Weighted Function Spaces Research Articles

Entropy numbers in weighted function spaces. The case of intermediate weights

The exact asymptotic behavior of the entropy numbers of compact embeddings of weighted Besov spaces is known in many cases, in particular for power-type weights and logarithmic weights. Here we consider intermediate weights that are strictly between these two scales; a typical example is \(w(x) = \exp \left( {\sqrt {\log (1 + |x|)} } \right)\). For such weights we prove almost optimal estimates of the entropy numbers ek(id: \(B_{p_2 q_1 }^{s_1 } (\mathbb{R}^d ,w) \to B_{p_2 q_2 }^{s_2 } (\mathbb{R}^d )\)).

Operators and Dynamical Systems on Weighted Function Spaces

Let X be a completely regular Hausdorff space, let V be a system of weights on X and let E be a locally convex Hausdorff space. Let CV0(X, E) and CVb(X, E) be the weighted locally convex spaces of vector-valued continuous functions with a topology generated by seminorms which are weighted analogue of the supremum norm. In the present paper, we characterize multiplication operators and weighted composition operators on the spaces CV0(X, E) and CVb(X, E) induced by scalar-valued and vector-valued mappings. A (linear) dynamical system on these weighted spaces is obtained as an application of the theory of multiplication operators.

Fundamental solutions and mapping properties of semielliptic operators

AbstractAn explicit formula is given for a fundamental solution for a class of semielliptic operators. The fundamental solution is used to investigate properties of these operators as mappings between weighted function spaces in ℝn . Necessary and sufficient conditions are given for such a mapping to be an isomorphism. Results apply, for example, to elliptic, parabolic, and generalized p ‐parabolic operators. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Good Lattice Rules in Weighted Korobov Spaces with General Weights

We study the problem of multivariate integration and the construction of good lattice rules in weighted Korobov spaces with general weights. These spaces are not necessarily tensor products of spaces of univariate functions. Sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found. These conditions are also necessary if the weights are such that the reproducing kernel of the weighted Korobov space is pointwise non-negative. The existence of a lattice rule which achieves the nearly optimal convergence order is proven. A component-by-component (CBC) algorithm that constructs good lattice rules is presented. The resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights. We also study special weights such as finite-order and order-dependent weights. For these special weights, the cost of the CBC algorithm is polynomial. Numerical computations show that the lattice rules constructed by the CBC algorithm give much smaller worst-case errors than the mean worst-case errors over all quasi-Monte Carlo rules or over all lattice rules, and generally smaller worst-case errors than the best Korobov lattice rules in dimensions up to hundreds. Numerical results are provided to illustrate the efficiency of CBC lattice rules and Korobov lattice rules (with suitably chosen weights), in particular for high-dimensional finance problems.

Constructing Embedded Lattice Rules for Multivariate Integration

Lattice rules are a family of equal-weight cubature formulae for approximating high-dimensional integrals. By now it is well established that good generating vectors for lattice rules having $n$ points can be constructed component-by-component for integrands belonging to certain weighted function spaces, and that they can achieve the optimal rate of convergence. Although the lattice rules constructed this way are extensible in dimension, they are not extensible in $n$; thus when $n$ is changed the generating vector needs to be constructed anew. In this paper we introduce a new algorithm for constructing good generating vectors for embedded lattice rules which can be used for a range of $n$ while still being extensible in dimension. By using an adaptation of the fast component-by-component construction algorithm (which makes use of fast Fourier transforms), we are able to obtain good generating vectors for thousands of dimensions and millions of points, under both product weight and order-dependent weight settings, at the cost of $\calO(d n (\log (n))^2)$ operations. With a sufficiently large number of points and good overall quality, these embedded lattice rules can be used for practical purposes in the same way as a low-discrepancy sequence. We show for a range of weight settings in an unanchored Sobolev space that our embedded lattice rules achieve the same (optimal) rate of convergence $\calO(n^{-1+\delta})$, $\delta>0$, as those constructed for a fixed number of points, and that the implied constant gains only a factor of 1.30 to 1.55.

