 | f(reiθ) | pdθ1 / p,0

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Geometric Optimization Problems Likely Not Contained in APX

Abstract. Maximizing geometric functionals such as the classical l _p -norms over polytopes plays an important role in many applications, hence it is desirable to know how efficiently the solutions can be computed or at least approximated. While the maximum of the l _∞ -norm over polytopes can be computed in polynomial time, for 2≤ p < ∞ the l _p -norm-maxima cannot be computed in polynomial time within a factor of 1.090 , unless P=NP. This result holds even if the polytopes are centrally symmetric parallelotopes. Quadratic Programming is a problem closely related to norm-maximization, in that in addition to a polytope P ⊂ R ⁿ , numbers c _ij , 1≤ i≤ j≤ n , are given and the goal is to maximize Σ _{1≤ i≤ j≤ n} c _ij x _i x _j over P . It is known that Quadratic Programming does not admit polynomial-time approximation within a constant factor, unless P=NP. Using the observation that the latter result remains true even if the existence of an integral optimal point is guaranteed, in this paper it is proved that analogous inapproximability results hold for computing the l _p -norm-maximum and various l _p -radii of centrally symmetric polytopes for 2≤ p < ∞.     

A remark on the behaviour ofL p-multipliers and the range of operators acting onL p-spaces

In this paper, new results are obtained concerning the uniform approximation property (UAP) inL ^p-spaces (p≠2,1,∞). First, it is shown that the “uniform approximation function” does not allow a polynomial estimate. This fact is rather surprising since it disproves the analogy between UAP-features and the presence of “large” euclidian subspaces in the space and its dual. The examples are translation invariant spaces on the Cantor group and this extra structure permits one to replace the problem with statements about the nonexistence of certain multipliers in harmonic analysis. Secondly, it is proved that the UAP-function has an exponential upper estimate (this was known forp=1, ∞). The argument uses Schauder’s fix point theorem. Its precise behaviour is left unclarified here. It appears as a difficult question, even in the translation invariant context.     

Coefficient multipliers of mixed norm space in the ball

In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (H _p,q (φ₁), H _u,v (φ₂)) for the values of p, q, u, v in three cases: (i) 0 < p ≤ u ≤ ∞, 0 < q ≤ min(1, v) ≤ ∞. (ii) v = ∞, 0 < p ≤ u ≤ ∞, 1 ≤ u, q ≤ ∞. (iii) 1 ≤ v ≤ 2 ≤ q ≤ ∞, and 0 < p ≤ u ≤ ∞ or 1 ≤ p, u ≤ ∞. The first case extends the result of Blasco, Jevtić, and Pavlović in one variable. The third case generalizes partly the results of Jevtić, Jovanović, and Wojtaszczyk to higher dimensions. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Almost simple groups with socle 3D4(q) act on finite linear spaces

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O’Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T ≤ G ≤ Aut(T) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to ³ D ₄(q), then T is line-transitive, where q is a power of the prime p.     

Bicolored Matchings in Some Classes of Graphs

We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R|=n+1 such that perfect matchings with k red edges exist for all k,0≤k≤n. Given two integers p<q we also determine the minimum cardinality of a set R of red edges such that there are perfect matchings with p red edges and with q red edges. For 3-regular bipartite graphs, we show that if p≤4 there is a set R with |R|=p for which perfect matchings M_k exist with |M_k∩R|≤k for all k≤p. For trees we design a linear time algorithm to determine a minimum set R of red edges such that there exist maximum matchings with k red edges for the largest possible number of values of k.     

Vector-valued walsh-paley martingales and geometry of banach spaces

The concept of Rademacher typep (1≤p≤2) plays an important role in the local theory of Banach spaces. In [3] Mascioni considers a weakening of this concept and shows that for a Banach spaceX weak Rademacher typep implies Rademacher typer for allr<p. As with Rademacher typep and weak Rademacher typep, we introduce the concept of Haar typep and weak Haar typep by replacing the Rademacher functions by the Haar functions in the respective definitions. We show that weak Haar typep implies Haar typer for allr<p. This solves a problem left open by Pisier [5]. The method is to compare Haar type ideal norms related to different index sets.     

Hardy Spaces on ? n with Pointwise Variable Anisotropy

In this work we develop highly geometric Hardy spaces, for the full range 0<p≤1. These spaces are constructed over multi-level ellipsoid covers of ℝ ⁿ that are highly anisotropic in the sense that the ellipsoids can change shape rapidly from point to point and from level to level. This generalizes previous work on anisotropic Hardy spaces where the geometry of the space was ‘fixed’ over ℝ ⁿ and extends Hardy spaces over spaces of homogeneous type, where the theory holds for p values that are ‘close’ to 1.     

Embedding into Rectilinear Spaces

We show that the problem whether a given finite metric space (X,d) can be embedded into the rectilinear space R ^m can be formulated in terms of m -colorability of a certain hypergraph associated with (X,d) . This is used to close a gap in the proof of an assertion of Bandelt and Chepoi [2] on certain critical metric spaces for this embedding problem. We also consider the question of determining the maximum number of equidistant points that can be placed in the m -dimensional rectilinear space and show that this number is equal to 2m for m ≤ 3 . Received March 19, 1996, and in revised form March 14, 1997.     

Approximation of Classes of Analytic Functions by Fourier Sums in Uniform Metric

We find asymptotic equalities for upper bounds of approximations by partial Fourier sums in the uniform metric on classes of Poisson integrals of periodic functions belonging to the unit balls in the spaces L _p , 1 ≤ p ≤ ∞. We generalize the results obtained to the classes of (ψ, )-differentiable (in the sense of Stepanets) functions that admit an analytic extension to a fixed strip of the complex plane. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1079 – 1096, August, 2005.     

Series of independent random variables in rearrangement invariant spaces: An operator approach

This paper studies series of independent random variables in rearrangement invariant spacesX on [0, 1]. Principal results of the paper concern such series in Orlicz spaces exp(L _p ), 1≤p≤∞ and Lorentz spacesA _Ψ. One by-product of our methods is a new (and simpler) proof of a result due to W. B. Johnson and G. Schechtman that the assumptionL _p ⊂X, p<∞ is sufficient to guarantee that convergence of such series inX (under the side condition that the sum of the measures of the supports of all individual terms does not exceed 1) is equivalent to convergence inX of the series of disjoint copies of individual terms. Furthermore, we prove the converse (in a certain sense) to that result. Research supported by the Australian Research Council.     

Spaces of analytic functions of Hardy-Bloch type

For 0<p≤∞ and 0<q≤∞, the space of Hardy-Bloch type ℬ(p,q) consists of those functionsf which are analytic in the unit diskD such that (1−r)M _p (r,f′)⊂L ^q (dr/(1−r)). We note that ℬ(∞,∞) coincides with the Bloch space ℬ and that ℬ⊂ℬ(p,∞) for allp. Also, the space ℬ(p,p) is the Dirichlet spaceD _p−1 ^p . We prove a number of results on decomposition of spaces with logarithmic weights which allow us to obtain sharp results about the mean growth of the ℬ(p,q). In particular, we prove that iff is an analytic function inD and 2≤p<∞, then the conditionM _p (r,f′)=O((1−r)⁻¹), asr→1, implies that