Weierstrass'' Theorem in Weighted Sobolev Spaces

We characterize the set of functions which can be approximated by polynomials with the following norm

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for a big class of weights w₀, w₁, …, w_k     

Estimates of Freud-Christoffel functions for some weights with the whole real line as support

Upper and lower bounds for generalized Christoffel functions, called Freud-Christoffel functions, are obtained. These have the form λ_n,p(W,j,x) = infPW_Lp(R)/|P^(j)(X)| where the infimum is taken over all polynomials P(x) of degree at most n − 1. The upper and lower bounds for λ_n,p(W,j,x) are obtained for all 0 < p ∞ and J = 0, 1, 2, 3,… for weights W(x) = exp(−Q(x)), where, among other things, Q(x) is bounded in [− A, A], and Q″ is continuous in β(−A, A) for some A > 0. For p = ∞, the lower bounds give a simple proof of local and global Markov-Bernstein inequalities. For p = 2, the results remove some restrictions on Q in Freud's work. The weights considered include W(x) = exp(− ¦x¦^α/2), α > 0, and W(x) = exp(− exp(¦x¦)), > 0.     

Asymptotics for Lp extremal polynomials on the unit circle

Let p > 1, and dμ a positive finite Borel measure on the unit circle Γ: = {z ε C: ¦z¦ = 1}. Define the monic polynomial φ_{n, p}(z)=zⁿ+…εP_n >(the set of polynomials of degree at most n) satisfying

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. Under certain conditions on dμ, the asymptotics of φ_{n, p}(z) for z outside, on, or inside Γ are obtained (cf. Theorems 2.2 and 2.4). Zero distributions of φ_{n, p} are also discussed (cf. Theorems 3.1 and 3.2).     

Asymptotic Behaviour of Best lp-Approximations from Affi ne Subspaces

In this paper we consider the problem of best approximation in ℓ_pⁿ, 1<p∞. If h_p, 1<p<∞, denotes the best ℓ_p-approximation of the element h ⁿ from a proper affine subspace K of ⁿ, hK, then lim_p→∞h_p=h_∞^*, where h_∞^* is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all r there are α_j ⁿ, 1jr, such that

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, with γ_p^{(r) n} and γ_p^(r)= (p^−r−1).     

Curves of positive solutions of boundary value problems on time-scales

Let TR be a time-scale, with a=infT, b=supT. We consider the nonlinear boundary value problem

(2)

(4)

u(a)=u(b)=0,

where λR₊:=[0,∞), and satisfies the conditions

We prove a strong maximum principle for the linear operator defined by the left-hand side of (1), and use this to show that for every solution (λ,u) of (1)–(2), u is positive on T a,b . In addition, we show that there exists λ_max>0 (possibly λ_max=∞), such that, if 0λ<λ_max then (1)–(2) has a unique solution u(λ), while if λλ_max then (1)–(2) has no solution. The value of λ_max is characterised as the principal eigenvalue of an associated weighted eigenvalue problem (in this regard, we prove a general existence result for such eigenvalues for problems with general, nonnegative weights).     

Joint Distributions of the Numbers of Visits for Finite-State Markov Chains

For a discrete-time Markov chain with finite state space {1, …, r} we consider the joint distribution of the numbers of visits in states 1, …, r−1 during the firstNsteps or before theNth visit tor. From the explicit expressions for the corresponding generating functions we obtain the limiting multivariate distributions asN→∞ when staterbecomes asymptotically absorbing and forj=1, …, r−1 the probability of a transition fromrtojis of order 1/N.     

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

     

The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

In this paper, we establish new asymptotic relations for the errors of approximation in L_p[−1,1], 0<p∞, of x^λ, λ>0, by the Lagrange interpolation polynomials at the Chebyshev nodes of the first and second kind. As a corollary, we show that the Bernstein constant

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is finite for λ>0 and .     

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.     

Markov-Type Inequalities for Products of Müntz Polynomials

Let Λ(λ_j)^∞_j=0 be a sequence of distinct real numbers. The span of {x^λ₀, x^λ₁, …, x^λ_n} over is denoted by M_n(Λ)span{x^λ₀, x^λ₁, …, x^λ_n}. Elements of M_n(Λ) are called Müntz polynomials. The principal result of this paper is the following Markov-type inequality for products of Müntz polynomials. T 2.1. LetΛ(λ_j)^∞_j=0andΓ(γ_j)^∞_j=0be increasing sequences of nonnegative real numbers. Let

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Then

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18(n+m+1)(λ_n+γ_m).In particular ,

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Under some necessary extra assumptions, an analog of the above Markov-type inequality is extended to the cases when the factor x is dropped, and when the interval [0, 1] is replaced by [a, b](0, ∞).     

