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As a generalization of cyclic codes, constacyclic codes is an important and interesting class of codes due to their nice algebraic structures and various applications in engineering. This paper is devoted to the study of the q -polynomial approach to constacyclic codes. Fundamental theory of this approach will be developed, and will be employed to construct some families of optimal and almost optimal codes in this paper. [ABSTRACT FROM AUTHOR]

Let f ∈ C [ − 1 , 1 ] and denote by E n ( f ) its degree of approximation by algebraic polynomials of degree < n . Assume that f changes its monotonicity, respectively, its convexity finitely many times, say s ≥ 2 times, in ( − 1 , 1 ) and we know that for q = 1 or q = 2 and some 1 < α ≤ 2 , such that q α ≠ 4 , we have E n ( f ) ≤ n − q α , n ≥ s + q + 1 . The purpose of this paper is to prove that the degree of comonotone, respectively, coconvex approximation, of f , by algebraic polynomials of degree < n , n ≥ N , is also ≤ c ( α , s ) n − q α , where the constant N depends only on the location of the extrema, respectively, inflection points in ( − 1 , 1 ) and on α . This answers, affirmatively, questions left open by the authors in papers with Kopotun (in Ukrainian Math. J.) and with Vlasiuk (see the list of references). [ABSTRACT FROM AUTHOR]

Abstract In this paper, we first determine the minimum possible size of an F q -linear set of rank k in PG (1 , q n). We obtain this result by relating it to the number of directions determined by a linearized polynomial whose domain is restricted to a subspace. We then use this result to find a lower bound on the number of points in an F q -linear set of rank k in PG (2 , q n). In the case k = n , this confirms a conjecture by Sziklai in [9]. [ABSTRACT FROM AUTHOR]

Zonotopal algebra is the study of a family of pairs of dual vector spaces of multivariate polynomials that can be associated with a list of vectors X . It connects objects from combinatorics, geometry, and approximation theory. The origin of zonotopal algebra is the pair ( D ( X ) , P ( X ) ) , where D ( X ) denotes the Dahmen–Micchelli space that is spanned by the local pieces of the box spline and P ( X ) is a space spanned by products of linear forms. The first main result of this paper is the construction of a canonical basis for D ( X ) . We show that it is dual to the canonical basis for P ( X ) that is already known. The second main result of this paper is the construction of a new family of zonotopal spaces that is far more general than the ones that were recently studied by Ardila–Postnikov, Holtz–Ron, Holtz–Ron–Xu, Li–Ron, and others. We call the underlying combinatorial structure of those spaces forward exchange matroid. A forward exchange matroid is an ordered matroid together with a subset of its set of bases that satisfies a weak version of the basis exchange axiom. [ABSTRACT FROM AUTHOR]

A tensor T , in a given tensor space, is said to be h -identifiable if it admits a unique decomposition as a sum of h rank one tensors. A criterion for h -identifiability is called effective if it is satisfied in a dense, open subset of the set of rank h tensors. In this paper we give effective h -identifiability criteria for a large class of tensors. We then improve these criteria for some symmetric tensors. For instance, this allows us to give a complete set of effective identifiability criteria for ternary quintic polynomials. Finally, we implement our identifiability algorithms in Macaulay2. [ABSTRACT FROM AUTHOR]

Modified de Bruijn sequences are created by removing a single zero from the longest run of zeros of de Bruijn sequences. There are few theoretical results on the minimal polynomial and linear complexity of modified de Bruijn sequences. Some preliminary results are presented in this paper. It shows that for the minimal polynomial of a modified de Bruijn sequence of order n there exists at least one irreducible factor of degree n . An equivalent condition on which a polynomial is the minimal polynomial of some modified de Bruijn sequence is derived, using the tool of rational fraction representation of periodic sequences. Based on the equivalent condition, the impossible linear complexity of modified de Bruijn sequences can be discussed. [ABSTRACT FROM AUTHOR]

