A continuous function space with a Faber basis

Let be compact with #S=∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (P_n)_n=0^∞ with deg P_n=n such that every fC(S) has a unique representation f=∑_i=0^∞ α_iP_i and call such a basis Faber basis. In the special case of , 0<q<1, we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, these polynomials state an example with A(S_q)≠U(S_q)=C(S_q), where A(S_q) is the so-called Wiener algebra and U(S_q) is the set of all fC(S_q) which are uniquely represented by its Fourier series.     

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 Lp-Generalization of a Theorem of Adamyan, Arov, and Kreın

The paper deals with problems relating to the theory of Hankel operators. Let G be a bounded simple connected domain with the boundary Γ consisting of a closed analytic Jordan curve. Denote by _n,p(G), 1p<∞, the class of all meromorphic functions on G that can be represented in the form h=β/α, where β belongs to the Smirnov class E_p(G), α is a polynomial degree at most n, α0. We obtain estimates of s-numbers of the Hankel operator A_f constructed from fL_p(Γ), 1p<∞, in terms of the best approximation Δ_n,p of f in the space L_p(Γ) by functions belonging to the class _n,p(G).     

Uniform estimates for polynomial approximation in domains with corners

Let be a domain with a Jordan boundary ∂G, consisting of l smooth curves Γ_j, such that {z_j}Γ_j-1∩Γ_j≠, j=1,…,l, where Γ₀Γ_l. Denote by α_jπ, 0<α_j2, the angles at z_j's between the curves Γ_j-1 and Γ_j, exterior with respect to G. Let Φ be a conformal mapping of the exterior of onto the exterior of the unit disk, normed by Φ^′(∞)>0. We assume that there is a neighborhood U of , such that , where

z≠z_j if α_j1. Set g_Gsup{|g(z)|:zG}. Then we prove Theorem. Let and 0βr. If a function f is analytic in G and f^(r)β_G<+∞, then for each nlr there is an algebraic polynomial P_n of degree <n, such that

     

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.     

Convergence of greedy approximation for the trigonometric system

Summary We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form G_m(f ) := ∑_kЄΛ f^(k) e ^{(i k,x)}, where ΛˆZ^d is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients f^ (k) of function f . Note that G_m(f ) gives the best m-term approximant in the L₂-norm and, therefore, for each f ЄL₂, ║f-G_m(f )║₂→0 as m →∞. It is known from previous results that in the case of p ≠2 the condition f ЄL_p does not guarantee the convergence ║f-G_m(f )║_p→0 as m →∞.. We study the following question. What conditions (in addition to f ЄL_p) provide the convergence ║f-G_m(f )║_p→0 as m →∞? In our previous paper [10] in the case 2< p ≤∞ we have found necessary and sufficient conditions on a decreasing sequence {A_n}_n=1^∞ to guarantee the L_p-convergence of {G_m(f )} for all f ЄL_p , satisfying a_n (f ) ≤A_n , where {a_n (f )} is a decreasing rearrangement of absolute values of the Fourier coefficients of f. In this paper we are looking for necessary and sufficient conditions on a sequence {M (m)} such that the conditions f ЄL_p and ║G_M(m)(f ) - G_m(f )║_p →0 as m →∞ imply ║f - G_m(f )║_p →0 as m →∞. We have found these conditions in the case when p is an even number or p = ∞.     

Weak Asymptotics for the Generating Polynomials of the Stirling Numbers of the Second Kind

For the horizontal generating functions P_n(z)=∑ⁿ_k=1 S(n, k) z^k of the Stirling numbers of the second kind, strong asymptotics are established, as n→∞. By using the saddle point method for Q_n(z)=P_n(nz) there are two main results: an oscillating asymptotic for z(−e, 0) and a uniform asymptotic on every compact subset of \[−e, 0]. Finally, an Airy asymptotic in the neighborhood of −e is deduced.     