Efficient Weighted Lattice Rules with Applications to Finance

Good lattice rules are an important type of quasi-Monte Carlo algorithm. They are known to have good theoretical properties, in the sense that they can achieve an error bound (or optimal error bound) that is independent of the dimension for weighted function spaces with suitably decaying weights. To use the theory of weighted function spaces for practical applications, one has to determine what weights should be used. After explaining that lattice rules based on figures of merit with classical weights (classical lattice rules) may not give good results for high-dimensional problems, we propose a "matching strategy," which chooses the weights in such a way that the typical functions of a weighted Korobov space are similar (in the sense of similar relative sensitivity indices) to those of the given function. The matching strategy relates the weights of the spaces to the sensitivity indices of the given function. We apply the Korobov construction and the component-by-component construction of lattice rules with the suitably chosen weights to the valuation of high-dimensional financial derivative securities and find that such weighted lattice rules improve dramatically on the results for the classical lattice rules; moreover, they are competitive with other types of low discrepancy sequences. We also demonstrate that lattice rules combined with variance reduction techniques and dimension reduction techniques increase the efficiency significantly. We showthat an acceptance-rejection approach to the construction of lattice rules can reduce the construction cost with almost no loss of accuracy.

On approximation numbers of Sobolev embeddings of weighted function spaces

We investigate asymptotic behaviour of approximation numbers of Sobolev embeddings between weighted function spaces of Sobolev-Hardy-Besov type with polynomials weights. The exact estimates are pro...

On approximation numbers of Sobolev embeddings of weighted function spaces

We investigate asymptotic behaviour of approximation numbers of Sobolev embeddings between weighted function spaces of Sobolev–Hardy–Besov type with polynomials weights. The exact estimates are proved in almost all cases.

Decay estimates for the damped wave equation with weighted odd initial data

In this paper, we consider the Cauchy problem to the damped wave equation. Several authors have shown the diffusive structure of the problem which has been represented by L p – L q estimates. We show the new L p – L q estimates to the problem when initial data are odd and belong to certain weighted function space. These L p – L q estimates imply the new decay estimates for the above Cauchy problem. Moreover, we apply the above decay estimates to the Cauchy problem for the semilinear wave equation with power nonlinearities lower than Fujita critical exponent. We show the existence of the time global solutions when the initial data are small and odd.

On the nonstationary linearized Navier–Stokes problem in domains with cylindrical outlets to infinity

The nonstationary linearized Navier–Stokes system is studied in the domain with cylindrical outlets to infinity in weighted function spaces with an exponential weight function. It is proved that under natural compatibility conditions there exists a unique solution with prescribed fluxes over the sections of outlets to infinity that exponentially tends in each outlet to the corresponding nonstationary Poiseuille flow.

Construction Algorithms for Digital Nets with Low Weighted Star Discrepancy

We introduce a new construction method for digital nets which yield point sets in the s-dimensional unit cube with low star discrepancy. The digital nets are constructed using polynomials over finite fields. It has long been known that there exist polynomials which yield point sets with low (unweighted) star discrepancy. This result was obtained by Niederreiter by the means of averaging over all polynomials. Hence concrete examples of good polynomials were not known in many cases. Here we show that good polynomials can be found by computer search. The search algorithm introduced in this paper is based on minimizing a quantity closely related to the star discrepancy. It has been pointed out that many integration problems can be modeled by weighted function spaces and it has been shown that in this case point sets with low weighted discrepancy are required. Hence it is particularly useful to be able to adjust a point set to some given weights. We are able to extend our results from the unweighted case to show that this can be done using our construction algorithms. This way we can find point sets with low weighted star discrepancy, making such point sets especially useful for many applications.

Wavelet bases and entropy numbers in weighted function spaces

AbstractThe aim of this paper is twofold. First we prove that inhomogeneous wavelets of Daubechies type are unconditional Schauder bases in weighted function spaces of Bspq and Fspq type. Secondly we use these results to estimate entropy numbers of compact embeddings between these spaces. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Cesàro summability of lagrange interpolation on Jacobi roots

The aim of this paper is to give such weighted function spaces in which the sequence of Cesàro means of Lagrange interpolatory polynomials on Jacobi roots are uniformly convergent.

On Korobov Lattice Rules in Weighted Spaces

This paper studies the error bounds of multivariate integration in weighted function spaces using lattice rules of the Korobov form, in which the generating vector for an n-point rule with n prime has the form (1,a,...,ad -1) (mod,n). With the parameter a chosen optimally, we establish new error bounds for Korobov lattice rules in weighted Korobov spaces. In particular, we prove that if the weights decay sufficiently fast, the optimal Korobov lattice rule has an error bound of order O(n^{-\alpha/2 +\delta})$ (for arbitrary $\delta \,{>}\, 0$), with the implied constant depending at worst polynomially on the dimension. Here $\alpha \,{>}\,1$ is the smoothness parameter of the weighted Korobov spaces. We generalize the construction to the case where n is a product of arbitrary distinct prime numbers, with the purpose of reducing the construction cost without sacrificing much of the quality of the lattice rules. A corresponding result is deduced for weighted Sobolev spaces of nonperiodic functions, using randomly shifted optimal Korobov lattice rules. A comparison of the worst-case errors for Korobov lattice rules and the recent component-by-component constructions is presented. The investigation establishes the usefulness of (shifted) optimal Korobov lattice rules for integration, even in high dimensions, if the weights which characterize the weighted spaces are suitably chosen.

