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Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a foundation of current science. In this paper, we find a new integral representation of the Whittaker function of the first kind and show a relevant summation formula for Kummer's confluent hypergeometric functions. We also perform the specifications of our identities to link to known and new results. [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: We study sharp estimates of integral functionals for operators on the set of real trigonometric polynomials of degree in terms of the uniform norm of the polynomials and similar questions for algebraic polynomials on the unit circle of the complex plane. P. Erdös, A.P. Calderon, G. Klein, L.V. Taikov, and others investigated such inequalities. In this paper, we, in particular, show that the sharp inequality holds on the set for the Weyl fractional derivatives of order for . For (), this fact was proved by Lizorkin (1965) [12]. For and positive integer , the inequality was proved by Taikov (1965) [23]; however, in this case, the inequality follows from results of an earlier paper by Calderon and Klein (1951) [6]. [Copyright &y& Elsevier]

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]

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]

Abstract: In the paper [Y. Xu, Lagrange interpolation on Chebyshev points of two variables, J. Approx. Theory 87 (1996) 220–238], the author introduced a set of Chebyshev-like points for polynomial interpolation (by a certain subspace of polynomials) in the square , and derived a compact form of the corresponding Lagrange interpolation formula. In [L. Bos, M. Caliari, S. De Marchi, M. Vianello, A numerical study of the Xu polynomial interpolation formula in two variables, Computing 76(3–4) (2005) 311–324], we gave an efficient implementation of the Xu interpolation formula and we studied numerically its Lebesgue constant, giving evidence that it grows like , n being the degree. The aim of the present paper is to provide an analytic proof to show that the Lebesgue constant does have this order of growth. [Copyright &y& Elsevier]

One of the most celebrated problems in dyadic harmonic analysis is the pointwise convergence of the Fejér (or (C,1)) means of functions on unbounded Vilenkin groups. There was no known positive result before the author''s paper appeared in 1999 (J. Approx. Theory 101(1) (1999) 1) with respect to the a.e. convergence of the one-dimensional (C,1) means of Lp (p>1) functions. This paper is concerned with the almost everywhere convergence of a subsequence of the two-dimensional Fejér means of functions in L log+ L. Namely, we prove the a.e. relation limn,k→∞ σMn,&Mtilde;kf=f (for the indices the condition |n-k|<α is provided, where α>0 is some constant). [Copyright &y& Elsevier]

In this paper, we investigate entropy numbers of multiplier operators Λ = { λ k } k ∈ Z d and Λ ∗ = { λ k ∗ } k ∈ Z d , Λ , Λ ∗ : L p ( T d ) → L q ( T d ) on the d -dimensional torus T d , where λ k = λ ( | k | ) and λ k ∗ = λ ( | k | ∗ ) for a function λ defined on the interval [ 0 , ∞ ) , with | k | = k 1 2 + ⋯ + k d 2 1 ∕ 2 and | k | ∗ = max 1 ≤ j ≤ d | k j | . In the first part, upper and lower bounds are established for entropy numbers of general multiplier operators. In the second part, we apply these results to the specific multiplier operators Λ ( 1 ) = | k | − γ ( ln | k | ) − ξ k ∈ Z d , Λ ∗ ( 1 ) = | k | ∗ − γ ( ln | k | ∗ ) − ξ k ∈ Z d , Λ ( 2 ) = e − γ | k | r k ∈ Z d and Λ ∗ ( 2 ) = e − γ | k | ∗ r k ∈ Z d for γ > 0 , 0 < r ≤ 1 and ξ ≥ 0 . We have that Λ ( 1 ) U p and Λ ∗ ( 1 ) U p are sets of finitely differentiable functions on T d , in particular, Λ ( 1 ) U p and Λ ∗ ( 1 ) U p are Sobolev-type classes if ξ = 0 , and Λ ( 2 ) U p and Λ ∗ ( 2 ) U p are sets of infinitely differentiable ( 0 < r < 1 ) or analytic ( r = 1 ) functions on T d , where U p denotes the closed unit ball of L p ( T d ) . In particular, we prove that, the estimates for the entropy numbers e n ( Λ ( 1 ) U p , L q ( T d ) ) , e n ( Λ ∗ ( 1 ) U p , L q ( T d ) ) , e n ( Λ ( 2 ) U p , L q ( T d ) ) and e n ( Λ ∗ ( 2 ) U p , L q ( T d ) ) are order sharp in various important situations. [ABSTRACT FROM AUTHOR]

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]

