Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on the Unit Circle

We study the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product on the unit circle

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where f(Z)=(f(z₁), …, f^(l₁)(z₁), …, f(z_m), …, f^(l_m)(z_m)), A is a M×M positive definite matrix or a positive semidefinite diagonal block matrix, M=l₁+…+l_m+m, dμ belongs to a certain class of measures, and |z_i|>1, i=1, 2, …, m.     

Characterization of generalized convex functions by their best approximation in sign-monotone norms

Let {u₀, u₁,… u_{n − 1}} and {u₀, u₁,…, u_n} be Tchebycheff-systems of continuous functions on [a, b] and let f ε C[a, b] be generalized convex with respect to {u₀, u₁,…, u_{n − 1}}. In a series of papers ([1], [2], [3]) D. Amir and Z. Ziegler discuss some properties of elements of best approximation to f from the linear spans of {u₀, u₁,…, u_{n − 1}} and {u₀, u₁,…, u_n} in the L_p-norms, 1 p ∞, and show (under different conditions for different values of p) that these properties, when valid for all subintervals of [a, b], can characterize generalized convex functions. Their methods of proof rely on characterizations of elements of best approximation in the L_p-norms, specific for each value of p. This work extends the above results to approximation in a wider class of norms, called “sign-monotone,” [6], which can be defined by the property: ¦ f(x)¦ ¦ g(x)¦,f(x)g(x) 0, a x b, imply f g . For sign-monotone norms in general, there is neither uniqueness of an element of best approximation, nor theorems characterizing it. Nevertheless, it is possible to derive many common properties of best approximants to generalized convex functions in these norms, by means of the necessary condition proved in [6]. For {u₀, u₁,…, u_n} an Extended-Complete Tchebycheff-system and f ε C⁽ⁿ⁾[a, b] it is shown that the validity of any of these properties on all subintervals of [a, b], implies that f is generalized convex. In the special case of f monotone with respect to a positive function u₀(x), a converse theorem is proved under less restrictive assumptions.     

Sharp inequalities for convolution-type operators

Let μ be a probability measure on [− a, a], a > 0, and let x₀ε[− a, a], f ε Cⁿ([−2a, 2a]), n 0 even. Using moment methods we derive best upper bounds to ¦∫_−a^a ([f(x₀ + y) + f(x₀ − y)]/2) μ(dy) − f(x₀)¦, leading to sharp inequalities that are attainable and involve the second modulus of continuity of f⁽ⁿ⁾ or an upper bound of it.     

Approximations of Two-Place Functions by Finite Sums of Terms with Separable Variables

It is known that if a smooth function h in two real variables x and y belongs to the class Σ_n of all sums of the form Σⁿ_k=1ƒ_k(x) g_k(y), then its (n + 1)th order "Wronskian" det[h_xⁱy^j]ⁿ_i,j=0 is identically equal to zero. The present paper deals with the approximation problem h(x, y) Σⁿ_k=1ƒ_k(x) g_k(y) with a prescribed n, for general smooth functions h not lying in Σ_n. Two natural approximation sums T=T(h) Σ_n, S=S(h) Σ_n are introduced and the errors |h-T|, |h-S| are estimated by means of the above mentioned Wronskian of the function h. The proofs utilize the technique of ordinary linear differential equations.     

Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

Let τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Q_n(x)}^∞_n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Q_n(x)}^∞_n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: L_N[y](x)=∑^N_i=1 ℓ_i(x) y⁽ⁱ⁾(x)=λ_ny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Q_n(x)}^∞_n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions.     

On Approximate ℓ 1 Systems in Banach Spaces

Let X be a real Banach space and let (f(n)) be a positive nondecreasing sequence. We consider systems of unit vectors (x_i)^∞_i=1 in X which satisfy ∑_iA±x_i|A|−f(|A|), for all finite A and for all choices of signs. We identify the spaces which contain such systems for bounded (f(n)) and for all unbounded (f(n)). For arbitrary unbounded (f(n)), we give examples of systems for which [x_i] is H.I., and we exhibit systems in all isomorphs of ℓ₁ which are not equivalent to the unit vector basis of ℓ₁. We also prove that certain lacunary Haar systems in L₁ are quasi-greedy basic sequences.     

