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Abstract Let n ≥ 2 and let A ∈ L (C n) be such that m (A) > 0. In this paper, we use a variational result for A -normalized univalent subordination chains, to deduce that every normalized univalent mapping which has A -parametric representation on B n can be approximated locally uniformly on B n by mappings which have A -parametric representation on B n and admit extensions, on one hand, to automorphisms of C n and, on the other hand, to quasiconformal diffeomorphisms of class C ∞ from C n onto C n , which are equal to the identity mapping in a neighborhood of ∞. Finally, we obtain approximation results of A -spirallike, starlike, and convex mappings by automorphisms of C n and quasiconformal diffeomorphisms of class C ∞ from C n onto itself, which are equal to the identity mapping in a neighborhood of ∞ , whose restrictions to B n have the same geometric property. We remark that even though these extensions exist for the same mapping on B n , they are not necessarily equal on C n ∖ B n ¯. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight w ( x ) = x α e − x − t / x , x ∈ ( 0 , ∞ ) , t > 0 and α > 0 . When the matrix size n → ∞ , we obtain an asymptotic formula for the Hankel determinants, valid uniformly for t ∈ ( 0 , d ] , d > 0 fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large- n asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift–Zhou nonlinear steepest descent method. [ABSTRACT FROM AUTHOR]

Abstract: The context of the paper is: a locally compact Hausdorff space ; the space , equipped with the uniform norm, of real continuous functions on which vanish at infinity; a linear subspace , of finite dimension , of the space ; and the set-valued metric projection of into the family of non-empty compact convex subsets of , defined by  the set of best uniform approximations to from . Those which are Chebyshev (that is the metric projection is single point valued) were characterised by Haar (1918). Those for which the metric projection is lower semi-continuous have been characterised by Wu Li (1989) and A.L. Brown (2005); the paper interprets and exploits the characterisation. A ‘Generalised Haar Condition’ which is necessary for to be lower semi-continuous is identified; in some circumstances it is also sufficient. The condition has a calculable determinantal form. The results of the paper include an essentially complete determination, in case is a compact space in which no net of components is convergent to a single point set, of those subspaces of for which is lower semi-continuous. Simple examples show that if the space is compact and totally disconnected, but not finite, the situation is essentially different. [Copyright &y& Elsevier]

Abstract: In the present paper, we study conditions under which the metric projection of a polyhedral Banach space onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if satisfies (a geometric property stronger than polyhedrality) and is any proximinal subspace, then the metric projection is Hausdorff continuous and is strongly proximinal (i.e., if , and , then ). One of the main results of a different nature is the following: if satisfies and is a closed subspace of finite codimension, then the following conditions are equivalent: (a) is strongly proximinal; (b) is proximinal; (c) each element of attains its norm. Moreover, in this case the quotient is polyhedral. The final part of the paper contains examples illustrating the importance of some hypotheses in our main results. [Copyright &y& Elsevier]

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]

In 1987 the author gave an example of a non convex Chebyshev set S in the incomplete inner product space E consisting of the vectors in l 2 which have at most a finite number of non zero terms. In this paper, we show that the closure of S in the Hilbert space completion l 2 of E is not Chebyshev in l 2 . [ABSTRACT FROM AUTHOR]

We prove a few interesting inequalities for Lorentz polynomials. A highlight of this paper states that the Markov-type inequality max x ∈ [ − 1 , 1 ] | f ′ ( x ) | ≤ n max x ∈ [ − 1 , 1 ] | f ( x ) | holds for all polynomials f of degree at most n with real coefficients for which f ′ has all its zeros outside the open unit disk. Equality holds only for f ( x ) : = c ( ( 1 ± x ) n − 2 n − 1 ) with a constant 0 ≠ c ∈ R . This should be compared with Erdős’s classical result stating that max x ∈ [ − 1 , 1 ] | f ′ ( x ) | ≤ n 2 ( n n − 1 ) n − 1 max x ∈ [ − 1 , 1 ] | f ( x ) | for all polynomials f of degree at most n having all their zeros in R ∖ ( − 1 , 1 ) . [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: In both Lyubarskii and Madych (1994) [7], and Schlumprecht and Sivakumar (2009) [9], it was shown that Paley–Wiener functions may be recovered from their values on a complete interpolating sequence. This paper explores the same phenomenon, and gives a sufficient condition on a function , called an interpolator, so that scattered translates of this function may be used to interpolate and recover any given Paley–Wiener function. [Copyright &y& Elsevier]

Abstract: In this paper our aim is to deduce some Turán type inequalities for -hypergeometric and -confluent hypergeometric functions. In order to obtain the main results we apply the methods developed in the case of classical Kummer and Gauss hypergeometric functions. [Copyright &y& Elsevier]

