Hyperfunctional Weights for Orthogonal Polynomials

The Chebychev polynomials associated to any given moments μ_{n ∞} ⁰ are formally orthogonal with respect to the formal δ-series $$w(x)= {\sum^\infty_0}(- 1)^{n}\mu_{{n}}\delta^{(n)}(x)/n!.$$ We show that this formal weight can be a true hyperfunctional weight if its Fourier transform is a slowly increasing holomorphic function in some tubular neighborhood of the real line. It provides a unifying treatment of real and complex orthogonality of Chebychev polynomials including all classical examples and characterizes Chebychev polynomials having Bessel type orthogonality.     

Representation of Polynomials by Linear Combinations of Radial Basis Functions

Let ${\mathcal {P}_{n}^{d}}$ denote the space of polynomials on ? ^d of total degree n. In this work, we introduce the space of polynomials ${\mathcal {Q}_{2 n}^{d}}$ such that ${\mathcal {P}_{n}^{d}}\subset {\mathcal {Q}_{2 n}^{d}}\subset\mathcal{P}_{2n}^{d}$ and which satisfy the following statement: Let h be any fixed univariate even polynomial of degree n and $\mathcal{A}$ be a finite set in ? ^d . Then every polynomial P from the space  ${\mathcal {Q}_{2 n}^{d}}$ may be represented by a linear combination of radial basis functions of the form h(∥x+a∥), $a\in \mathcal{A}$ , if and only if the set $\mathcal{A}$ is a uniqueness set for the space  ${\mathcal {Q}_{2 n}^{d}}$ .     

Stability of Roots of Polynomials Under Linear Combinations of Derivatives

Let T=α ₀ I+α ₁ D+⋅⋅⋅+α _n D ⁿ , where D is the differentiation operator and a₀ 1 0\alpha_{0}\not=0 , and let f be a square-free polynomial with large minimum root separation. We prove that the roots of Tf are close to the roots of f translated by −α ₁/α ₀.     

Orthogonal Polynomials Associated with Related Measures and Sobolev Orthogonal Polynomials

Connection between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), are looked at. The results are applied to obtain information regarding Sobolev orthogonal polynomials associated with certain pairs of measures.     

Average Number of Real Roots of Random Harmonic Equations

Let {g_k}be a sequence of normally distributed independent random variables with mathematical expectation zero and variance unity. Let _k (t ) (k = 0, 1, 2,...) be the normalized Jacobi polynomials orthogonal with respect to the interval [ – 1, 1 ]. Then it is proved that the average number of real roots of the random equations, _k=0 ⁿ g_kk(1)=C where Cis a constant, is asymptotically equal to n/in the same interval when nis large and even for C as long as C=O (n ²).     

Contracted Zero Distributions of Extremal Polynomials Associated with Slowly Decaying Weights

We consider the asymptotic zero behavior of polynomials that are extremal with respect to slowly decaying weights on [0, ∈fty) , such as the log-normal weight \exp(-γ ² log  ² x) . The zeros are contracted by taking the appropriate d _n th roots with d _n →∈fty . The limiting distribution of the contracted zeros is described in terms of the solution of an extremal problem in logarithmic potential theory with a circular symmetric external field. November 23, 1998. Date revised: February 8, 1999. Date accepted: March 2, 1999.     

Real Zeros of Algebraic Polynomials with Dependent Random Coefficients

The expected number of real zeros of polynomials a ₀ + a ₁ x + a ₂ x ² +…+a _n?1 x ^n?1 with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π)log n. For the dependent cases studied so far it is shown that this asymptotic value remains O(log n). In this article, we show that when cov(a _i , a _j ) = 1 ? |i ? j|/n, for i = 0,…, n ? 1 and j = 0,…, n ? 1, the above expected number of real zeros reduces significantly to O(log n)^1/2.     

Geometry of Truncated Symmetric Products and Real Roots of Real Polynomials

We study the conditions for truncated symmetric products ofmanifolds to be manifolds. In particular, we show that suitablydefined spaces of systems of real roots of real polynomialsare homeomorphic to real projective spaces. 1991 MathematicsSubject Classification 57N99, 26C10.     

Jost Asymptotics for Matrix Orthogonal Polynomials on the Real Line

We obtain matrix-valued Jost asymptotics for block Jacobi matrices under an L ¹-type condition on Jacobi coefficients, and give a necessary and sufficient condition for an analytic matrix-valued function to be the Jost function of a block Jacobi matrix with exponentially converging parameters. This establishes the matrix-valued analogue of Damanik and Simon (Int. Math. Res. Not. 32:19396, 2006). The above results allow us to fully characterize the Weyl?CTitchmarsh m-functions of Jacobi matrices with exponentially converging parameters.     

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Let denote the linear space over spanned by . Define the (real) inner product , where V satisfies: (i) V is real analytic on ; (ii) ; and (iii) . Orthogonalisation of the (ordered) base with respect to yields the even degree and odd degree orthonormal Laurent polynomials , and . Define the even degree and odd degree monic orthogonal Laurent polynomials: and . Asymptotics in the double-scaling limit such that of (in the entire complex plane), , and (in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on , and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].     

Random Linear Combinations of Functions from L 1

In this paper, properties of random polynomials with respect to a general system of functions are studied. Some lower bounds for the mathematical expectation of the uniform norm and the recently introduced integral-uniform norm of random polynomials are established.     

A Borg-Type Theorem Associated with Orthogonal Polynomials on the Unit Circle

We prove a general Borg-type result for reflectionless unitaryCMV operators U associated with orthogonal polynomials on theunit circle. The spectrum of U is assumed to be a connectedarc on the unit circle. This extends a recent result of Simonin connection with a periodic CMV operator with spectrum thewhole unit circle. In the course of deriving the Borg-type result we also use exponentialHerglotz representations of Caratheodory functions to provean infinite sequence of trace formulas connected with the CMVoperator U.     

On the Expected Condition Number of Linear Programming Problems

Summary.   Let A be an n×m real matrix and consider the linear conic system

Linear Graininess Time Scales and Ladder Operators of Orthogonal Polynomials

In this paper we introduce linear graininess (LG) time scales. We further study orthogonal polynomials (OPs) with the weight function supported on LG time scales and derive the raising and lowering ladder operators by using the time scales calculus. We also derive a second order dynamic equation satisfied by these polynomials. The notion of an LG time scale encompasses the cases of the reals, the h-equidistant grid, the q-grid and, more general, a mixed (q, h)-grid. This allows a unified treatment of the ladder operators theory for classical OPs on these time scales. Moreover we will explain, why exclusively LG time scales provide the right framework for general OP theory.