On existence of singular solutions

In the paper sufficient conditions are given under which the differential equation y⁽ⁿ⁾=f(t,y,…,y⁽ⁿ⁻²⁾)g(y⁽ⁿ⁻¹⁾) has a singular solution y :[T,τ)→R, τ<∞ fulfilling

     

On sharp Burkholder–Rosenthal-type inequalities for infinite-degree U-statistics

In this paper, we present a method that allows one to obtain a number of sharp inequalities for expectations of functions of infinite-degree U-statistics. Using the approach, we prove, in particular, the following result: Let D be the class of functions f :R₊→R₊ such that the function f(x+z)−f(x) is concave in xR₊ for all zR₊. Then the following estimate holds: for all fD and all U-statistics ∑_1i1<<ilnY_i1,…,il(X_i1,…,X_il) with nonnegative kernels Y_i1,…,il :R^l→R₊, 1i_kn; i_r≠i_s, r≠s; k,r,s=1,…,l; l=0,…,m, in independent r.v.'s X₁,…,X_n. Similar inequality holds for sums of decoupled U-statistics. The class D is quite wide and includes all nonnegative twice differentiable functions f such that the function f″(x) is nonincreasing in x>0, and, in particular, the power functions f(x)=x^t, 1<t2; the power functions multiplied by logarithm f(x)= (x+x₀)^t ln(x+x₀), 1<t<2, x₀max(e^{(3t2−6t+2)/(t(t−1)(2−t))},1); and the entropy-type functions f(x)=(x+x₀)ln(x+x₀), x₀1. As an application of the results, we determine the best constants in Burkholder–Rosenthal-type inequalities for sums of U-statistics and prove new decoupling inequalities for those objects. The results obtained in the paper are, to our knowledge, the first known results on the best constants in sharp moment estimates for U-statistics of a general type.     

Persistence, contractivity and global stability in logistic equations with piecewise constant delays

We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N^*=1/(a+∑_i=0^mb_i) of the following differential equation with piecewise constant arguments:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, ∑_i=0^mb_i>0, b_i0, i=0,1,2,…,m, and a+∑_i=0^mb_i>0. These new conditions depend on a,b₀ and ∑_i=1^mb_i, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m=0 and r(t)≡r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, 1−ax−g(x,x,…,x)=0 has a unique solution x^*>0 and g(x₀,x₁,…,x_m)C¹[(0,+∞)×(0,+∞)××(0,+∞)].     

A Minimax Problem Admitting the Equioscillation Characterization of Bernstein and Erd?s

This paper shows that under certain conditions a solution of the minimax problem min_{a<x1<…<xn<b} max_1in+1 f_i(x₁, …, x_n) admits the equioscillation characterizations of Bernstein and Erd s and has strong uniqueness. This problem includes as a particular example the optimal Lagrange interpolation problem.     

On the convergence of averaging Hermite interpolators

We investigate two sequences of polynomial operators, H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x), of degrees 2n − 2 and 2n − 3, respectively, defined by interpolatory conditions similar to those of the classical Hermite-Féjer interpolators H_{2n − 1}(f, x). If H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x) are based on the zeros of the jacobi polynomials P_n^(α,β)(x), their convergence behaviour is similar to that of H_{2n − 1}(f;, x). If they are based on the zeros of (1 − x²)T_{n − 2}(x), their convergence behaviour is better, in some sense, than that of H_{2n − 1}(f, x).     

Global integration of differential equations through Lobatto quadrature

For the numerical solution of the initial value problemy=f(x,y), –1x1;y(–1)=y ₀ a global integration method is derived and studied. The method goes as follows.At first the system of nonlinear equations is solved. The matrix (A _i,k ⁽ⁿ⁾ ) of quadrature coefficients is nearly lower left triangular and the pointsx _k,n ,k=1,2,...,n are the zeros ofP _n –P _n–2, whereP _n is the Legendre polynomial of degreen. It is showed that the errors From the valuesf(x _i,n ,y _i,n ),i=1,2,...,n an approximation polynomial is constructed. The approximation is Chebyshevlike and the error at the end of the interval of integration is particularly small.     

Application of generalized differential transform method to multi-order fractional differential equations

In a recent paper [Odibat Z, Momani S, Erturk VS. Generalized differential transform method: application to differential equations of fractional order, Appl Math Comput. submitted for publication] the authors presented a new generalization of the differential transform method that would extended the application of the method to differential equations of fractional order. In this paper, an application of the new technique is applied to solve fractional differential equations of the form y^(μ)(t)=f(t,y(t),y^(β₁)(t),y^(β₂)(t),…,y^(β_n)(t)) with μ>β_n>β_n-1>…>β₁>0, combined with suitable initial conditions. The fractional derivatives are understood in the Caputo sense. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the new generalization.     

