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Register a new userThe Marcinkiewicz-type discretization theorems for the hyperbolic cross polynomials
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V.N. Temlyakov
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The main goal of this paper is to study the discretization problem for the hyperbolic cross trigonometric polynomials. This important problem turns out to be very difficult. In this paper we begin a systematic study of this problem and demonstrate two different techniques -- the probabilistic and the number theoretical techniques.
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The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. This problem is very important in applications but there is no systematic study of it. We present here a new technique, which works well for discretization of the integral norm. It is a combination of probabilistic technique, based on chaining, with results on the entropy numbers in the uniform norm.
The paper studies the sampling discretization problem for integral norms on subspaces of $L^p(mu)$. Several close to optimal results are obtained on subspaces for which certain Nikolskii-type inequality is valid. The problem of norms discretization is connected with the probabilistic question about the approximation with high probability of marginals of a high dimensional random vector by sampling. As a byproduct of our approach we refine the result of O. Gu$acute{e}$don and M. Rudelson concerning the approximation of marginals. In particular, the obtained improvement recovers a theorem of J. Bourgain, J. Lindenstrauss, and V. Milman concerning embeddings of finite dimensional subspaces of $L^p[0, 1]$ into $ell_p^m$. The proofs in the paper use the recent developments of the chaining technique by R. van Handel.
In this paper we consider the numerical approximation of systems of Boussinesq-type to model surface wave propagation. Some theoretical properties of these systems (multi-symplectic and Hamiltonian formulations, well-posedness and existence of solitary-wave solutions) were previously analyzed by the authors in Part I. As a second part of the study, considered here is the construction of geometric schemes for the numerical integration. By using the method of lines, the geometric properties, based on the multi-symplectic and Hamiltonian structures, of different strategies for the spatial and time discretizations are discussed and illustrated.
Let $G$ be a finite union of disjoint and bounded Jordan domains in the complex plane, let $mathcal{K}$ be a compact subset of $G$ and consider the set $G^star$ obtained from $G$ by removing $mathcal{K}$; i.e., $G^star:=Gsetminus mathcal{K}$. We refer to $G$ as an archipelago and $G^star$ as an archipelago with lakes. Denote by ${p_n(G,z)}_{n=0}^infty$ and ${p_n(G^star,z)}_{n=0}^infty$, the sequences of the Bergman polynomials associated with $G$ and $G^star$, respectively; that is, the orthonormal polynomials with respect to the area measure on $G$ and $G^star$. The purpose of the paper is to show that $p_n(G,z)$ and $p_n(G^star,z)$ have comparable asymptotic properties, thereby demonstrating that the asymptotic properties of the Bergman polynomials for $G^star$ are determined by the boundary of $G$. As a consequence we can analyze certain asymptotic properties of $p_n(G^star,z)$ by using the corresponding results for $p_n(G,z)$, which were obtained in a recent work by B. Gustafsson, M. Putinar, and two of the present authors. The results lead to a reconstruction algorithm for recovering the shape of an archipelago with lakes from a partial set of its complex moments.
The main objective of this paper is to develop a general method of geometric discretization for infinite-dimensional systems and apply this method to the EPDiff equation. The method described below extends one developed by Pavlov et al. for incompressible Euler fluids. Here this method is presented in a general case applicable to all, not only divergence-free, vector fields. Also, a different (pseudospectral) representation of the velocity field is used. We will apply this method to the one-dimensional EPDiff equation and present numerical results.
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