Let ( λ n ) be a strictly increasing sequence of positive integers. Inspired by the notions of topological multiple recurrence and disjointness in dynamical systems, Costakis and Tsirivas have recently established that there exist power series ∑ k ≥ 0 a k z k with radius of convergence 1 such that the pairs of partial sums { ( ∑ k = 0 n a k z k , ∑ k = 0 λ n a k z k ) : n = 1 , 2 , … } approximate all pairs of polynomials uniformly on compact subsets K ⊂ { z ∈ C : | z | ≥ 1 } , with connected complement, if and only if lim sup n λ n n = + ∞ . In the present paper, we give a new proof of this statement avoiding the use of tools of potential theory. It allows to study the case of doubly universal infinitely differentiable functions. We obtain also the algebraic genericity of the set of such power series. Further we show that the Cesàro means of partial sums of power series with radius of convergence 1 cannot be frequently universal. [ABSTRACT FROM AUTHOR]

Permutation polynomials having the form ( x 2 i + x + δ ) s 1 + ( x 2 i + x + δ ) s 2 + x over finite fields of even characteristic are investigated in this paper. Eight classes of such polynomials are proved to be permutations. [ABSTRACT FROM AUTHOR]

Abstract: This is a follow-up paper of “Liberating the dimension for function approximation”, where we studied approximation of infinitely variate functions by algorithms that use linear information consisting of finitely many linear functionals. In this paper, we study similar approximation problems, however, now the algorithms can only use standard information consisting of finitely many function values. We assume that the cost of one function value depends on the number of active variables. We focus on polynomial tractability, and occasionally also study weak tractability. We present non-constructive and constructive results. Non-constructive results are based on known relations between linear and standard information for finitely variate functions, whereas constructive results are based on Smolyak’s construction generalized to the case of infinitely variate functions. Surprisingly, for many cases, the results for standard information are roughly the same as for linear information. [Copyright &y& Elsevier]

Abstract: We say that the polynomial sequence is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product where is a semiclassical linear functional, is the differential, the difference or the -difference operator, and is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional . The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time. [ABSTRACT FROM AUTHOR]

Abstract: We introduce in this paper a method to calculate the Hessenberg matrix of a sum of measures from the Hessenberg matrices of the component measures. Our method extends the spectral techniques used by G. Mantica to calculate the Jacobi matrix associated with a sum of measures from the Jacobi matrices of each of the measures. We apply this method to approximate the Hessenberg matrix associated with a self-similar measure and compare it with the result obtained by a former method for self-similar measures which uses a fixed point theorem for moment matrices. Results are given for a series of classical examples of self-similar measures. Finally, we also apply the method introduced in this paper to some examples of sums of (not self-similar) measures obtaining the exact value of the sections of the Hessenberg matrix. [ABSTRACT FROM AUTHOR]

Abstract: We present some general properties of the field of constants of monomial derivations of , where k is a field of characteristic zero. The main result of this paper is a description of all monomial derivations of with trivial field of constants. In this description a crucial role plays the classification result of Moulin Ollagnier for Lotka–Volterra derivations with strict Darboux polynomials. Several applications of our description are also given in this paper. [Copyright &y& Elsevier]

Abstract: This paper treats in detail the life and work of Otto Blumenthal, one of the most tragic figures of the 188 emigré mathematicians from Germany and the Nazi-occupied continent. Blumenthal, the first doctoral student of David Hilbert, was crucial in the publication and communication system of German mathematics between the two World Wars. There has been an unusual revival of interest in his mathematical work in the last three decades. Thus his work on orthogonal polynomials whose zeros are dense in intervals, called the Blumenthal theorem by T.S. Chihara (1972), lead to over two dozen recent papers in the field. The Blumenthal–Nevai theorem, with applications to scattering theory in physics, is one example. In modern work on Hilbert modular forms, increasingly being called Hilbert–Blumenthal modular forms, many recent papers even contain the word Blumenthal in their titles. This paper contains 212 references. [Copyright &y& Elsevier]