Supercyclic and chaotic translation semigroups

We give a necessary and sufficient condition for a translation semigroup to be supercyclic and to be chaotic in a weighted function space.

Stability of Self-Similar Solutions of the Dafermos Regularization of a System of Conservation Laws

In contrast to a viscous regularization of a system of n conservation laws, a Dafermos regularization admits many self-similar solutions of the form \( u=u(\frac{X}{T})\). In particular, it is known in many cases that Riemann solutions of a system of conservation laws have nearby self-similar smooth solutions of an associated Dafermos regularization. We refer to these smooth solutions as {\em Riemann--Dafermos solutions}. In the coordinates $x=\frac{X}{T}$, $t=\ln T$, Riemann--Dafermos solutions become stationary, and their time-asymptotic stability as solutions of the Dafermos regularization can be studied by linearization. We study the stability of Riemann--Dafermos solutions near Riemann solutions consisting of n Lax shock waves. We show, by studying the essential spectrum of the linearized system in a weighted function space, that stability is determined by eigenvalues only. We then use asymptotic methods to study the eigenvalues and eigenfunctions. We find there are fast eigenvalues of order $\frac{1}{\epsilon}$ and slow eigenvalues of order 1. The fast eigenvalues correspond to eigenvalues of the viscous profiles for the individual shock waves in the Riemann solution; these have been studied by other authors using Evans function methods. The slow eigenvalues are related to inviscid stability conditions that have been obtained by various authors for the underlying Riemann solution.

On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications

Generalising an analysis of Corvino and Schoen, we study surjectivity properties of the constraint map in general relativity in a large class of weighted Sobolev spaces. As a corollary we prove several perturbation, gluing, and extension results: we show existence of non-trivial, singularityfree, vacuum space-times which are stationary in a neighborhood of i; for small perturbations of parity-covariant initial data sufficiently close to those for Minkowski space-time this leads to space-times with a smooth global I ; we prove existence of initial data for many black holes which are exactly Kerr – or exactly Schwarzschild – both near infinity and near each of the connected components of the apparent horizon; under appropriate conditions we obtain existence of vacuum extensions of vacuum initial data across compact boundaries; we show that for generic metrics the deformations in the Isenberg-Mazzeo-Pollack gluings can be localised, so that the initial data on the connected sum manifold coincide with the original ones except for a small neighborhood of the gluing region; we prove existence of asymptotically flat solutions which are static or stationary up to r−m terms, for any fixed m, and with multipole moments freely prescribable within certain ranges. ∗Partially supported by a Polish Research Committee grant 2 P03B 073 24; email piotr@ gargan.math.univ-tours.fr †Visiting Scientist. Permanent address: Departement de Mathematiques, Faculte des Sciences, Parc de Grandmont, F37200 Tours, France. ‡Partially supported by the ACI program of the French Ministry of Research; email delay@ gargan.math.univ-tours.fr

EXISTENCE AND DECAY OF WEAK SOLUTIONS TO DEGENERATE DAVEY-STEWARTSON EQUATIONS

In this paper, the authors consider a system of degenerate Davey-Stewartson equations. They prove the global existence of weak solutions in some weighted function spaces and the decay of weak solutions in some anisotropic spaces for appropriate initial data.

THE MATHEMATICAL MODEL OF COMPRESSIBLE FLUID FLOW

In this note we consider the mathematical model of the isothermal compressible fluid flow in an exterior domain O ⊂ R3. In order to solve this problem we apply a decomposition scheme and reduce the nonlinear problem to an operator equation with a contraction operator. After the decomposition the nonlinear problem splits into three linear problems: Neumann‐like problem, modified Stokes problem and transport equation. These linear problems are solved in weighted function spaces with detached asymptotics.

Multipliers on weighted function spaces over locally compact Vilenkin groups

In this note, we consider the multipliers on weighted function spaces over totally disconnected locally compact obelian groups (Vilenkin groups). Firstly we show an (H1, Lq) multiplier result. We also give an (H a p , H a p ) multiplier result under a similiar condition of Lu Yang type. In section 2, we obtain a result about the boundedness of multipliers on weighted Besov spaces.