Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights w_n(x)dx = e^−nV(x) dx on the line as n → ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [19, 20]. The Riemann‐Hilbert problem is analyzed in turn using the steepest‐descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμ_V for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.     

On the shape of the ground state eigenvalue density of a random Hill's equation

Consider the Hill's operator Q = ?d²/dx² + q(x) in which q(x), 0 ≤ x ≤ 1, is a white noise. Denote by f(μ) the probability density function of ?λ₀(q), the negative of the ground state eigenvalue, at μ. We prove the detailed asymptotics as μ → + ∞. This result is based on a precise Laplace analysis of a functional integral representation for f(μ) established by S. Cambronero and H. P. McKean in 5 . © 2005 Wiley Periodicals, Inc.     

The ground state eigenvalue of Hill's equation with white noise potential

The distribution of the ground state eigenvalue λ₀(Q) of Hill's operator Q = −d²/dx² + q(x) on the circle of perimeter 1 is expressed in two different ways in case the potential q is standard white noise. Let WN be the associated white noise measure, and let CBM be the measure for circular Brownian motion p(x), 0 ≤ x < 1, formed from the standard Brownian motion b(x), 0 ≤ x ≤ 1, starting at b(0) = a, by conditioning so that b(1) = a, and distributing this common level over the line according to the measure da. The connection is based upon the Ricatti correspondence q(x) = λ + p′ (x) + p²(x). The two versions of the distribution are (1) in which $\overline{p}$ is the mean value ∫ pdx, and (2) the left‐hand side of (2) being the density for (1) and CBM₀ the conditional circular Brownian measure on $\overline{p}$ = 0. (1) and (2) are related by the divergence theorem in function space as suggested by the recognition of the Jacobian factor the outward‐pointing normal component of the vector field v(x) = ∂Δ(λ)/∂q(x), 0 ≤ x < 1, Δ being the Hill's discriminant for Q. The Ricatti correspondence prompts the idea that (1) and (2) are instances of the Cameron‐Martin formula, but it is not so: The latter has to do with the initial value problem for Ricatti, but it is the periodic problem that figures here, so the proof must be done by hand, by finite‐dimensional approximation. The adaptation of 1 and 2 to potentials of Ornstein‐Uhlenbeck type is reported without details. © 1999 John Wiley & Sons, Inc.     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators I

In this paper we study weighted function spaces of type B(?ⁿ, Q(x)) and F(?ⁿ, Q(x)), where Q(x) is a weight function of at most polynomial growth. Of special interest are the weight functions Q(x) = (1 + |x|²)^α/2 with α ? ?. The main result deals with estimates for the entropy numbers of compact embeddings between spaces of this type.     

Conversion of Polynomials between Different Polynomial Bases

The purpose of this paper is to derive a recursive scheme forthe evaluation of the coefficients in the expansion , in terms of the coefficients in the expansion , where both q_k(x) and Q_k(x) are polynomials in xof degree k, and where both q_k(x) and Q_k{x} satisfy recursionformulae of the type satisfied by orthogonal polynomials. Thesets {Q_k(x)} and {q_k(x)} need not be orthogonal polynomials,though they usually are in the applications. An applicationis made to the evaluation of integrals with oscillatory andsingular integrands.     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators II

This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type Ψ with χ χ ? ? and 0 ≤ γ ≤ 1 in the weighted function spaces B (?ⁿ, w(x)) and F (?ⁿ, w(x)) treated in [17]. Furthermore, we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b₂(x) b(x, D) b₁(x), where b₁(x) and b₂(x) are appropriate functions and b(x, D) ? Ψ. Finally, on the basis of the Birman-Schwinger principle, we deal with the “negative spectrum” (bound states) of related symmetric operators in L₂.     

Tangential Convergence of Temperatures and Harmonic Functions in Besov and in Triebel-Lizorkin Spaces

We study the maximal function Mf(x) = sup |f(x + y, t)| when Ω is a region in the (y,t) Ω upper half space R and f(x, t) is the harmonic extension to R₊^N+1 of a distribution in the Besov space B^α_p,q(R^N) or in the Triebel-Lizorkin space F^α_p,q(R^N). In particular, we prove that when Ω= {|_y|^{N/ (N-αp)} < t < 1} the operator M is bounded from F (R^N) into L^p (R^N). The admissible regions for the spaces B (R^N) with p < q are more complicated.     