Abstract: The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching influence in approximation theory since publication of their paper in 1973. In this paper, we further develop their idea and investigate quasi-projection operators. We give sharp estimates in terms of moduli of smoothness for approximation with scaled shift-invariant spaces by means of quasi-projection operators. In particular, we provide error analysis for approximation of quasi-projection operators with Lipschitz spaces. The study of quasi-projection operators has many applications to various areas related to approximation theory and wavelet analysis. [Copyright &y& Elsevier]

An error in Corollary 2.7 of Ostafe and Sha (2015) [1] is corrected, which also triggers several changes in some of the main results of the paper. [ABSTRACT FROM AUTHOR]

Several properties of stationary subdivision schemes are nowadays well understood. In particular, it is known that the polynomial generation and reproduction capability of a stationary subdivision scheme is strongly connected with sum rules, its convergence, smoothness and approximation order. The aim of this paper is to show that, in the non-stationary case, exponential polynomials and approximate sum rules play an analogous role of polynomials and sum rules in the stationary case. Indeed, in the non-stationary univariate case we are able to show the following important facts: (i) reproduction of N exponential polynomials implies approximate sum rules of order N ; (ii) generation of N exponential polynomials implies approximate sum rules of order N , under the additional assumption of asymptotical similarity and reproduction of one exponential polynomial; (iii) reproduction of an N -dimensional space of exponential polynomials and asymptotical similarity imply approximation order N ; (iv) the sequence of basic limit functions of a non-stationary scheme reproducing one exponential polynomial converges uniformly to the basic limit function of the asymptotically similar stationary scheme. [ABSTRACT FROM AUTHOR]

The D -invariant polynomial subspaces play a crucial role in ideal interpolation. In this paper, we analyze the structure of a second-degree D -invariant polynomial subspace P 2 . As an application for ideal interpolation, we solve the discrete approximation problem for δ z P 2 ( D ) under certain conditions, i.e., we compute pairwise distinct points, such that the limiting space of the evaluation functionals at these points is the given space δ z P 2 ( D ) , as the evaluation sites all coalesce at one site z . [ABSTRACT FROM AUTHOR]

We study approximation properties of sequences of centered random elements X d , d ∈ N , with values in separable Hilbert spaces. We focus on sequences of tensor product-type random elements, which have covariance operators of corresponding tensor product form. The average case approximation complexity n X d ( ε ) is defined as the minimal number of evaluations of arbitrary linear functionals that is needed to approximate X d with relative 2 -average error not exceeding a given threshold ε ∈ ( 0 , 1 ) . The growth of n X d ( ε ) as a function of ε − 1 and d determines whether a sequence of corresponding approximation problems for X d , d ∈ N , is tractable or not. Different types of tractability were studied in the paper by Lifshits et al. (J. Complexity, 2012), where for each type the necessary and sufficient conditions were found in terms of the eigenvalues of the marginal covariance operators. We revise the criterion of quasi-polynomial tractability and provide a simplified version. We illustrate our result by applying it to random elements corresponding to tensor products of squared exponential kernels. We also extend a recent result of Xu (2014) concerning weighted Korobov kernels. [ABSTRACT FROM AUTHOR]

In this paper, by the exponential complete Bell polynomials, we establish two general approximations for the gamma function, where the coefficients in the series of the approximations can be determined by recurrences. These two general approximations include as special cases some well-known results such as those under the names De Moivre, Gosper, Gosper–Smith, Laplace, Luschny, Ramanujan and Wehmeier, and some recently published results such as those due to Batir, Chen, Lu, Mortici, Nemes, Paris, et al. By these two general approximations, we give unified approaches to dealing with many known approximations of the gamma function and to establishing new ones. [ABSTRACT FROM AUTHOR]

In this paper the problem of density in the space C ( X ) , for a compact set X ⊂ C , of polynomial modules of the type { p + z ¯ d q : p , q ∈ C [ z ] } for integer d > 1 , as well as several related problems are studied. We obtain approximability criteria for Carathéodory compact sets using the concept of a d -Nevanlinna domain, which is a new special analytic characteristic of planar simply connected domains. In connection with this concept we study the problem of taking roots in the model spaces, that is, in the subspaces of the Hardy space H 2 which are invariant under the backward shift operator. [ABSTRACT FROM AUTHOR]