Comonotone Jackson's Inequality

Let 2s points y_i=−πy_2s<…<y₁<π be given. Using these points, we define the points y_i for all integer indices i by the equality y_i=y_i+2s+2π. We shall write fΔ⁽¹⁾(Y) if f is a 2π-periodic continuous function and f does not decrease on [y_i, y_i−1], if i is odd; and f does not increase on [y_i, y_i−1], if i is even. In this article the following Theorem 1—the comonotone analogue of Jackson's inequality—is proved. 1. If fΔ⁽¹⁾(Y), then for each nonnegative integer n there is a trigonometric polynomial τ_n(x) of order n such that τ_nΔ⁽¹⁾(Y), and |f(x)−π_n(x)|c(s) ω(f; 1/(n+1)), x , where ω(f; t) is the modulus of continuity of f, c(s)=const. Depending only on s.     

On numerical improvement of Gauss–Legendre quadrature rules

Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule∫₋₁¹f(x) dx∑_i=1ⁿw_if(x_i)has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n−1, where nodes x_i are zeros of Legendre polynomial P_n(x), and w_i's are corresponding weights.In this paper we are going to estimate numerical values of nodes x_i and weights w_i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ε₀, say ε₀=10⁻⁸, for monomial functionsf(x)=x^j, j=0,1,…,2n+1.(Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules.     

Centres of Convex Sets inLMetrics

It is shown that for each convex bodyARⁿthere exists a naturally defined family _AC(Sⁿ⁻¹) such that for everyg _A, and every convex functionf: R→Rthe mappingy∫_Sⁿ⁻¹ f(g(x)−y, x) dσ(x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toL^pmetrics is discussed.     

Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions

A function f(x) defined on = ₁ × ₂ × … × _n where each _i is totally ordered satisfying f(x y) f(x y) ≥ f(x) f(y), where the lattice operations and refer to the usual ordering on , is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies −DΣ⁻¹D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

Schur functions and the invariant polynomials characterizing U(n) tensor operators

We give a direct formulation of the invariant polynomials _μG_q⁽ⁿ⁾(, Δ_i,;, x_{i,i + 1},) characterizing U(n) tensor operators p, q, …, q, 0, …, 0 in terms of the symmetric functions S_λ known as Schur functions. To this end, we show after the change of variables Δ_i = γ_i − δ_i and x_{i, i + 1} = δ_i − δ_{i + 1} that_μG_q⁽ⁿ⁾(,Δ_i;, x_{i, i + 1},) becomes an integral linear combination of products of Schur functions S_α(, γ_i,) · S_β(, δ_i,) in the variables {γ₁,…, γ_n} and {δ₁,…, δ_n}, respectively. That is, we give a direct proof that _μG_q⁽ⁿ⁾(,Δ_i,;, x_{i, i + 1},) is a bisymmetric polynomial with integer coefficients in the variables {γ₁,…, γ_n} and {δ₁,…, δ_n}. By making further use of basic properties of Schur functions such as the Littlewood-Richardson rule, we prove several remarkable new symmetries for the yet more general bisymmetric polynomials _μ^mG_q⁽ⁿ⁾(γ₁,…, γ_n; δ₁,…, δ_m). These new symmetries enable us to give an explicit formula for both _μ^mG₁⁽ⁿ⁾(γ; δ) and ₁G₂⁽ⁿ⁾(γ; δ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas for _μ^mG_q⁽ⁿ⁾(γ; δ).     

L1-optimal estimates for a regression type function in R

Let X₁, X₂, …, X_n be random vectors that take values in a compact set in R^d, d ≥ 1. Let Y₁, Y₂, …, Y_n be random variables (“the responses”) which conditionally on X₁ = x₁, …, X_n = x_n are independent with densities f(y | x_i, θ(x_i)), i = 1, …, n. Assuming that θ lives in a sup-norm compact space Θ_q,d of real valued functions, an optimal L₁-consistent estimator of θ is constructed via empirical measures. The rate of convergence of the estimator to the true parameter θ depends on Kolmogorov's entropy of Θ_q,d.     

Delay optimization of linear depth boolean circuits with prescribed input arrival times

We consider boolean circuits C over the basis Ω={,} with inputs x₁, x₂,…,x_n for which arrival times are given. For 1in we define the delay of x_i in C as the sum of t_i and the number of gates on a longest directed path in C starting at x_i. The delay of C is defined as the maximum delay of an input.Given a function of the form

f(x₁,x₂,…,x_n)=g_n−1(g_n−2(…g₃(g₂(g₁(x₁,x₂),x₃),x₄)…,x_n−1),x_n)