Let { p k } k = 0 ∞ be the orthogonal polynomials with certain exponential weights. In this paper, we prove that under certain mild conditions on exponential weights class, a series of the form ∑ b k p k converges uniformly and absolutely on compact subsets of an open strip in the complex plane, and diverges at every point outside the closure of this strip. [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]

Let ω φ k ( f , δ ) w , L q be the Ditzian–Totik modulus with weight w , M k be the cone of k -monotone functions on ( − 1 , 1 ) , i.e., those functions whose k th divided differences are nonnegative for all selections of k + 1 distinct points in ( − 1 , 1 ) , and denote E ( X , P n ) w , q : = sup f ∈ X inf P ∈ P n ‖ w ( f − P ) ‖ L q , where P n is the set of algebraic polynomials of degree at most n . Additionally, let w α , β ( x ) : = ( 1 + x ) α ( 1 − x ) β be the classical Jacobi weight, and denote by S p α , β the class of all functions such that ‖ w α , β f ‖ L p = 1 . In this paper, we determine the exact behavior (in terms of δ ) of sup f ∈ S p α , β ∩ M k ω φ k ( f , δ ) w α , β , L q for 1 ≤ p , q ≤ ∞ (the interesting case being q < p as expected) and α , β > − 1 / p (if p < ∞ ) or α , β ≥ 0 (if p = ∞ ). It is interesting to note that, in one case, the behavior is different for α = β = 0 and for ( α , β ) ≠ ( 0 , 0 ) . Several applications are given. For example, we determine the exact (in some sense) behavior of E ( M k ∩ S p α , β , P n ) w α , β , L q for α , β ≥ 0 . [ABSTRACT FROM AUTHOR]

The paper is concerned with orders of the Kolmogorov, Gelfand and linear widths of function classes on sets with tree-like structure. As examples we consider weighted Sobolev classes on a John domain and some function classes on trees. [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]

In this paper we show that any positive definite matrix V with measurable entries can be written as V=UΛU*, where the matrix Λ is diagonal, the matrix U is unitary, and the entries of U and Λ are measurable functions (U* denotes the transpose conjugate of U). This result allows to obtain results about the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which products of derivatives of different order appear. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials. [ABSTRACT FROM AUTHOR]

In any quasi-metric space of homogeneous type, Auscher and Hytönen recently gave a construction of orthonormal wavelets with Hölder-continuity exponent η>0. However, even in a metric space, their exponent is in general quite small. In this paper, we show that the Hölder-exponent can be taken arbitrarily close to 1 in a metric space. We do so by revisiting and improving the underlying construction of random dyadic cubes, which also has other applications. [ABSTRACT FROM AUTHOR]

Abstract: In the paper, we obtain a discrete universality theorem on approximation of analytic functions by discrete shifts of the Hurwitz zeta-function under a new hypothesis on the parameter generalizing the transcendence condition. [Copyright &y& Elsevier]

Abstract: In this paper two important classes of orthogonal polynomials in higher dimensions using the framework of Clifford analysis are considered, namely the Clifford–Hermite and the Clifford–Gegenbauer polynomials. For both classes an explicit generating function is obtained. [Copyright &y& Elsevier]

Abstract: In this paper, the authors give a sufficient condition, on the parameters , , , and , which ensures that the embeddings of radial and subradial subspaces of Besov-type spaces and into Morrey spaces are compact. As corollaries, some sufficient conditions for the compactness of the embeddings from the radial and the subradial subspaces of Sobolev–Morrey spaces into Morrey spaces are also obtained. [Copyright &y& Elsevier]

Abstract: In the paper, the authors prove that the generalized sine function and the generalized hyperbolic sine function are respectively geometrically concave and geometrically convex. Consequently, the authors verify a conjecture posed by B. A. Bhayo and M. Vuorinen. [Copyright &y& Elsevier]

Abstract: In this paper we study the mapping properties of some Hardy-type operators with general bounds of integration. In particular, some estimates for Schatten–von Neumann norms of these Hardy–Steklov operators are derived and discussed. [Copyright &y& Elsevier]

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: Fourier-like systems are formed by multiplying a class of exponentials with a set of window functions. Via the Fourier transform they are equivalent to shift-invariant systems. We present sufficient and easily verifiable conditions for such systems to form a frame with a dual frame having the same structure. An attractive class of frames is formed by letting the window functions be trigonometric polynomials, restricted to compact intervals. We prove, under weak conditions, that such systems generate a frame with a dual that is also generated by a trigonometric polynomial. For polynomial windows, a result of this type does not hold. Throughout the paper the results are related to the well established theory for Gabor systems. [Copyright &y& Elsevier]