Let E be a compact set in C with connected complement, and let A ( E ) be the class of all complex continuous function on E that are analytic in the interior E 0 of E . Let f ∈ A ( E ) be zero free on E 0 . By Mergelyan’s theorem f can be uniformly approximated on E by polynomials, but is it possible to realize such approximation by polynomials that are zero-free on E ? This natural question has been proposed by J. Andersson and P. Gauthier. So far it has been settled for some particular sets E . The present paper describes classes of functions for which zero free approximation is possible on an arbitrary E . [ABSTRACT FROM AUTHOR]

Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schönhage in 1982. The main drawbacks of Pan's method are that it is quite involved 2 2 In Victor Pan's own words: “Our algorithms are quite involved, and their implementation would require a non-trivial work, incorporating numerous known implementation techniques and tricks”. In fact, we are not aware of any implementation of Pan's method. and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity. In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted A New Dsc , which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. A New Dsc can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task. [ABSTRACT FROM AUTHOR]

In the paper, the authors find two closed forms involving the Stirling numbers of the second kind and in terms of a determinant of combinatorial numbers for the Bernoulli polynomials and numbers. [ABSTRACT FROM AUTHOR]

In this paper, we study some special subfields of C called Mahler fields. These fields are generated over Q by a set of Mahler's U -numbers having approximation in a fixed algebraic number field. We completely classify their finite extensions. We provide a necessary condition for the non-zero polynomial image of a U m -number is a U m -number. Using this result, we give another proof of the fact that the set of U m -numbers are non-empty for each m ≥ 1 . The famous Schanuel's conjecture states that, for any Q -linearly independent complex numbers ξ 1 , … , ξ n , the transcendence degree of the field Q ( ξ 1 , … , ξ n ) over Q is at least n . Here, we prove that for any Q -linearly independent complex numbers ξ 1 , … , ξ n , there exist uncountably many U -numbers c such that the transcendence degree of the field Q ( c ξ 1 , … , c ξ n , e c ξ 1 , … , e c ξ n ) over Q is at least n . [ABSTRACT FROM AUTHOR]

In this paper, we present algorithms for the approximation of multivariate periodic functions by trigonometric polynomials. The approximation is based on sampling of multivariate functions on rank-1 lattices. To this end, we study the approximation of periodic functions of a certain smoothness. Our considerations include functions from periodic Sobolev spaces of generalized mixed smoothness. Recently an algorithm for the trigonometric interpolation on generalized sparse grids for this class of functions was investigated by Griebel and Hamaekers (2014). The main advantage of our method is that the algorithm is based mainly on a single one-dimensional fast Fourier transform, and that the arithmetic complexity of the algorithm depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. Therefore, we investigate trigonometric polynomials with frequencies supported on hyperbolic crosses and energy norm based hyperbolic crosses in more detail. Furthermore, we present an algorithm for sampling multivariate functions on perturbed rank-1 lattices and show the numerical stability of the suggested method. Numerical results are presented up to dimension d = 10 , which confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we present error estimates for the approximation of multivariate periodic functions in periodic Hilbert spaces of isotropic and dominating mixed smoothness by trigonometric polynomials. The approximation is based on sampling of the multivariate functions on rank-1 lattices. We use reconstructing rank-1 lattices with generating vectors of Korobov form for the sampling and generalize the technique from Temlyakov (1986), in order to show that the aliasing error of that approximation is of the same order as the error of the approximation using the partial sum of the Fourier series. The main advantage of our method is that the computation of the Fourier coefficients of such a trigonometric polynomial, which we use as approximant, is based mainly on a one-dimensional fast Fourier transform, cf. Kämmerer et al. (2013), Kämmerer (2014). This means that the arithmetic complexity of the computation depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. Numerical results are presented up to dimension d = 10 . [ABSTRACT FROM AUTHOR]

