Widths of weighted Sobolev classes on the ball

We study the Kolmogorov n-widths and the linear n-widths of weighted Sobolev classes on the unit ball B^d in L_q,μ, where L_q,μ, 1≤q≤∞, denotes the weighted L_q space of functions on B^d with respect to weight . Optimal asymptotic orders of and as n→∞ are obtained for all 1≤p,q≤∞ and μ≥0.     

A quantitative version of Trotter's approximation theorem

A quantitative version, based on modified K-functionals, of the classical Trotter's theorem concerning the approximation of C₀-semigroups is presented. The result is applied to the study of the degree of convergence of the iterated Bernstein operators on the N-dimensional simplex to their limiting semigroup.     

Gradient estimates and Jackson’s theorem in spaces related to measures

Jackson’s theorem is established in a new kind of holomorphic function space Q_μ related to measures in any starlike circular domain in . Particularly, the result covers many spaces including BMOA, Q_p, Q_K, and F(p,q,s) spaces in the unit ball of . Moreover, we construct integral operators which give pointwise estimates for the gradient of the difference in terms of the gradient on the boundary. The gradient estimates are independent of the measures in question and give rise to Jackson’s theorem.     

Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and -minimal polynomials,

Let be a sequence of polynomials with real coefficients such that uniformly for [α-δ,β+δ] with G(eⁱ)≠0 on [α,β], where 0α<βπ and δ>0. First it is shown that the zeros of are dense in [α,β], have spacing of precise order π/n and are interlacing with the zeros of p_n+1(cos) on [α,β] for every nn₀. Let be another sequence of real polynomials with uniformly on [α-δ,β+δ] and on [α,β]. It is demonstrated that for all sufficiently large n the zeros of p_n(cos) and strictly interlace on [α,β] if on [α,β]. If the last expression is zero then a weaker kind of interlacing holds. These interlacing properties of the zeros are new for orthogonal polynomials also. For instance, for large n a simple criteria for interlacing of zeros of Jacobi polynomials on [-1+,1-], >0, is obtained. Finally it is shown that the results hold for wide classes of weighted L_q-minimal polynomials, q[1,∞], linear combinations and products of orthogonal polynomials, etc.     

On -convergence of Fourier series

In this paper we consider the trigonometric Fourier series with the β-general monotone coefficients. Necessary and sufficient conditions of L₁-convergence for such a series, that is f−S_n=o(1), are obtained in terms of coefficients.     

The behavior of best -approximations as . A counterexample of convergence

In the space of summable sequences we give an example of a one-dimensional affine subspace C such that the best L_p-approximations of 0 from C fail to converge as p↓1. We thus give an answer to this problem of convergence in infinite measure spaces.     

On Bernstein–Markov-type inequalities for multivariate polynomials in -norm

Bernstein–Markov-type inequalities provide estimates for the norms of derivatives of algebraic and trigonometric polynomials. They play an important role in Approximation Theory since they are widely used for verifying inverse theorems of approximation. In the past decades these inequalities were extended to the multivariate setting, but the main emphasis so far was on the uniform norm. It is considerably harder to derive Bernstein–Markov-type inequalities in the L_q-norm, and it requires introduction of new methods. In this paper we verify certain Bernstein–Markov-type inequalities in L_q-norm on convex and star-like domains. Special attention is given to the question of how the geometry of the domain affects the corresponding estimates.     

The convergence of three spline methods for scattered data interpolation and fitting

The convergences of three L₁ spline methods for scattered data interpolation and fitting using bivariate spline spaces are studied in this paper. That is, L₁ interpolatory splines, splines of least absolute deviation, and L₁ smoothing splines are shown to converge to the given data function under some conditions and hence, the surfaces from these three methods will resemble the given data values.     

Strong Haagerup inequalities with operator coefficients

We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, denotes the subspace of the von Neumann algebra of a free group F_I spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on , which improves and generalizes previous results by Kemp–Speicher (in the scalar case) and Buchholz and Parcet–Pisier (in the non-holomorphic setting). Namely the norm of an element of the form ∑_{i=(i1,…,id)}a_iλ(g_i1g_id) is less than , where M₀,…,M_d are d+1 different block-matrices naturally constructed from the family (a_i)_iI^d for each decomposition of I^dI^l×I^d−l with l=0,…,d. It is also proved that the same inequality holds for the norms in the associated non-commutative L_p spaces when p is an even integer, pd and when the generators of the free group are more generally replaced by *-free -diagonal operators. In particular it applies to the case of free circular operators. We also get inequalities for the non-holomorphic case, with a rate of growth of order d+1 as for the classical Haagerup inequality. The proof is of combinatorial nature and is based on the definition and study of a symmetrization process for partitions.     

