Shifted Jacobi multiplier sequences and transplantation between sine and cosine series

It is shown that the multiplier norm of a shifted Jacobi multiplier sequence can be estimated by the (same) multiplier norm of the original sequence uniformly with respect to the shift. Muckenhoupt’s transplantation theorem for Jacobi series is used essentially, for which also a functional analytic understanding is given in terms of the minimality of the Jacobi system in weighted L ^p -spaces.      

A functional calculus in hilbert space based on operator valued analytic functions

A homomorphic map is defined from the algebra of norm bounded analyticN-operator valued functions in the unit disc into the algebra of bounded operators in Hilbert spaces represented as left invariant subspaces ofH ²(N), and the spectral properties of the map are studied. The subclass of functions having norm bound one in the disc is characterized in terms of the power series coefficients. This paper was partially supported by the National Science Foundation under contract NSF GP-5455.     

BMO estimates for lacunary series

We proveBMO andL ^p norm inequalities inR ⁿ for lacunary Walsh and generalized trigonometric series.     

Convergence of the Neumann Series in Higher Norms

Abstract

Natural conditions on an operator A are given so that the Neumann series for (Id + A)^?1 converges in higher norm topologies.     

Conditions for uniqueness of a projector with unit norm

Suppose that in a normed linear space B there exists a projector with unit norm onto a subspace D. A sufficient condition for this projector to be unique is the existence of a setM D ^* which is total on D, each functional in which attains its norm on the unit sphere in D and has a unique extension onto B with preservation of norm. As corollaries to this fact, we obtain a series of sufficient conditions for uniqueness (some of which were previously known) as well as a necessary and sufficient condition for uniqueness.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 45–49, July, 1977.The author thanks M. I. Kadets for useful discussions.     

Near-best multivariate approximation by Fourier series, Chebyshev series and Chebyshev interpolation

A set of results concerning goodness of approximation and convergence in norm is given for L_∞ and L₁ approximation of multivariate functions on hypercubes. Firstly the trigonometric polynomial formed by taking a partial sum of a multivariate Fourier series and the algebraic polynomials formed either by taking a partial sum of a multivariate Chebyshev series of the first kind or by interpolating at a tensor product of Chebyshev polynomial zeros are all shown to be near-best L_∞ approximations. Secondly the trigonometric and algebraic polynomials formed by taking, respectively, a partial sum of a multivariate Fourier series and a partial sum of a multivariate Chebyshev series of the second kind are both shown to be hear-best L₁ approximations. In all the cases considered, the relative distance of a near-best approximation from a corresponding best approximation is shown to be at most of the order of Π log n_j, where n_j (j = 1, 2,…, N) are the respective degrees of approximation in the N individual variables. Moreover, convergence in the relevant norm is established for all the sequences of near-best approximations under consideration, subject to appropriate restrictions on the function space.     

Coefficient Condition forL1-Convergence of Walsh–Fourier Series

In this paper we give a condition with respect to Walsh–Fourier coefficients that implies theL₁-convergence of the corresponding Walsh–Fourier series. We show that theL₁-convergence class induced by this condition contains each one of the previously known convergence classes as a proper subset. We also show that our condition implies not only theL₁-convergence but also the convergence in the dyadic Hardy norm if the function represented by the series belongs to the dyadic Hardy space.     

Minimum norm solutions for cooperative games

A solution f for cooperative games is a minimum norm solution, if the space of games has a norm such that f(v) minimizes the distance (induced by the norm) between the game v and the set of additive games. We show that each linear solution having the inessential game property is a minimum norm solution. Conversely, if the space of games has a norm, then the minimum norm solution w.r.t. this norm is linear and has the inessential game property. Both claims remain valid also if solutions are required to be efficient. A minimum norm solution, the least square solution, is given an axiomatic characterization.      

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.     

A posteriori error analysis for numerical approximations of Friedrichs systems

The global error of numerical approximations for symmetric positive systems in the sense of Friedrichs is decomposed into a locally created part and a propagating component. Residual-based two-sided local a posteriori error bounds are derived for the locally created part of the global error. These suggest taking the -norm as well as weaker, dual norms of the computable residual as local error indicators. The dual graph norm of the residual is further bounded from above and below in terms of the norm of where h is the local mesh size. The theoretical results are illustrated by a series of numerical experiments. Received January 10, 1997 / Revised version received March 5, 1998     

Controlling Orthogonal Series with Irrational Rotations

We compute covering numbers associated to the set of partial sums of orthogonal series by means of irrational rotations. The device is a new metric inequality linking the increment's norm of partial sums to the one of ergodic averages of rotations acting on a suitable L ²-element of the torus. This allows to compute the number of balls covering the whole set of partial sums, by means of rotations.     