This paper examines the asymptotic behaviour of entropy and approximation numbers of compact embeddings of weighted Sobolev spaces into Lebesgue spaces. We consider admissible weights multiplied by a polynomial. The main results are related to the distribution of eigenvalues of some degenerate elliptic operators. [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]

Abstract: In this paper, we introduce a variant of the Descartes method to isolate the real roots of a square-free polynomial with arbitrary real coefficients. It is assumed that each coefficient of F can be approximated to any specified error bound. Our algorithm uses approximate arithmetic only, nevertheless, it is certified, complete and deterministic. We further provide a bound on the complexity of our method which exclusively depends on the geometry of the roots and not on the complexity of the coefficients of F. For the special case, where F is a polynomial of degree n with integer coefficients of maximal bitsize τ, our bound on the bit complexity writes as . Compared to the complexity of the classical Descartes method from Collins and Akritas (based on ideas dating back to Vincent), which uses exact rational arithmetic, this constitutes an improvement by a factor of n. The improvement mainly stems from the fact that the maximal precision that is needed for isolating the roots of F is by a factor n lower than the precision needed when using exact arithmetic. [Copyright &y& Elsevier]

Text: We consider the analogue, when Z is replaced with Fq[T] of the elementary cyclotomic polynomials and prove an analogue of Suzuki's Theorem. Video: For a video summary of this paper, please visit http://youtu.be/mVBw82N34NE. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we construct an embedding of holomorphic functions in two complex variables into the unit ball in . This leads to a closed subspace of the -functions on the unit sphere spanned by quaternionic polynomials for which we construct orthonormal bases and study the related convergence properties. [Copyright &y& Elsevier]

Abstract: In this paper we study the convergence of the famous Weierstrass method for simultaneous approximation of polynomial zeros over a complete normed field. We present a new semilocal convergence theorem for the Weierstrass method under a new type of initial conditions. Our result is obtained by combining ideas of Weierstrass (1891) and Proinov (2010). A priori and a posteriori error estimates are also provided under the new initial conditions. [Copyright &y& Elsevier]

Abstract: In this paper we introduce and study a new class of elliptic second-order differential operators on a convex compact subset K of , , which are associated with a Markov operator T on and which degenerate on a suitable subset of K containing its extreme points. Among other things, we show that the closures of these operators generate Markov semigroups. Moreover, we prove that these semigroups can be approximated by means of iterates of suitable positive linear operators, which are referred to as the Bernstein–Schnabl operators associated with T. As a consequence we show that the semigroups preserve polynomials of a given degree as well as Hölder continuity, which gives rise to some spatial regularity properties of the solutions of the relevant evolution equations. [Copyright &y& Elsevier]

Abstract: In this paper, the expression of Weierstrass operators as differential operators on polynomials is used for the construction of associated quasi-interpolants. Then the convergence properties of these operators are studied. [Copyright &y& Elsevier]

Abstract: In this paper, we prove an extension of Mahler’s theorem on interpolation series, a celebrated result of -adic analysis. Mahler’s original result states that a function from to is uniformly continuous for the -adic metric if and only if it can be uniformly approximated by polynomial functions. We prove the same result for functions from a free monoid to , where is replaced by the pro- metric, the profinite metric on defined by -groups. [Copyright &y& Elsevier]

Abstract: In this paper, we present an algebraic perspective of the de Rham transform of a binary subdivision scheme and propose an elegant strategy for constructing dual -ary approximating subdivision schemes of de Rham-type, starting from two primal schemes of arity and 2, respectively. On the one hand, this new strategy allows us to show that several existing dual corner-cutting subdivision schemes fit into a unified framework. On the other hand, the proposed strategy provides a straightforward algorithm for constructing new dual subdivision schemes having higher smoothness and higher polynomial reproduction capabilities with respect to the two given primal schemes. [Copyright &y& Elsevier]