On Positive and Copositive Polynomial and Spline Approximation inLp[−1, 1], 0<p<∞

For a functionfL_p[−1, 1], 0<p<∞, with finitely many sign changes, we construct a sequence of polynomialsP_nΠ_nwhich are copositive withfand such that f−P_npCω(f, (n+1)⁻¹)_p, whereω(f, t)_pdenotes the Ditzian–Totik modulus of continuity inL_pmetric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, thenωcannot be replaced byω²if 1<p<∞. In fact, we show that even for positive approximation and all 0<p<∞ the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained.     

Orthogonal Rational Functions and Nested Disks

In Akhiezer's book [“The Classical Moment Problem and Some Related Questions in Analysis,” Oliver & Boyd, Edinburghasol;London, 1965] the uniqueness of the solution of the Hamburger moment problem, if a solution exists, is related to a theory of nested disks in the complex plane. The purpose of the present paper is to develop a similar nested disk theory for a moment problem that arises in the study of certain orthogonal rational functions. Let {α_n}^∞_n=0be a sequence in the open unit disk in the complex plane, let

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( /|α_k|=−1 whenα_k=0), and let

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We consider the following “moment” problem: Given a positive-definite Hermitian inner product ·, · on × , find a non-decreasing functionμon [−π, π] (or a positive Borel measureμon [−π,π)) such that

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In particular we give necessary and sufficient conditions for the uniqueness of the solution in the case that If this series diverges the solution is always unique.     

Remarks Concerning Completeness of Translates in Function Spaces

This note explains how to translate the author's old result on cyclic vectors of the multiple shift operator into the language of completeness theorems for integer translates. This translation, together with those results, turns out to be a source for many completeness theorems. In particular, there follows the existence of functions f whose positive integer translates f(x−k), where k ₊ are complete in the spaces C^l₀( ), L^p( ), W^l_p( ), 2<p<∞, l=0, 1, …, as well as in their weighted and/or vector-valued analogues.     

Quadrature formulae connected to σ-orthogonal polynomials

Let dλ(t) be a given nonnegative measure on the real line , with compact or infinite support, for which all moments exist and are finite, and μ₀>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes

where σ=σ_n=(s₁,s₂,…,s_n) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness d_max=2∑_ν=1ⁿs_ν+2n−1 if and only if

The proof of the uniqueness of the extremal nodes τ₁,τ₂,…,τ_n was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term R(f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τ_ν, ν=1,2,…,n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.     

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.     

Weighted Hardy inequalities

For bounded Lipschitz domains D in it is known that if 1<p<∞, then for all β[0,β₀), where β₀=p−1>0, there is a constant c<∞ with

for all . We show that if D is merely assumed to be a bounded domain in that satisfies a Whitney cube-counting condition with exponent λ and has plump complement, then the same inequality holds with β₀ now taken to be . Further, we extend the known results (see [H. Brezis, M. Marcus, Hardy's inequalities revisited, Dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997–1998) 217–237; M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, A geometrical version of Hardy's inequality, J. Funct. Anal. 189 (2002) 537–548; J. Tidblom, A geometrical version of Hardy's inequality for W^1,p(Ω), Proc. Amer. Math. Soc. 132 (2004) 2265–2271]) concerning the improved Hardy inequality

c=c(n,p), by showing that the class of domains for which the inequality holds is larger than that of all bounded convex domains.     

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 Sharp Inequality of Markov Type for Polynomials Associated with Laguerre Weight

The best possible constant A_n in an inequality of Markov type

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, where ·_[0, ∞) denotes the sup-norm on the half real line [0, ∞) and p_n is an arbitrary polynomial of degree at most n, is determined in terms of the weighted Chebyshev polynomials associated with the Laguerre weight e^−x on [0, ∞).     

On Rado numbers for

For all integers m3 and all natural numbers a₁,a₂,…,a_m−1, let n=R(a₁,a₂,…,a_m−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to

a₁x₁+a₂x₂++a_m−1x_m−1=x_m.