An algebraic structure for Faber polynomials

Let Σ be the set of functions, convergent for all |z|>1, with a Laurent series of the form f(z)=z+∑_n?0a_nz^-n. In this paper, we prove that the set of Faber polynomial sequences over Σ and the set of their normalized kth derivative sequences form groups which are isomorphic to the hitting time subgroup and the Bell^(k) subgroup of the Riordan group, respectively. Further, a relationship between such Faber polynomial sequences and Lucas and Sheffer polynomial sequences is derived.     

Perturbations on the subdiagonals of Toeplitz matrices

Let L be an Hermitian linear functional defined on the linear space of Laurent polynomials. It is very well known that the Gram matrix of the associated bilinear functional in the linear space of polynomials is a Toeplitz matrix. In this contribution we analyze some linear spectral transforms of L such that the corresponding Toeplitz matrix is the result of the addition of a constant in a subdiagonal of the initial Toeplitz matrix. We focus our attention in the analysis of the quasi-definite character of the perturbed linear functional as well as in the explicit expressions of the new monic orthogonal polynomial sequence in terms of the first one.     

Invariant Hyperfunction Solutions to Invariant Differential Equations on the Space of Real Symmetric Matrices

The real special linear group of degree n naturally acts on the vector space of n×n real symmetric matrices. How to determine invariant hyperfunction solutions of invariant linear differential equations with polynomial coefficients on the vector space of n×n real symmetric matrices is discussed in this paper. We prove that every invariant hyperfunction solution is expressed as a linear combination of Laurent expansion coefficients of the complex power of the determinant function with respect to the parameter of the power. Then the problem is reduced to the determination of Laurent expansion coefficients.     

Asymptotics of orthogonal L-polynomials for log-normal distributions

Asymptotic properties of orthogonal Laurent (L-) polynomialsV _n (z), associated with log-normal distributions, are derived by constructive methods. It is shown that the sequences {V _2n (z)} and {V _2n+1 (z)} converge separately (asn→∞) to functionsV ⁽⁰⁾ (z) andV ⁽¹⁾ (z), respectively, both holomorphic in 0<|z|<∞. Explicit Laurent series expansions are obtained, from which it follows that each limit function has essential, isolated singularities atz=0 andz=∞.     

An extended class of orthogonal polynomials defined by a Sturm-Liouville problem

We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as X₁-Jacobi and X₁-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval [−1,1] or the half-line [0,∞), respectively, and they are a basis of the corresponding L² Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second-order operator has a complete set of polynomial eigenfunctions , then it must be either the X₁-Jacobi or the X₁-Laguerre Sturm-Liouville problem. A Rodrigues-type formula can be derived for both of the X₁ polynomial sequences.     

Radicals in skew polynomial and skew Laurent polynomial rings

We provide a general procedure for characterizing radical-like functions of skew polynomial and skew Laurent polynomial rings under grading hypotheses. In particular, we are able to completely characterize the Wedderburn and Levitzki radicals of skew polynomial and skew Laurent polynomial rings in terms of ideals in the coefficient ring. We also introduce the T-nilpotent radideals, and perform similar characterizations.     

Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality

We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e−φ(x), giving a unified treatment for the so-called Freud (i.e., when φ has polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.     

Extension theorems for linear lattices with positive algebraic basis

An ordered linear spaceV with positive wedgeK is said to satisfy extension property (E1) if for every subspaceL ₀ ofV such thatL ₀ ∩K is reproducing inL ₀, and every monotone linear functionalf ₀ defined onL ₀,f ₀ has a monotone linear extension to all ofV. A linear latticeX is said to satisfy extension property (E2) if for every sublatticeL ofX, and every linear functionalf defined onL which is a lattice homomorphism,f has an extensionf′ to all ofX which is also a linear functional and a lattice homomorphism. In this paper it is shown that a linear lattice with a positive algebraic basis has both extension property (E1) and (E2). In obtaining this result it is shown that the linear span of a lattice idealL and an extremal element not inL is again a lattice ideal. (HereX does not have to have a positive algebraic basis.) It is also shown that a linear lattice which possesses property (E2) must be linearly and lattice isomorphic to a functional lattice. An example is given of a function lattice which has property (E2) but does not have a positive algebraic basis. Yudin [12] has shown a reproducing cone in ann-dimensional linear lattice to be the intersection of exactlyn half-spaces. Here it is shown that the positive wedge in ann-dimensional archimedean ordered linear space satisfying the Riesz decomposition property must be the intersection ofn half-spaces, and hence the space must be a linear lattice with a positive algebraic basis. The proof differs from those given for the linear lattice case in that it uses no special techniques, only well known results from the theory of ordered linear space.     

The sharp Markov-Nikol’skii inequality for algebraic polynomials in the spaces L q and L 0 on a closed interval

In this paper, an inequality between the L _q -mean of the kth derivative of an algebraic polynomial of degree n ≥ 1 and the L ₀-mean of the polynomial on a closed interval is obtained. Earlier, the author obtained the best constant in this inequality for k = 0, q ∈ [0,∞] and 1 ≤ k ≤ n, q ∈ {0} ∪ [1,∞]. Here a newmethod for finding the best constant for all 0 ≤ k ≤ n, q ∈ [0,∞], and, in particular, for the case 1 ≤ k ≤ n, q ∈ (0, 1), which has not been studied before is proposed. We find the order of growth of the best constant with respect to n as n → ∞ for fixed k and q.     

