An <Emphasis Type="Italic">L</Emphasis><Subscript>\infty</Subscript>/<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-Constrained Quadratic Optimization Problem with Applications to Neural Networks

Pattern formation in associative neural networks is related to a quadratic optimization problem. Biological considerations imply that the functional is constrained in the L _\infty norm and in the L ₁ norm. We consider such optimization problems. We derive the Euler–Lagrange equations, and construct basic properties of the maximizers. We study in some detail the case where the kernel of the quadratic functional is finite-dimensional. In this case the optimization problem can be fully characterized by the geometry of a certain convex and compact finite-dimensional set.     

Estimating L ∞ Norms by L 2k Norms for Functions on Orbits

Abstract. Let G be a compact group acting in a real vector space V . We obtain a number of inequalities relating the L ^∞ norm of a matrix element of the representation of G with its L ^2k norm for a positive integer k . As an application, we obtain approximation algorithms to find the maximum absolute value of a given multivariate polynomial over the unit sphere (in which case G is the orthogonal group) and for the assignment problem of degree d , a hard problem of combinatorial optimization generalizing the quadratic assignment problem (in which case G is the symmetric group).     

Best Basis Selection for Approximation in L p

We study the approximation of a function class F in L _p by choosing first a basis B and then using n -term approximation with the elements of B . Into the competition for best bases we enter all greedy (i.e., democratic and unconditional [20]) bases for L _p . We show that if the function class F is well-oriented with respect to a particular basis B then, in a certain sense, this basis is the best choice for this type of approximation. Our results extend the recent results of Donoho [9] from L ₂ to L _p , p\neq 2 .     

L p -dual Quermassintegral sums

In this paper, we first introduce a concept of L _p -dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for L _p -dual Quermassintegral sums. Moreover, by using Lutwak’s width-integral of index i, we establish the L _p -Brunn-Minkowski inequality for the polar mixed projection bodies. As applications, we prove some interrelated results. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10271071), Zhejiang Provincial Natural Science Foundation of China (Grant No. Y605065) and Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

Best Simultaneous Approximation in L(I,E)

Let G be a reflexive subspace of the Banach space E and let L^p(I,E) denote the space of all p-Bochner integrable functions on the interval I=[0,1] with values in E, 1p∞. Given any norm N( , ) on R², N nondecreasing in each coordinate on the set R²₊, we prove that L^p(I,G) is N-simultaneously proximinal in L^p(I,E). Other results are also obtained.     

Walsh Series Analysis of the L 2-Discrepancyof Symmetrisized Point Sets

 We present a method to estimate the L ₂-discrepancy of symmetrisized point sets from above and from below with the help of Walsh series analysis. We apply the method to a class of two-dimensional net-type point sets, thereby generalizing results of Halton and Zaremba and of Proinov.     

The convergence of the best discrete linear Lp approximation as p → 1

It is well known that the best discrete linear L_p approximation converges to a special best Chebyshev approximation as p → ∞. In this paper it is shown that the corresponding result for the case p → 1 is also true. Furthermore, the special best L₁ approximation obtained as the limit is characterized as the unique solution of a nonlinear programming problem on the set of all L₁ solutions.     

The L2 norm of the approximation error for Bernstein polynomials

Wassily Hoeffding (J. Approximation Theory 4 (1971), 347–356) obtained a convergence rate for the L₁ norm of the approximation error, using Bernstein polynomials for a wide class of functions. Here, by a different method of proof, a similar result is obtained for the L₂ norm.     

(Almost) primitivity of Hecke L -functions

Let f be a cusp form of the Hecke space \frak M₀(l,k,e){\frak M}_0(\lambda,k,\epsilon) and let L _f be the normalized L-function associated to f. Recently it has been proved that L _f belongs to an axiomatically defined class of functions [`(S)]<sup>\sharp</sup>\bar{\cal S}^\sharp . We prove that when λ ≤ 2, L <sub>f</sub> is always almost primitive, i.e., that if L <sub>f</sub> is written as product of functions in [`(S)]^\sharp\bar{\cal S}^\sharp , then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if l ? {?2,?3,2}\lambda\notin\{\sqrt{2},\sqrt{3},2\} then L _f is also primitive, i.e., that if L _f = F ₁ F ₂ then F ₁ (or F ₂) is constant; for l ? {?2,?3,2}\lambda\in\{\sqrt{2},\sqrt{3},2\} the factorization of non-primitive functions is studied and examples of non-primitive functions are given. At last, the subset of functions f for which L _f belongs to the more familiar extended Selberg class S^\sharp{\cal S}^\sharp is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in S^\sharp{\cal S}^\sharp .     

L∞-boundedness of L2-projections on splines for a geometric mesh

In this paper we complete the investigations started by [14.]. We study C. de Boor's conjecture of the L_∞-boundedness of the L₂-projection P on smooth splines in the special case of a geometric mesh x = {q^v}. A connection to the interpolation projection is established and the uniform boundedness of ¦|P¦|_π with respect to q is proved.     

