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In this paper, we study the critical Sobolev embeddings W 1 , p ( ⋅ ) ( Ω ) ⊂ L p ⁎ ( ⋅ ) ( Ω ) for variable exponent Sobolev spaces from the point of view of the Γ-convergence. More precisely we determine the Γ-limit of subcritical approximation of the best constant associated with this embedding. As an application we provide a sufficient condition for the existence of extremals for the best constant. [ABSTRACT FROM AUTHOR]

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]

Padé approximation has two natural extensions to vector rational approximation through the so-called type I and type II Hermite–Padé approximants. The convergence properties of type II Hermite–Padé approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite–Padé approximants for Nikishin systems of functions. [ABSTRACT FROM AUTHOR]

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: The aim of this paper is to show how solutions to the one-dimensional compressible Euler equations can be approximated by solutions to an enlarged hyperbolic system with a strong relaxation term. The enlarged hyperbolic system is linearly degenerate and is therefore suitable to build an efficient approximate Riemann solver. From a theoretical point of view, the convergence of solutions to the enlarged system towards solutions to the Euler equations is proved for local in time smooth solutions. We also show that arbitrarily large shock waves for the Euler equations admit smooth shock profiles for the enlarged relaxation system. In the end, we illustrate these results of convergence by proposing a numerical procedure to solve the enlarged hyperbolic system. We test it on various cases. [Copyright &y& Elsevier]

Abstract: The homogeneous approximation property (HAP) for frames is useful in practice and has been developed recently. In this paper, we study the HAP for the continuous wavelet transform. We show that every pair of admissible wavelets possesses the HAP in sense, while it is not true in general whenever pointwise convergence is considered. We give necessary and sufficient conditions for the pointwise HAP to hold, which depends on both wavelets and functions to be reconstructed. [Copyright &y& Elsevier]

Abstract: This paper estimates upper and lower bounds for the approximation rates of iterated Boolean sums of multivariate Bernstein polynomials. Both direct and inverse inequalities for the approximation rate are established in terms of a certain -functional. From these estimates, one can also determine the class of functions yielding optimal approximations to the iterated Boolean sums. [Copyright &y& Elsevier]

Abstract: In the present paper, we investigate the convergence and the approximation order of the localized Szsz–Mirakjan operators, and obtain some new results to improve the results due to Omey [Note on operators of Szsz–Mirakjan type, J. Approx. Theory 47 (1986) 246–254]. [Copyright &y& Elsevier]