Weakly convergent sequences of functions and orthogonal polynomials

Measures and sequences of functions on locally compact spaces are studied, and a condition is given that, for a sequence of functions that is weakly convergent in L¹, ensures the strong convergence of a related sequence of functions. This result, together with a new integral formula for the reflection coefficients Φ_n(0) for the monic orthogonal polynomial Φ_n associated with a measure on the unit circle, is used to investigate convergence properties of orthogonal polynomials.     

Constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain

In this paper we show that the orthogonal complement of a subspace in the polynomial space of degree n over d  -dimensional simplex domain with respect to the _L2$L_2$-inner product and the weighted Euclidean inner product of BB (Bézier–Bernstein) coefficients are equal. Using it we also prove that the best constrained degree reduction of polynomials over the simplex domain in BB form equals the best approximation of weighted Euclidean norm of coefficients of given polynomial in BB form from the coefficients of polynomials of lower degree in BB form.     

Orthogonality and recurrence for ordered Laurent polynomial sequences

We consider orderings of nested subspaces of the space of Laurent polynomials on the real line, more general than the balanced orderings associated with the ordered bases {1,z⁻¹,z,z⁻²,z²,…} and {1,z,z⁻¹,z²,z⁻²,…}. We show that with such orderings the sequence of orthonormal Laurent polynomials determined by a positive linear functional satisfies a three-term recurrence relation. Reciprocally, we show that with such orderings a sequence of Laurent polynomials which satisfies a recurrence relation of this form is orthonormal with respect to a certain positive functional.     

On sequences of Hurwitz polynomials related to orthogonal polynomials

ABSTRACT

In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials.     

Sur la suite de polynômes orthogonaux associée à la formeu=δ c +λ(x−c) −1 L

We show that, ifL is regular, semi-classical functional, thenu is also regular and semi-classical for every complex λ, except for a discrete set of numbers depending onL andc. We give the second order linear differential equation satisfied by each polynomial of the orthogonal sequence associated withu. The cases whereL is either a classical functional (Hermite, Laguerre, Bessel, Jacobi) or a functional associated with generalized Hermite polynomials are treated in detail.

     

Orthogonal polynomials and laurent polynomials related to the Hahn-Extonq-Bessel function

Laurent polynomials related to the Hahn-Extonq-Bessel function, which areq-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurentq-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orthogonal polynomials, the orthogonality measure is determined using the three-term recurrence relation as a starting point. The relation between Chebyshev polynomials of the second kind and the Laurentq-Lommel polynomials and related functions is used to obtain estimates for the latter.     

Computation of Gauss-type quadrature formulas

Gaussian formulas for a linear functional L (such as a weighted integral) are best computed from the recursion coefficients relating the monic polynomials orthogonal with respect to L. In Gauss-type formulas, one or more extraneous conditions (such as pre-assigning certain nodes) replace some of the equations expressing exactness when applied to high-order polynomials. These extraneous conditions may be applied by modifying the same number of recursion coefficients. We survey the methods of computing formulas from recursion coefficients, methods of obtaining recursion coefficients and modifying them for Gauss-type formulas, and questions of existence and numerical accuracy associated with those computations.     

Some extremal problems for trigonometric polynomials

Three extremal problems for trigonometric polynomials are studied in this paper. The first was initiated by Maiorov. It relates to the trigonometric polynomials with n nonzero harmonics. The L_p-norm of the Weyl derivative is compared with the L_q-norm of the polynomial. The other two problems have appeared in the recent paper by Oswald. They deal with the polynomials of degree n. The minimum of L_p-norm with respect to the choice of phases is compared with l_q-norm of its coefficients.     

Some properties of two analytic functions associated with complex polynomials

Two basic analytic functions α(z) and β(z) defined in domains depending on the location of the zeros of a complex polynomial P(z) are given by $\text{P′}\text{P}\text{=}\text{n}\text{(z ? α)}$ and P = (z ? β)ⁿ. These functions are studied with respect to their growth and their Laurent expansion coefficients. Applications to the location of zeros of complex polynomials are indicated.     

The lattice of integer partitions

In this paper we study the lattice L_n of partitions of an integer n ordered by dominance. We show L_n to be isomorphic to an infimum subsemilattice under the component ordering of certain concave nondecreasing (n+1)-tuples. For L_n, we give the covering relation, maximal covering number, minimal chains, infimum and supremum irreducibles, a chain condition, distinguished intervals; and show that partition conjugation is a lattice antiautomorphism. L_n is shown to have no sublattice having five elements and rank two, and we characterize intervals generated by two cocovers. The Möbius function of L_n is computed and shown to be 0,1 or -1. We then give methods for studying classes of (0,1)-matrices with prescribed row and column sums and compute a lower bound for their cardinalities.     

A Hardy field extension of Szemerédi's theorem

In 1975 Szemerédi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form p(n) where p(n) is any integer polynomial with zero constant term. We produce a variety of new results of this type related to sequences that are not polynomial. We show that the common difference of the progression in Szemerédi's theorem can be of the form [nδ] where δ is any positive real number and [x] denotes the integer part of x. More generally, the common difference can be of the form [a(n)] where a(x) is any function that is a member of a Hardy field and satisfies a(x)/xk→∞ and a(x)/xk+1→0 for some non-negative integer k. The proof combines a new structural result for Hardy sequences, techniques from ergodic theory, and some recent equidistribution results of sequences on nilmanifolds.