A Remez-Type Inequality for Non-dense Müntz Spaces with Explicit Bound

LetΛ :=(λ_k)^∞_k=0be a sequence of distinct nonnegative real numbers withλ₀ :=0 and ∑^∞_k=1 1/λ_k<∞. Let(0, 1) and(0, 1−) be fixed. An earlier work of the authors shows that [formula]is finite. In this paper an explicit upper bound forC(Λ, , ) is given. In the special caseλ_k :=k^α,α>1, our bounds are essentially sharp.     

Completeness of Trigonometric System with Integer Indices {e; x }

Necessary and sufficient conditions are given which ensure the completeness of the trigonometric systems with integer indices; {e^inx; x }^∞_n=−∞ or {e^inx; x }^∞_n=1 in L^α(μ,  ), α1. If there exists a support Λ of the measure μ which is a wandering set, that is, Λ+2kπ, k=0, ±1, ±2, … are mutually disjoint for different k's, then the linear span of our trigonometric system {e^inx; x }^∞_n=−∞ is dense in L^α(μ,  ) α1. The converse statement is also true.     

On the Rate of Approximation of Closed Jordan Curves by Lemniscates

As proved by Hilbert, it is, in principle, possible to construct an arbitrarily close approximation in the Hausdorff metric to an arbitrary closed Jordan curve Γ in the complex plane {z} by lemniscates generated by polynomials P(z). In the present paper, we obtain quantitative upper bounds for the least deviations H _n (Γ) (in this metric) from the curve Γ of the lemniscates generated by polynomials of a given degree n in terms of the moduli of continuity of the conformal mapping of the exterior of Γ onto the exterior of the unit circle, of the mapping inverse to it, and of the Green function with a pole at infinity for the exterior of Γ. For the case in which the curve Γ is analytic, we prove that H _n (Γ) = O(q _n ), 0 ≤ q = q(Γ) < 1, n → ∞.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 861–876.Original Russian Text Copyright ©2005 by O. N. Kosukhin.     

Littlewood–Paley and Multiplier Theorems on Weighted Spaces over Locally Compact Vilenkin Groups

We extend the Littlewood–Paley theorem toL^p_w(G), whereGis a locally compact Vilenkin group andware weights satisfying the MuckenhouptA_pcondition. As an application we obtain a mixed-norm type multiplier result onL^p_w(G) and prove the sharpness of our result. We also obtain a sufficient condition for φ L^∞(Γ) to be a multiplier on the power weightedL^p_α(G) in terms of its smoothness condition.     

On the asymptotic behaviour of a semigroup of linear operators

Let T = {T(t)}_{t ≥ 0} be a C₀-semigroup on a Banach space X. In this paper, we study the relations between the abscissa ω_Lp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ω^p_R(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ω_β(T), β ≥ 0, of T. Our main results are as follows.

1. (i) Let T be a C₀-semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω_{1 − ε}(T) < 0 for some ε > 0.

2. (ii) Let T be a C₀-semigroup on L^p such that the resolvent of the generator is uniformly bounded in the right half plane. Then ω_β(T) < 0 for all β>¦1/p − 1/p′¦, 1/p + 1/p′ = 1.

3. (iii) Let 1 ≤ p ≤ 2 and let T be a weakly L^p-stable C₀-semigroup on a Banach space X. Then for all β>1/p we have ω_β(T) ≤ 0.

Further, we give sufficient conditions in terms of ω^q_R(A) for the existence of L^p-solutions and W^1,p-solutions (1 ≤ p ≤ ∞) of the abstract Cauchy problem for a general class of operators A on X.     

Special Uniform Approximations of Continuous Vector-Valued Functions. Part I: Special Approximations in CX(T)

In this paper, we give special uniform approximations of functions u from the spaces C_X(T) and C_∞(T,X), with elements of the tensor products C_Γ(T)X, respectively C₀(T,Γ)X, for a topological space T and a Γ-locally convex space X. We call an approximation special, if satisfies additional constraints, namely supp vu⁻¹(X\{0}) and (T) co(u(T)) (resp. co(u(T){0})). In Section 3, we give three distinct applications, which are due exactly to these constraints: a density result with respect to the inductive limit topology, a Tietze–Dugundji's type extension new theorem and a proof of Schauder–Tihonov's fixed point theorem.     