Abstract: In this paper, an algorithm is given for determining the topology of an algebraic space curve and to compute a certified rational parametric approximation of the algebraic space curve. The algorithm works by extending to dimension one the local generic position method for solving zero-dimensional polynomial equation systems. Here, certified means that the approximation curve and the original curve have the same topology and their Hausdorff distance is smaller than a given precision. The main advantage of the algorithm, inherited from the local generic position method, is that the topology computation and approximation for a space curve are directly reduced to the same tasks for two plane curves. In particular, an error bound of the approximation space curve is deduced explicitly from the error bounds of the approximation plane curves. The complexity of the algorithm is also analyzed. Its effectivity is shown on some non-trivial examples. [Copyright &y& Elsevier]

Abstract: The and its modifications have been proposed for selecting the degree in a polynomial growth curve model under a large-sample framework when the sample size is large, but the dimension is fixed. In this paper, first we propose a high-dimensional (denoted by ) which is an asymptotic unbiased estimator of the -type risk function defined by the expected log-predictive likelihood or equivalently the Kullback–Leibler information, under a high-dimensional framework such that . It is noted that our new criterion gives an estimator with small biases in a wide range of and . Next we derive asymptotic distributions of and under the high-dimensional framework. Through a Monte Carlo simulation, we note that these new approximations are more accurate than the approximations based on a large-sample framework. [Copyright &y& Elsevier]

Abstract: In computing approximate Gröbner bases, it is not easy to trace precision losses of floating-point coefficients of intermediate approximate polynomials. The measured precision losses are usually much larger than their genuine values. One reason causing this phenomenon is that most existing methods for tracing precision losses do not consider the dependence of such precision losses in any polynomial (as an equation). In this paper, we define an algebraic structure called PL-space (precision loss space) for a polynomial (as an equation) and set up a theory for it. We prove that any PL-space has a finite weak basis and a strong basis and show how they effect on tracing precision losses by an example. Based on the study of minimal strong bases, we propose the concept of dependence number which reveals the complexity of the dependence of precision losses in a polynomial. [Copyright &y& Elsevier]

Abstract: The computation of approximate Gröbner bases is reported to be highly unstable in the literature. Selecting a suitable length of floats is helpful for stabilizing this computation. In this paper, we present a method to compute such a suitable length of floats. We concentrate on a family of polynomial systems sharing the same support and study the relation between the lengths of floats and the coefficients of relevant polynomials. Then we give a reference length of floats for all the polynomial systems in the family. One feature of our method is that it need not utilize numerical algorithms of Gröbner bases. Hence, our method can avoid the influence of the instabilities of the existing numerical algorithms and thus will be helpful for designing stable ones (e.g., stable Shirayanagiʼs algorithm). Experiments show that our method can work out reliable and reasonably large lengths of floats for most of the tested benchmarks. [Copyright &y& Elsevier]

Abstract: The paper gives a partial answer to a question posed by Johan Andersson and Paul Gauthier. Namely, we prove that every function continuous on a locally connected compact set with connected complement, analytic in its interior and not vanishing in the interior can be approximated by polynomials with zeros outside of . [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: Let be a separable polynomial over a local field. The Montes algorithm computes certain approximations to the different irreducible factors of , with strong arithmetic properties. In this paper, we develop an algorithm to improve any one of these approximations, till a prescribed precision is attained. The most natural application of this “single-factor lifting” routine is to combine it with the Montes algorithm to provide a fast polynomial factorization algorithm. Moreover, the single-factor lifting algorithm may be applied as well to accelerate the computational resolution of several global arithmetic problems in which the improvement of an approximation to a single local irreducible factor of a polynomial is required. [Copyright &y& Elsevier]

Abstract: The manner in which two random variables influence one another often depends on covariates. A way to model this dependence is via a conditional copula function. This paper contributes to the study of semiparametric estimation of conditional copulas by starting from a parametric copula function in which the parameter varies with a covariate, and leaving the marginals unspecified. Consequently, the unknown parts in the model are the parameter function and the unknown marginals. The authors use a local pseudo-likelihood with nonparametrically estimated marginals approximating the unknown parameter function locally by a polynomial. Under this general setting, they prove the consistency of the estimators of the parameter function as well as its derivatives; they also establish asymptotic normality. Furthermore, they derive an expression for the theoretical optimal bandwidth and discuss practical bandwidth selection. They illustrate the performance of the estimation procedure with data-driven bandwidth selection via a simulation study and a real-data case. [Copyright &y& Elsevier]