Stability of Cahn‐Hilliard fronts

We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. © 1999 John Wiley & Sons, Inc.     

Representations for the first associated q-classical orthogonal polynomials

Let {p_k(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation D_q(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {p_k⁽¹⁾(x;q)} be associated polynomials of the polynomials {p_k(x; q)}. Explicit forms of the coefficients b_n,k and c_n,k in the expansions

are given in terms of basic hypergeometric functions. Here _k(x) equals x^k if σ⁺(0)=0, or (x;q)_k if σ⁺(1)=0, where σ⁺(x)σ(x)+(q−1)xτ(x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous q-difference equation satisfied by p_n−1⁽¹⁾(x;q) in a special form, recurrence relations (in k) for b_n,k and c_n,k are obtained in terms of σ and τ.     

Designs with mutually orthogonal resolutions and decompositions of edge‐colored graphs

Two resolutions R and R^′ of a combinatorial design are called orthogonal if |R_i∩R|≤1 for all R_i∈R and R∈R^′. A set Q={R¹, R², …, R^d} of d resolutions of a combinatorial design is called a set of mutually orthogonal resolutions (MORs) if the resolutions of Q are pairwise orthogonal. In this paper, we establish necessary and sufficient conditions for the asymptotic existence of a (v, k, 1)‐BIBD with d mutually orthogonal resolutions for d≥2 and k≥3 and necessary and sufficient conditions for the asymptotic existence of a (v, k, k?1)‐BIBD with d mutually orthogonal near resolutions for d≥2 and k≥3. We use complementary designs and the most general form of an asymptotic existence theorem for decompositions of edge‐colored complete digraphs into prespecified edge‐colored subgraphs. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 425–447, 2009     

On generalized Fibonacci and Lucas polynomials

Let h(x) be a polynomial with real coefficients. We introduce h(x)-Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h(x)-Fibonacci polynomials. We also introduce h(x)-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q_h(x) that generalizes the Q-matrix whose powers generate the Fibonacci numbers.     

Isomorphism criterion for monomial graphs

Let q be a prime power, ??_q be the field of q elements, and k, m be positive integers. A bipartite graph G = G_q(k, m) is defined as follows. The vertex set of G is a union of two copies P and L of two‐dimensional vector spaces over ??_q, with two vertices (p₁, p₂) ∈ P and [ l₁, l₂] ∈ L being adjacent if and only if p₂ + l₂ = pl. We prove that graphs G_q(k, m) and G_q′(k′, m′) are isomorphic if and only if q = q′ and {gcd (k, q ? 1), gcd (m, q ? 1)} = {gcd (k′, q ? 1),gcd (m′, q ? 1)} as multisets. The proof is based on counting the number of complete bipartite INFgraphs in the graphs. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 322–328, 2005     

Random walks on the torus with several generators

Given n vectors {_i} ∈ [0, 1)^d, consider a random walk on the d‐dimensional torus ??^d = ?^d/?^d generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*^k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*^k) ≥ C₁k^?n/2, where C₁ = C(n, d) is a constant. If the vectors are badly approximated by rationals (in a sense we will define), then D(Q*^k) ≤ C₂k^?n/2d for C₂ = C(n, d, _j) a constant. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Hardy-Littlewood Maximal Operator,Singular Integral Operators and the Riesz Potentials on Generalized Morrey Spaces

Any continuous linear operator T: L^p → L^q has a natural vector-valued extension T: L^p(l) → L^q(l) which is automatically continuous. Relations between the norms of these operators in the cases of p = q and r = 2 were considered by Marcinkiewicz -Zygmund [28], Herz [14] and Krivine [19] - [21]. In this paper we study systematically these relations and given some applications. It turns out that some known results can be proved in a simple way as a consequence of these developments.     