Archimedean L -factors on GL ( n ) × GL ( n ) and generalized Barnes integrals

The Rankin-Selberg method associates, to each local factorL(s, π _v × π _v ^′ ) of an automorphicL-function onGL(n) ×GL(n), a certain local integral of Whittaker functions for π _v and _v ^′ . In this paper we show that, if ν is archimedean, and π _v and _v ^′ are spherical principal series representations with trivial central character, then the localL-factor and local integral are, in fact, equal. This result verifies a conjecture of Bump, which predicts that the archimedean situation should, in the present context, parallel the nonarchimedean one. We also derive, as prerequisite to the above result, some identities for generalized Barnes integrals. In particular, we deduce a new transformation formula for certain single Barnes integrals, and a multiple-integral analog of the classical Barnes’ Lemma.     

L 1→ L q Poincaré Inequalities for 0 < q < 1 Imply Representation Formulas

Given two doubling measures μ and ν in a metric space (S, ρ) of homogeneous type, let B ₀⊂S be a given ball. It has been a well-known result by now (see [1–4]) that the validity of an L ¹→L ¹ Poincaré inequality of the following form: for all metric balls B⊂B ₀⊂S, implies a variant of representation formula of fractional integral type: for ρ-a.e. x∈B ₀, One of the main results of this paper shows that an L ¹ to L ^q Poincaré inequality for some 0 < q < 1, i.e., for all metric balls B⊂B ₀, will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition, also implies the same formula. Analogous theorems related to high-order Poincaré inequalities and Sobolev spaces in metric spaces are also proved. Received December 27, 2000, Accepted May 28, 2001     

Near-best L1 approximations on circular and elliptical contours

Polynomial approximations are obtained to analytic functions on circular and elliptical contours by forming partial sums of order n of their expansions in Taylor series and Chebyshev series of the second kind, respectively. It is proved that the resulting approximations converge in the L₁ norm as n → ∞, and that they are near-best L₁ approximations within relative distances of the order of log n. Practical implications of the results are discussed, and they are shown to provide a theoretical basis for polynomial approximation methods for the evaluation of indefinite integrals on contours.     

Lp-Approximable Sequences of Vectors and Limit Distribution of Quadratic Forms of Random Variables

The properties of L₂-approximable sequences established here form a complete toolkit for statistical results concerning weighted sums of random variables, where the weights are nonstochastic sequences approximated in some sense by square-integrable functions and the random variables are “two-wing” averages of martingale differences. The results constitute the first significant advancement in the theory of L₂-approximable sequences since 1976 when Moussatat introduced a narrower notion of L₂-generated sequences. The method relies on a study of certain linear operators in the spaces L_p and l_p. A criterion of L_p-approximability is given. The results are new even when the weight generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method.     

Some remarks on the Schr?dinger equation with a potential in LrtLsx

We study the dispersive properties of the linear Schr?dinger equation with a time-dependent potential V(t,x). We show that an appropriate integrability condition in space and time on V, i.e. the boundedness of a suitable L^r_tL^s_x norm, is sufficient to prove the full set of Strichartz estimates. We also construct several counterexamples which show that our assumptions are optimal, both for local and for global Strichartz estimates, in the class of large unsigned potentials V ∈L^r_tL^s_x. Support. The authors are partially supported by the Research Training Network (RTN) HYKE and by grant HPRN-CT-2002-00282 from the European Union. The third author is supported also by INDAM     

Best polynomial and durrmeyer approximation in Lp(S)

On a simplex SR^d, the best polynomial approximation is E_n()_Lp(S)=Inf{P_n−_Lp(S): P_n of total degree n}. The Durrmeyer modification, M_n, of the Bernstein operator is a bounded operator on L_p(S) and has many “nice” properties, most notably commutativity and self-adjointness. In this paper, relations between M_n−z.dfnc;_Lp(S) and E_[√n]()_Lp(S) will be given by weak inequalities will imply, for 0<α<1 and 1≤p≤∞, E_n()_Lp(S)=O(n^-2α)M_n−z.dfnc;_Lp(S)=O(n^-α). We also see how the fact that P(D)εL_p(S) for the appropriate P(D) affects directional smoothness.     

L p - L q estimates of solutions to the damped wave equation in 3-dimensional space and their application

 It has been asserted that the damped wave equation has the diffusive structure as t→∞. In this paper we consider the Cauchy problem in 3-dimensional space for the linear damped wave equation and the corresponding parabolic equation, and obtain the L ^p −L ^q estimates of the difference of each solution, which represent the assertion precisely. Explicit formulas of the solutions are analyzed for the proof. The second aim is to apply the L ^p −L ^q estimates to the semilinear damped wave equation with power nonlinearity. If the power is larger than the Fujita exponent, then the time global existence of small weak solution is proved and its optimal decay order is obtained. Received: 8 June 2001; in final form: 12 August 2002 / Published online: 1 April 2003 Mathematical Subject Classification (2000): 35L15.     

Constrained Lp-Approximation by Generalized n-Convex Functions Induced by ECT-Systems

The problem of finding a best L_p-approximation (1 ≤ p < ∞) to a function in L_p from a special subcone of generalized n-convex functions induced by an ECT-system is considered. Tchebycheff splines with a countably infinite number of knots are introduced and best approximations are characterized in terms of local best approximations by these splines. Various properties of best approximations and their uniqueness in L₁ are investigated. Some special results for generalized monotone and convex cases are obtained.