Uniformly Gâteaux smooth approximations on

Every Lipschitz mapping from c₀(Γ) into a Banach space Y can be uniformly approximated by Lipschitz mappings that are simultaneously uniformly Gâteaux smooth and C^∞-Fréchet smooth.     

On the sequence of powers of a stochastic matrix with large exponent

We consider the class of primitive stochastic n×n matrices A, whose exponent is at least (n²−2n+2)/2+2. It is known that for such an A, the associated directed graph has cycles of just two different lengths, say k and j with k>j, and that there is an α between 0 and 1 such that the characteristic polynomial of A is λⁿ−αλ^n−j−(1−α)λ^n−k. In this paper, we prove that for any mn, if α1/2, then A^m+k−A^m_∞A^m−1w^T_∞, where 1 is the all-ones vector and w^T is the left-Perron vector for A, normalized so that w^T1=1. We also prove that if jn/2, n31 and , then A^m+j−A^m_∞A^m−1w^T_∞ for all sufficiently large m. Both of these results lead to lower bounds on the rate of convergence of the sequence A^m.     

Dimension Theory and Nonstable K1 of Quadratic Modules

Employing Bak's dimension theory, we investigate the nonstable quadratic K-group K _1,2n (A, ) = G _2n (A, )/E _2n (A, ), n 3, where G _2n (A, ) denotes the general quadratic group of rank n over a form ring (A, ) and E _2n (A, ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G _2n (A, ) G _2n ⁰(A, ) ; G _2n ¹(A, ) ... E _2n (A, ) of the general quadratic group G _2n (A, ) such that G _2n (A, )/G _2n ⁰(A, ) is Abelian, G _2n ⁰(A, ) G _2n ¹(A, ) ... is a descending central series, and G _2n ^d(A)(A, ) = E _2n (A, ) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K _1,2n (A, ) is solvable when d(A) < .     

Équations du type Monge-Ampère sur les variétés Riemanniennes compactes, I

Let (V_n, g) be a C^∞ compact Riemannian manifold. For a suitable function on V_n, let us consider the change of metric: g′ = g + Hess(), and the function, as a ratio of two determinants, M() = ¦g′¦ ¦g¦⁻¹. Using the method of continuity, we first solve in C^∞ the problem: Log M() = λ + ƒ, λ > 0, ƒ ε C^∞. Then, under weak hypothesis on F, we solve the general equation: Log M() = F(P, ), F in C^∞(V_n × ¦α, β¦), using a method of iteration. Our study gives rise to an interesting a priori estimate on ¦¦, which does not occur in the complex case. This estimate should enable us to solve the equation above when λ 0, providing we can overcome difficulties related to the invertibility of the linearised operator. This open question will be treated in our next article.     

A Singular Riesz Product in the Nevai Class and Inner Functions with the Schur Parameters in ∩p>2 l

There exist singular Riesz products dσ=∏^∞_κ=1 (1+Re(α_κζ^n_κ)) on the unit circle with the parameters (a_n)_n0 of orthogonal polynomials in L²(dσ) satisfying ∑^∞_n=0 |a_n|^p<+∞ for every p, p>2. The Schur parameters of the inner factor of the Cauchy integral ∫ (ζ−z)⁻¹ dσ(ζ), σ being such a Riesz product, belong to ∩_p>2 l^p.     

Removable Singularities for Lu=Ψ(u) and Orlicz Capacities

Suppose L is a second order elliptic differential operator in ^d and let α>1. Baras and Pierre have proved in 1984 that Γ is removable for Lu=u^α if and only if its Bessel capacity Cap_2, α′(Γ)=0. We extend this result to a general equation Lu=Ψ(u) where Ψ(u) is an increasing convex function subject to Δ₂ and ₂ conditions. Namely, we prove that Γ is removable for Lu=Ψ(u) if and only if its Orlicz capacity is zero, that is, the integral ∫_B dx Ψ(∫_Γ |x−y|^2−d ν(dy)) is equal to 0 or ∞ for every measure ν concentrated on Γ, where B stands for any ball containing Γ.     

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.