Entropy numbers of Sobolev embeddings of radial Besov spaces

Let be the radial subspace of the Besov space . We prove the independence of the asymptotic behavior of the entropy numbers

from the difference s₀−s₁ as long as the embedding itself is compact. In fact, we shall show that

This is in a certain contrast to earlier results on entropy numbers in the context of Besov spaces B_p,q^s(Ω) on bounded domains Ω.     

Boundedness of Hausdorff operators on the power weighted Hardy spaces

In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space H_(|x|α)~1(R~2) ( -1 ≤α≤0), defined by H_(Φ,A)f(x)=∫R~2Φ(u)f(A(u)x)du,where Φ∈L_loc~1(R~2),A(u) = (α_(ij)(u))_(i,j=1)~2 is a 2×2 matrix, and each α_(i,j) is a measurablefunction.We obtain that HΦ,A is bounded from H_(|x|~α)~1(R~2) ( -1≤α≤0) to itself, if∫R2|Φ(u)‖det A~(-1)(u)|‖A(u)‖~(-α)ln(1+‖A~(-1)(u)‖~2/|det A~(-1)(u)|)du∞.This result improves some known theorems, and in some sense it is sharp.     

Local convergence of some iterative methods for generalized equations

We study generalized equations of the following form:

(render)

0f(x)+g(x)+F(x),

where f is Fréchet differentiable in a neighborhood of a solution x^* of (*) and g is Fréchet differentiable at x^* and where F is a set-valued map acting in Banach spaces. We prove the existence of a sequence (x_k) satisfying

which is super-linearly convergent to a solution of (*). We also present other versions of this iterative procedure that have superlinear and quadratic convergence, respectively.     

Characterizations of spaces in the unit ball of

Let B denote the unit ball of . For 0<p<∞, the holomorphic function spaces Q_p and Q_p,0 on the unit ball of are defined as

and

In this paper, we give some derivative-free, mixture and oscillation characterizations for Q_p and Q_p,0 spaces in the unit ball of .     

Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes

With the notation ,

we prove the following result.Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfies

||p||_L2(K)An^1/2 and ||p′||_L2(K)Bn^3/2.

Then

M₄(p)−M₂(p)M₂(p)

with

We also prove that

and

M₂(p)−M₁(p)10⁻³¹M₂(p)

for every , where denotes the collection of all trigonometric polynomials of the form

     

Weakly irregular quasiperiodic solutions of nonlinear Pfaff systems

We consider the nonlinear quasiperiodic Pfaff system

$$\frac{{\partial x}}{{\partial t_j }} = F^{(j)} (t,x) + G^{(j)} (t,x)(j = 1,...,m).$$

Let K ^(j) be a frequency basis with respect to t _j of the functions F ⁽¹⁾,...,F ^(m), and let L ^(j) be a frequency basis with respect to t _j of the functions G ⁽¹⁾,...,G ^(m). Suppose that the set K ^(j) ∪ L ^(j) of numbers is rationally linearly independent. We obtain necessary and sufficient conditions for the existence of quasiperiodic solutions with frequency bases L ⁽¹⁾,..., L ^(m).

     

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

     

Mapping properties of certain oscillatory integrals on modulation spaces

Suppose β1 α1 ≥0,β2 α2 ≥ 0 and(k,j) ∈R2. In this paper, we mainly investigate the mapping properties of the operator T_αβf(x,y,z)=∫_Q~2f(x-t,y-s,z-t~ks~j)e~(-2πit-β1_s-β2)t~(-1-α1)s~(-1-α2)dtds on modulation spaces, where Q~2 = [0,1] x [0,1] is the unit square in two dimensions.     

Bijective counting of plane bipolar orientations and Schnyder woods

A bijection Φ is presented between plane bipolar orientations with prescribed numbers of vertices and faces, and non-intersecting triples of upright lattice paths with prescribed extremities. This yields a combinatorial proof of the following formula due to Baxter for the number Θ_ij of plane bipolar orientations with i non-polar vertices and j inner faces:

In addition, it is shown that Φ specializes into the bijection of Bernardi and Bonichon between Schnyder woods and non-crossing pairs of Dyck words.This is the extended and revised journal version of a conference paper with the title “Bijective counting of plane bipolar orientations”, which appeared in Electr. Notes in Discr. Math. pp. 283–287 (Proceedings of Eurocomb’07, 11–15 September 2007, Sevilla).     

Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations

We consider the numerical discretization of singularly perturbed Volterra integro-differential equations (VIDE)

(*)

and Volterra integral equations (VIE)

(**)

by tension spline collocation methods in certain tension spline spaces, where is a small parameter satisfying 0<1, and q₁, q₂, g and K are functions sufficiently smooth on their domains to ensure that Eqs. (*) and (**) posses a unique solution.We give an analysis of the global convergence properties of a new tension spline collocation solution for 0<1 for singularly perturbed VIDE and VIE; thus, extending the existing theory for =1 to the singularly perturbed case.     

Perturbation of chains of de Branges spaces

We investigate the structure of the set of de Branges spaces of entire functions which are contained in a space L²(μ). Thereby, we follow a perturbation approach. The main result is a growth dependent stability theorem. Namely, assume that measures μ₁ and μ₂ are close to each other in a sense quantified relative to a proximate order. Consider the sections of corresponding chains of de Branges spaces C₁ and C₂ which consist of those spaces whose elements have finite (possibly zero) type with respect to the given proximate order. Then either these sections coincide or one is smaller than the other but its complement consists of only a (finite or infinite) sequence of spaces.