Abstract: Bernstein inequalities and inverse theorems are a recent development in the theory of radial basis function (RBF) approximation. The purpose of this paper is to extend what is known by deriving Bernstein inequalities for RBF networks on . These inequalities involve bounding a Bessel-potential norm of an RBF network by its corresponding norm in terms of the separation radius associated with the network. The Bernstein inequalities will then be used to prove the corresponding inverse theorem. [Copyright &y& Elsevier]

Abstract: We give necessary and sufficient conditions on the Fourier transform of a generator function of a principal shift-invariant subspace of providing density order . Our starting point is a paper by de Boor, DeVore, and Ron [C. de Boor, R.A. DeVore, A. Ron, Approximation from shift-invariant subspaces of , Trans. Amer. Math. Soc. 341 (2) (1994) 787–806] and the new conditions that we present here involve the classical notion of the point of approximate continuity. In addition, we study properties of the approximation of a shift-invariant subspace of when it is dilated by a diagonalizable expansive linear map . Indeed, we present a necessary and sufficient condition on a principal shift-invariant subspace such that its union with itself dilated by integer powers of is dense in . [Copyright &y& Elsevier]

Abstract: Cartoon-like images, i.e., functions which are smooth apart from a discontinuity curve, have by now become a standard model for measuring sparse (nonlinear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit sparse approximations within this model, which are optimal up to a log-factor. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications. In this paper, we present the first complete proof of optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements. This class will be chosen as a subset of (non-tight) shearlet frames with shearlet generators having compact support and satisfying some weak directional vanishing moment conditions. [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 obtain a new strong type of Steckin inequality for the linear combinations of Bernstein–Kantorovich operators, which gives the optimal approximation rate. On the basis of this inequality, we further obtain the lower estimate for these operators. [Copyright &y& Elsevier]

Abstract: Let be a closed subset of the unit circle and let . We investigate the problem of uniform approximation of on by polynomials which are uniformly bounded on the unit disk . In a particular case when is a closed arc of , the problem was solved by L. Zalcman in 1982, who has also pointed out the possibility of considering more general approximation sets instead of an arc. The present paper gives a necessary and sufficient solution of the above problem. In fact we show that the (simple) description of given by Zalcman for the case of an arc also holds in the general case. [Copyright &y& Elsevier]

Abstract: This paper determines the exact value of the -term approximation of a diagonal linear operator from to , using an elementary method. [Copyright &y& Elsevier]

Abstract: We continue the study of the so-called thematic factorizations of admissible very badly approximable matrix functions. These factorizations were introduced by V.V. Peller and N.J. Young for studying superoptimal approximation by bounded analytic matrix functions. Even though thematic indices associated with a thematic factorization of an admissible very badly approximable matrix function are not uniquely determined by the function itself, R.B. Alexeev and V.V. Peller showed that the thematic indices of any monotone non-increasing thematic factorization of an admissible very badly approximable matrix function are uniquely determined. In this paper, we prove the existence of monotone non-decreasing thematic factorizations for admissible very badly approximable matrix functions. It is also shown that the thematic indices appearing in a monotone non-decreasing thematic factorization are not uniquely determined by the matrix function itself. Furthermore, we show that the monotone non-increasing thematic factorization gives rise to a great number of other thematic factorizations. [Copyright &y& Elsevier]

Abstract: We study multivariate approximation with the error measured in and weighted norms. We consider the worst case setting for a general reproducing kernel Hilbert space of functions of variables with a bounded or integrable kernel. Here can be arbitrarily large. We analyze algorithms that use standard information consisting of function values, and we are especially interested in the optimal order of convergence, i.e., in the maximal exponent for which the worst case error of such an algorithm is of order . We prove that for weighted approximation and for approximation, where is the optimal order of convergence for weighted approximation among all algorithms that may use arbitrary linear functionals, as opposed to function values only. Under a mild assumption on the reproducing kernels we have . It was shown in our previous paper that the optimal order for approximation and linear information is . We do not know if our bounds are sharp for standard information. We also study tractability of multivariate approximation, i.e., we analyze when the worst case error bounds depend at most polynomially on and . We present necessary and sufficient conditions on tractability and illustrate our results for the weighted Korobov spaces with arbitrary smoothness and for the weighted Sobolev spaces with the Wiener sheet kernel. Tractability conditions for these spaces are given in terms of the weights defining these spaces. [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]

Abstract: In the paper we prove that the maximal operator of the -means of cubical partial sums of -dimensional Walsh–Fourier series is of weak type (1,1). Moreover, the -means of the function converge a.e. to as . [Copyright &y& Elsevier]