Gibbs-Wilbraham splines

If a function with a jump discontinuity is approximated in the norm ofL ²[–1,1] by a periodic spline of orderk with equidistant knots, a behavior analogous to the Gibbs-Wilbraham phenomenon for Fourier series occurs. A set of cardinal splines which play the role of the sine integral function of the classical phenomenon is introduced. It is then shown that ask becomes large, the phenomenon for splines approaches the classical phenomenon.Communicated by Ronald A. DeVore.     

Approximation by Dirichlet Series with Nonnegative Coefficients

The problem of approximating a given function by Dirichlet series with nonnegative coefficients is associated with the discrete spectral representation of the relaxation modulus in rheology. The main result of this paper is that if a function can be approximated arbitrarily closely by Dirichlet series with nonnegative coefficients in supremum norm or L_p-norm, 1p<∞, then it must be completely monotonic.     

BMO and Lipschitz Norm Estimates for Composite Operators

In this paper, we obtain Lipschitz and BMO norm inequalities for the composition G ∘ T of the homotopy operator T and Green’s operator G applied to differential forms. We also investigate the relationship among Lipschitz norm, BMO norm and L ^s norm.     

Triangular Dirichlet Kernels and Growth of L p Lebesgue Constants

Let P be a polygon in ℤ² and consider the mapping of an L¹(\mathbbT²)L^{1}(\mathbb{T}^{2}) function into the partial sum of its Fourier series determined by the dilate of P by the integer N. If the image space is endowed with the L ^p norm, 1<p<∞, then the operator norm will be given by the L ^p norm of ∑_(m,n)∈NP e ^{2π i(mx+ny)}. The size of this operator norm is shown to be O(N ^2(1−1/p)) when the polygon is a triangle. The estimate is independent of the shape of the triangle. For a k sided polygon the corresponding estimate is O(kN ^2(1−1/p)).     

Operator norm localization property of metric spaces under finite decomposition complexity

The notions of operator norm localization property and finite decomposition complexity were recently introduced in metric geometry to study the coarse Novikov conjecture and the stable Borel conjecture. In this paper we show that a metric space X has weak finite decomposition complexity with respect to the operator norm localization property if and only if X itself has the operator norm localization property. It follows that any metric space with finite decomposition complexity has the operator norm localization property. In particular, we obtain an alternative way to prove a very recent result by E. Guentner, R. Tessera and G. Yu that all countable linear groups have the operator norm localization property.     

A Concentration Theorem for the Heat Equation

 We compare the solution of to the solution of the same equation where f is replaced by a “concentrated” source . As a result we derive some estimates on the solution in spatial norm, locally uniformly in t, with respect to the norm of for any integer . In the case we obtain a critical inequality relating the norm of to an exponential norm of u. (Received 1 September 2000; in revised form 17 January 2001)     

Transitivity of the norm on Banach spaces¶ having a Jordan structure

We study transitivity conditions on the norm of JB ^*-triples, C ^*-algebras, JB-algebras, and their preduals. We show that, for the predual X of a JBW ^*-triple, each one of the following conditions i) and ii) implies that X is a Hilbert space. i) The closed unit ball of X has some extreme point and the norm of X is convex transitive. ii) The set of all extreme points of the closed unit ball of X is non rare in the unit sphere of X. These results are applied to obtain partial affirmative answers to the open problem whether every JB ^*-triple with transitive norm is a Hilbert space. We extend to arbitrary C ^*-algebras previously known characterizations of transitivity [20] and convex transitivity [36] of the norm on commutative C ^*-algebras. Moreover, we prove that the Calkin algebra has convex transitive norm. We also prove that, if X is a JB-algebra, and if either the norm of X is convex transitive or X has a predual with convex transitive norm, then X is associative. As a consequence, a JB-algebra with almost transitive norm is isomorphic to the field of real numbers. Received: 9 June 1999 / Revised version: 20 February 2000     

Analysis of finite element methods for second order boundary value problems using mesh dependent norms

This paper presents a new approach to the analysis of finite element methods based onC ⁰-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL ₂ norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL ₂ norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.     

Norms of the singular integral operator with Cauchy Kernel along certain contours

The norm of the above-mentioned operatorS is computed on the unions of parallel lines or concentric circles. The upper bound is found for its norm on the ellipse. In case of weighted spaces on the unit circle, the exact norm is found for some rational weights, and necessary and sufficient conditions on the weight are established, under which the essential norm ofS equals 1.