Abstract: A closed subspace of a normed linear space is said to be ball proximinal if the closed unit ball of is proximinal in . In this paper, we characterize ball proximinal and strongly ball proximinal hyperplanes. [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: In this paper, we discuss asymptotic relations for the approximation of in by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind. [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: In this paper, the monotonicity property for a function involving -gamma and -digamma functions is investigated for . An application of this result provides a sharp inequality for the -gamma function. Our results are shown to be a generalization of results which were obtained by Alzer and Batir (2007) . [Copyright &y& Elsevier]

Abstract: In this paper, we obtain the exact values of -widths of some classes of periodic differentiable functions in the space , satisfying the condition where . Some further generalizations are included. [Copyright &y& Elsevier]

Abstract: In this paper, we present several bounds for the perimeter of an ellipse, which improve some well-known results. [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: The generalized Stieltjes–Wigert polynomials depending on parameters and are discussed. By removing the mass at zero of an N-extremal solution concentrated in the zeros of the -function from the Nevanlinna parametrization, we obtain a discrete measure , which is uniquely determined by its moments. We calculate the coefficients of the corresponding orthonormal polynomials . As noticed by Chihara, these polynomials are the shell polynomials corresponding to the maximal parameter sequence for a certain chain sequence. We also find the minimal parameter sequence, as well as the parameter sequence corresponding to the generalized Stieltjes–Wigert polynomials, and compute the value of related continued fractions. The mass points of have been studied in recent papers of Hayman, Ismail–Zhang and Huber. In the special case of , the maximal parameter sequence is constant and the determination of and gives an answer to a question posed by Chihara in 2001. [Copyright &y& Elsevier]

Abstract: In this paper, we prove that for and the inequality is valid if and reversed if and that the power is the best possible, where is the Euler gamma function. This extends the result of [Y. Yu, An inequality for ratios of gamma functions, J. Math. Anal. Appl. 352 (2) (2009) 967–970] and resolves an open problem posed in [B.-N. Guo, F. Qi, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math. 7 (2) (2003) 239–247]. [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: The main subject of the paper is an in-depth analysis of Weyl matrix balls which are associated with a finite moment problem for rational matrix functions in the nondegenerate case. Thereby, the investigations tie in with preceding studies on a class of extremal solutions of the moment problem in question. We will point out that each member of this class is also extremal concerning the parameters of Weyl matrix balls. The considerations lead to characterizations of these particular solutions within the whole solution set of the problem. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get that insight. [Copyright &y& Elsevier]

Abstract: A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the previous ones, the coefficients of this linear combination being self-reciprocal rational functions. We show that, under very general conditions on the self-reciprocal coefficients, this new sequence satisfies orthogonality conditions as well as a recurrence relation. Further, we identify the Carathéodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients. The new class under study includes the associated rational functions as a particular case. As a consequence of the previous general analysis, we obtain explicit representations for the associated rational functions of arbitrary order, as well as for the related Carathéodory function. Such representations are used to find new properties of the associated rational functions. [Copyright &y& Elsevier]

Abstract: New sufficient conditions for the representation of a function via an absolutely convergent Fourier integral are obtained in the paper. In the main result, this is controlled by the behavior near infinity of both the function and its derivative. This result is extended to any dimension . [Copyright &y& Elsevier]

Abstract: In this paper we obtain the following sharp inequalities for moduli of smoothness of fractional order: and or . Corresponding estimates for -functionals and corresponding embedding theorems for the Lipschitz spaces are investigated. [Copyright &y& Elsevier]

Abstract: In this paper we characterize the behavior of the sequence of the -norm of primitives of a function by its spectrum (the support of its Fourier transform). [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: The classical Heun equation has the form where is a cubic complex polynomial, is a polynomial of degree at most and is at most linear. In the second half of the nineteenth century E. Heine and T. Stieltjes initiated the study of the set of all for which the above equation has a polynomial solution of a given degree . The main goal of the present paper is to study the union of the roots of the latter set of ’s when . We provide an explicit description of this limiting set and give a substantial amount of preliminary and additional information about it obtained using a certain technique developed by A.B.J. Kuijlaars and W. Van Assche. [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: In this paper we study the asymptotic behaviour of polynomials orthogonal with respect to a Sobolev-type inner product where and . We will focus our attention on the outer relative asymptotics with respect to the standard Laguerre polynomials as well as on an analog of the Mehler–Heine formula for the rescaled polynomials. [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: In this paper we obtain some explicit three term recurrence relations for the determination of multivariate orthogonal polynomials. These formulas allow us to obtain evaluation algorithms of finite series of these polynomials. [Copyright &y& Elsevier]