Abstract: The notion of p-compact sets arises naturally from Grothendieckʼs characterization of compact sets as those contained in the convex hull of a norm null sequence. The definition, due to Sinha and Karn (2002), leads to the concepts of p-approximation property and p-compact operators (which form an ideal with its ideal norm ). This paper examines the interaction between the p-approximation property and certain space of holomorphic functions, the p-compact analytic functions. In order to understand these functions we define a p-compact radius of convergence which allows us to give a characterization of the functions in the class. We show that p-compact holomorphic functions behave more like nuclear than compact maps. We use the ϵ-product of Schwartz, to characterize the p-approximation property of a Banach space in terms of p-compact homogeneous polynomials and in terms of p-compact holomorphic functions with range on the space. Finally, we show that p-compact holomorphic functions fit into the framework of holomorphy types which allows us to inspect the -approximation property. Our approach also allows us to solve several questions posed by Aron, Maestre and Rueda (2010). [Copyright &y& Elsevier]

Abstract: This paper is devoted to studying the approximation of multivariate periodic functions in the average case setting. We equip the space of multivariate periodic functions with a Gaussian measure μ such that its Cameron–Martin space is the anisotropic multivariate periodic space. With respect to this Gaussian measure, we discuss the best approximation of functions by trigonometric polynomials with harmonics from parallelepipeds and the approximation by the corresponding anisotropic Fourier partial summation operators and Vallée-Poussin operators, and get the average error estimation. We shall show that, in the average case setting, with the average being with respect to this Gaussian measure μ, the anisotropic trigonometric polynomial subspaces are order optimal in the metric for , and the anisotropic Fourier partial summation operators and Vallée-Poussin operators are the order optimal linear operators, which are as good as optimal nonlinear operators in the metric for . [Copyright &y& Elsevier]

Abstract: Symbolic numeric algorithms for polynomials are very important, especially for practical computations since we have to operate with empirical polynomials having numerical errors on their coefficients. Recently, for those polynomials, a number of algorithms have been introduced, such as approximate univariate GCD and approximate multivariate factorization for example. However, for polynomials over integers having coefficients rounded from empirical data, changing their coefficients over reals does not remain them in the polynomial ring over integers; hence we need several approximate operations over integers. In this paper, we discuss computing a polynomial GCD of univariate or multivariate polynomials over integers approximately. Here, “approximately” means that we compute a polynomial GCD over integers by changing their coefficients slightly over integers so that the input polynomials still remain over integers. [Copyright &y& Elsevier]

Abstract: We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in with . Here is a -parallelepiped in with sides parallel to the coordinate axes. We consider the error of best approximation of a function by algebraic polynomials of fixed degree at most in variable , and relate it to a so-called total mixed modulus of smoothness appropriate to characterizing the convergence rate of the approximation error. This theorem is derived from a Johnen type theorem on equivalence between a certain K-functional and the total mixed modulus of smoothness which is proved in the present paper. [Copyright &y& Elsevier]

Abstract: The four-point interpolatory subdivision scheme of Dubuc and its generalizations to irregularly spaced data studied by Warren and by Daubechies, Guskov, and Sweldens are based on fitting cubic polynomials locally. In this paper, we analyze the convergence of the scheme by viewing the limit function as the limit of piecewise cubic functions arising from the scheme. This allows us to recover the regularity results of Daubechies et al. in a simpler way and to obtain the approximation order of the scheme and its first derivative. [Copyright &y& Elsevier]

Abstract: A tournament is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance on vertices. In fact, given any fixed , the kernelized instance has at most vertices. Our result improves the previous known bound of on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST. [Copyright &y& Elsevier]