Among other situations, we apply—and refine—this general theorem in two important particular situations

1. (1)
   the measures μ₁ and μ₂ differ in essence only on a compact set; then stability of whole chains rather than sections can be shown
    
2. (2)
   the linear space of all polynomials is dense in L²(μ₂); then conditions for density of polynomials in the space L²(μ₂) are obtained.
    

In the proof of the main result, we employ a method used by P. Yuditskii in the context of density of polynomials. Another vital tool is the notion of the index of a chain, which is a generalisation of the index of determinacy of a measure having all power moments. We undertake a systematic study of this index, which is also of interest on its own right.

     

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, ∞).     

Meromorphic Jost functions and asymptotic expansions for Jacobi parameters

We show that the parameters a _n , b _n of a Jacobi matrix have a complete asymptotic expansion

$a_n^2 - 1 = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n} + O(R^{ - 2n} ),} b_n = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n + 1} + O(R^{ - 2n} )} $

, where 1 < |µ_j| < R for j ? K(R) and all R, if and only if the Jost function, u, written in terms of z (where E = z + z ^?1) is an entire meromorphic function. We relate the poles of u to the µ_j’s.

     

Existence of multiple positive solutions for nonlinear m-point boundary-value problems

In this paper, we afford some sufficient conditions to guarantee the existence of multiple positive solutions for the nonlinear m-point boundary-value problem for the one-dimensional p-Laplacian

(φ_p(u^′))^′+f(t,u)=0, t(0,1),

     

Positive solutions for one-dimensional -Laplacian boundary value problems with sign changing nonlinearity

This paper deals with the existence of positive solutions for the one-dimensional p-Laplacian

subject to the boundary value conditions:

where _p(s)=|s|^p−2s,p>1. We show that it has at least one or two positive solutions under some assumptions by applying the fixed point theorem. The interesting points are that the nonlinear term f is involved with the first-order derivative explicitly and f may change sign.     

Optimal embeddings and compact embeddings of Bessel-potential-type spaces

First, we establish necessary and sufficient conditions for embeddings of Bessel potential spaces \({H^{\sigma}X(\mathbb R^n)}\) with order of smoothness less than one, modelled upon rearrangement invariant Banach function spaces \({X(\mathbb R^n)}\), into generalized Hölder spaces. To this end, we derive a sharp estimate of modulus of smoothness of the convolution of a function \({f\in X(\mathbb R^n)}\) with the Bessel potential kernel g _σ , 0 < σ < 1. Such an estimate states that if \({g_{\sigma}}\) belongs to the associate space of X, then

$\omega(f*g_{\sigma},t)\precsim \int\limits_0^{t^n}s^{\frac{\sigma}{n}-1}f^*(s)\,ds \quad {\rm for\,all} \quad t\in(0,1) \quad {\rm and\,every}\quad f\in X(\mathbb R^n).$

Second, we characterize compact subsets of generalized Hölder spaces and then we derive necessary and sufficient conditions for compact embeddings of Bessel potential spaces \({H^{\sigma}X(\mathbb R^n)}\) into generalized Hölder spaces. We apply our results to the case when \({X(\mathbb R^n)}\) is the Lorentz–Karamata space \({L_{p,q;b}(\mathbb R^n)}\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces \({H^{\sigma}L_{p,q;b}(\mathbb R^n)}\) into generalized Hölder spaces and also compact embeddings of spaces in question. Applications cover both superlimiting and limiting cases.

     

Solvability of the -Laplacian with nonlocal boundary conditions

Rather mild sufficient conditions are provided for the existence of positive solutions of a boundary value problem of the form

which unify several cases discussed in the literature. In order to formulate these conditions one needs to know only properties of the homeomorphism and have information about the level of growth of the response operator F. No metric information concerning the linear operators L₀,L₁ in the boundary conditions is used, except that they are positive and continuous and such that L_j(1)<1 j{0,1}.     

Free resolutions in multivariable operator theory

Let be the complex polynomial ring in d variables. A contractive -module is Hilbert space equipped with an action such that for any ,

||z₁ξ₁+z₂ξ++z_dξ_d||²||ξ₁||²+||ξ₂||²++||ξ_d||².

Such objects have been shown to be useful for modeling d-tuples of mutually commuting operators acting on a Hilbert space. There is a subclass of the category of contractive modules whose members play the role of free objects. Given a contractive -module, one can construct a free resolution, i.e. an exact sequence of partial isometries of the following form:

(*)

where is a free module for each i0. The notion of a localization of a free resolution will be defined, in which for each λB_d there is a vector space complex of linear maps derived from (*):

We shall show that the homology of this complex is isomorphic to the homology of the Koszul complex of the d-tuple (¹,²,…,^d), of where ⁱ is the ith coordinate function of a Möbius transform on B_d such that (λ)=0.     

Prophet inequalities for parallel processes

Generalizations of prophet inequalities for single sequences are obtained for optimal stopping of several parallel sequences of independent random variables. For example, if {X_{i, j}, 1 ≤ i ≤ n, 1 ≤ j < ∞} are independent non-negative random variables, then

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and this bound is best possible. Applications are made to comparisons of the optimal expected returns of various alternative methods of stopping of parallel processes.