Iteratively reweighted least squares minimization for sparse recovery

Under certain conditions (known as the restricted isometry property, or RIP) on the m × N matrix Φ (where m < N), vectors x ∈ ?^N that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := Φx even though Φ^?1(y) is typically an (N ? m)—dimensional hyperplane; in addition, x is then equal to the element in Φ^?1(y) of minimal ??₁‐norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x, as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w, the element in Φ^?1(y) with smallest ??₂(w)‐norm. If x⁽ⁿ⁾ is the solution at iteration step n, then the new weight w⁽ⁿ⁾ is defined by w := [|x|² + ε]^?1/2, i = 1, …, N, for a decreasing sequence of adaptively defined ε_n; this updated weight is then used to obtain x^{(n + 1)} and the process is repeated. We prove that when Φ satisfies the RIP conditions, the sequence x⁽ⁿ⁾ converges for all y, regardless of whether Φ^?1(y) contains a sparse vector. If there is a sparse vector in Φ^?1(y), then the limit is this sparse vector, and when x⁽ⁿ⁾ is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast (linear convergence in the terminology of numerical optimization). The same algorithm with the “heavier” weight w = [|x|² + ε]^?1+τ/2, i = 1, …, N, where 0 < τ < 1, can recover sparse solutions as well; more importantly, we show its local convergence is superlinear and approaches a quadratic rate for τ approaching 0. © 2009 Wiley Periodicals, Inc.     

On polynomials orthogonal to all powers of a given polynomial on a segment

In this paper we investigate the following “polynomial moment problem”: for a given complex polynomial P(z) and distinct a,b∈C to describe polynomials q(z) orthogonal to all powers of P(z) on [a,b]. We show that for given P(z), q(z) the condition that q(z) is orthogonal to all powers of P(z) is equivalent to the condition that branches of the algebraic function Q(P⁻¹(z)), where , satisfy a certain system of linear equations over Z. On this base we provide the solution of the polynomial moment problem for wide classes of polynomials. In particular, we give the complete solution for polynomials of degree less than 10.     

On Σ1-definable Functions Provably Total in I ∏

We prove the following theorem: Let φ(x) be a formula in the language of the theory PA^? of discretely ordered commutative rings with unit of the form ?yφ′(x,y) with φ′ and let ∈ Δ₀ and let fφ: ? → ? such that f_φ(x) = y iff φ′(x,y) & (?z < y) φ′(x,z). If I ∏ ∈(?x ≥ 0), φ then there exists a natural number K such that I ∏ ? ?y?x(x > y ? ?φ(x) < x^K). Here I ∏₁^? denotes the theory PA^? plus the scheme of induction for formulas φ(x) of the form ?yφ′(x,y) (with φ′) with φ′ ∈ Δ₀.     

A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations

This article analyzes the solution of the integrated forms of fourth‐order elliptic differential equations on a rectilinear domain using a spectral Galerkin method. The spatial approximation is based on Jacobi polynomials P (x), with α, β ∈ (?1, ∞) and n the polynomial degree. For α = β, one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α = β = ?½, α = β = 0, the Chebyshev of the first and second kinds and Legendre polynomials respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases α = ?β = ±½ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two‐dimensional version of the approximations is obtained by tensor products of the one‐dimensional bases. The various matrix systems resulting from these discretizations are carefully investigated, especially their condition number. An algebraic preconditioning yields a condition number of O(N), N being the polynomial degree of approximation, which is an improvement with respect to the well‐known condition number O(N⁸) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to N^d+1 for a d‐dimensional problem. This operational count is the best one can achieve with a spectral method. The numerical results illustrate the theory and constitute a convincing argument for the feasibility of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Enumeration of graphs by degree sequence

The degree sequence (d₀, d₁, …, d_p-1) of a graph G of order p is defined by d_k = the number of points of G of degree k. Methods of Robinson are extended to produce a generating function F(x₀, x₁, x₂, …) where the coefficient of x…x is the number of graphs of order p having degree sequence (d₀, …, d_p-1).     

Blowup for nonlinear wave equations describing boson stars

We consider the nonlinear wave equation modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u₀(x) ∈ C (?³), with negative energy, we prove blowup of u(t, x) in the H^1/2‐norm within a finite time. Physically this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blowup in external, spherically symmetric potentials, and we consider more general Hartree‐type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0. © 2007 Wiley Periodicals, Inc.