Search

Abstract: The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets. In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well. [Copyright &y& Elsevier]

Abstract: In a previous paper, the author introduced a class of multivariate rational interpolants, which are called optimal Padé-type approximants (OPTA). The main goal of this paper is to extend classical results on convergence both in measure and in capacity of sequences of Padé approximants to the multivariate case using OPTA. To this end, we obtain some estimations of the size of multivariate polynomial lemniscates in terms of the Hausdorff content, which we also think are of some interest. [Copyright &y& Elsevier]

The Douglas-Rachford splitting algorithm is a classical optimization method that has found many applications. When specialized to two normal cone operators, it yields an algorithm for finding a point in the intersection of two convex sets. This method for solving feasibility problems has attracted a lot of attention due to its good performance even in nonconvex settings. In this paper, we consider the Douglas-Rachford algorithm for finding a point in the intersection of two subspaces. We prove that the method converges strongly to the projection of the starting point onto the intersection. Moreover, if the sum of the two subspaces is closed, then the convergence is linear with the rate being the cosine of the Friedrichs angle between the subspaces. Our results improve upon existing results in three ways: First, we identify the location of the limit and thus reveal the method as a best approximation algorithm; second, we quantify the rate of convergence, and third, we carry out our analysis in general (possibly infinite-dimensional) Hilbert space. We also provide various examples as well as a comparison with the classical method of alternating projections. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate -average sampling numbers of a periodic Sobolev space with a Gaussian measure in the metric for and , and obtain their asymptotic orders. Moreover, we show that in the average case setting, the operators , which are the Lagrange interpolating operators, are asymptotically optimal in the metric for all . It is interesting to note that in the worst case setting, are not asymptotically optimal algorithms in the metric for or . [Copyright &y& Elsevier]

Abstract: Regularization schemes with an -regularizer often produce sparse representations for objects in approximation theory, image processing, statistics and learning theory. In this paper, we study a kernel-based learning algorithm for regression generated by regularization schemes associated with the -regularizer. We show that convergence rates of the learning algorithm can be independent of the dimension of the input space of the regression problem when the kernel is smooth enough. This confirms the effectiveness of the learning algorithm. Our error analysis is carried out by means of an approximation theory approach using a local polynomial reproduction formula and the norming set condition. [Copyright &y& Elsevier]

Abstract: In this paper the two-dimensional -fraction with independent variables which is the generalization of the continued -fraction is considered. The correspondence between the formal double power series and the above mentioned fraction is studied. As a result the algorithm for the expansion of the formal double power series into the corresponding two-dimensional -fraction with independent variables has been constructed and the conditions for the existence of such an algorithm have been established. The expansion of some functions into the corresponding two-dimensional -fraction with independent variables is constructed and the efficiency of the obtained expansion by using approximants is shown. [Copyright &y& Elsevier]

Abstract: The aim of the paper is the construction and the analysis of nonlinear and non-separable multiscale representations for multivariate functions defined using a non-diagonal dilation matrix . We show that a function in or Besov spaces can be characterized by means of its multiscale representation. We also study the stability of these representations, a key issue to design adaptive algorithms. [Copyright &y& Elsevier]

Abstract: This paper concerns systems with small coherence parameter. Simple greedy-type algorithms perform well on these systems, which are also useful in the construction of compressed sensing matrices. We discuss the following problems for both and . How large can a dictionary be, if we prescribe the coherence parameter? How small could the resulting coherence parameter be, if we impose a size on the dictionary? How could we construct such a system? Several fundamental results from different areas of mathematics shed light on these important problems with far-reaching implications in approximation theory. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate nonlinear -term approximation with regard to orthogonal dictionaries. We consider this problem in the periodic multivariate case for generalized Besov classes under the condition where is a univariate function. We prove that the well-known dictionary which consists of trigonometric polynomials (shifts of the Dirichlet kernels) is nearly optimal among orthogonal dictionaries. Moreover, it is established that for these classes near-best -term approximation, with regard to , can be achieved by simple greedy-type algorithms. [Copyright &y& Elsevier]

Abstract: In this paper we deal with the approximation of a given function on by special bilinear forms via the so-called cross approximation. In particular we are interested in estimating the error function of the corresponding algorithm in the maximum norm. There is a large amount of publications available that successfully deal with similar matrix algorithms in applied situations, for example in connection with -matrices (see Boerm and Grasedyck (2003)  or Hackbusch (2007)  for many references). But as they do not give satisfactory error estimates, we concentrate on the theoretical issues of the problem in the language of functions. We connect it with related results from other areas of analysis in a historical survey and give a lot of references. Our main result is the connection of the error of our algorithm with the error of best approximation by arbitrary bilinear forms. This will be compared with the different approach in Bebendorf (2008) . [Copyright &y& Elsevier]