Hyperbolic topology of normed linear spaces

In a previous paper [H. Tsuiki, Y. Hattori, Lawson topology of the space of formal balls and the hyperbolic topology of a metric space, Theoret. Comput. Sci. 405 (2008) 198–205], the authors introduced the hyperbolic topology on a metric space, which is weaker than the metric topology and naturally derived from the Lawson topology on the space of formal balls. In this paper, we characterize spaces L_p(Ω,Σ,μ) on which the hyperbolic topology induced by the norm _p coincides with the norm topology. We show the following:

(1) The hyperbolic topology and the norm topology coincide for 1<p<∞.

(2) They coincide on L₁(Ω,Σ,μ) if and only if μ(Ω)=0 or Ω has a finite partition by atoms.

(3) They coincide on L_∞(Ω,Σ,μ) if and only if μ(Ω)=0 or there is an atom in Σ.

Positive solutions for Robin problem involving the -Laplacian

Consider Robin problem involving the p(x)-Laplacian on a smooth bounded domain Ω as follows

Applying the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that there exists λ_*>0 such that the problem has at least two positive solutions if λ(0,λ_*), has at least one positive solution if λ=λ_*<+∞ and has no positive solution if λ>λ_*. To prove the results, we prove a norm on W^1,p(x)(Ω) without the part of ||_L^p(x)(Ω) which is equivalent to usual one and establish a special strong comparison principle for Robin problem.     

Linear information versus function evaluations for -approximation

We study algorithms for the approximation of functions, the error is measured in an L₂ norm. We consider the worst case setting for a general reproducing kernel Hilbert space of functions. We analyze algorithms that use standard information consisting in n function values and we are interested in the optimal order of convergence. This is the maximal exponent b for which the worst case error of such an algorithm is of order n^-b.Let p be the optimal order of convergence of all algorithms that may use arbitrary linear functionals, in contrast to function values only. So far it was not known whether p>b is possible, i.e., whether the approximation numbers or linear widths can be essentially smaller than the sampling numbers. This is (implicitly) posed as an open problem in the recent paper [F.Y. Kuo, G.W. Wasilowski, H. Woźniakowski, On the power of standard information for multivariate approximation in the worst case setting, J. Approx. Theory, to appear] where the authors prove that implies . Here we prove that the case and b=0 is possible, hence general linear information can be exponentially better than function evaluation. Since the case is quite different, it is still open whether b=p always holds in that case.     

Kernel estimates and -spectral independence of generators of -semigroups

After the appearance of W. Arendt's result that “Gaussian estimate of a semigroup implies the L^p-spectral independence of the generator,” various generalizations have been obtained. This paper shows that a certain kernel estimate of a semigroup implies the L^p-spectral independence of the generator, generalizing the case of upper Gaussian estimate and “Gaussian estimate of order α(0,1] [S. Miyajima, H. Shindoh, Gaussian estimates of order α and L^p-spectral independence of generators of C₀-semigroups, Positivity 11 (1) (2007) 15–39], Definition 3.1.” The proof uses S. Karrmann's result about the L^p-spectral independence and B.A. Barnes' theorem about the spectrum of integral operators. As an application, the L^p-spectral independence of −[(−Δ)^α+V] (α(0,1]) for a suitable V is proved with the help of a recent result by V. Liskevich, H. Vogt and J. Voigt [V. Liskevich, H. Vogt, J. Voigt, Gaussian bounds for propagators perturbed by potentials, J. Funct. Anal. 238 (2006) 245–277].     

A regularization semismooth Newton method based on the generalized Fischer–Burmeister function for -NCPs

We consider a regularization method for nonlinear complementarity problems with F being a P₀-function which replaces the original problem with a sequence of the regularized complementarity problems. In this paper, this sequence of regularized complementarity problems are solved approximately by applying the generalized Newton method for an equivalent augmented system of equations, constructed by the generalized Fischer–Burmeister (FB) NCP-functions φ_p with p>1. We test the performance of the regularization semismooth Newton method based on the family of NCP-functions through solving all test problems from MCPLIB. Numerical experiments indicate that the method associated with a smaller p, for example p[1.1,2], usually has better numerical performance, and the generalized FB functions φ_p with p[1.1,2) can be used as the substitutions for the FB function φ₂.     

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}.     

Bidual renormings of Banach spaces

Given a separable Banach space X with no isomorphic copies of ℓ₁ and a separable subspace Y of its bidual, we provide a sufficient condition on Y to ensure that X admits an equivalent norm such that the restriction to Y of the corresponding bidual norm is midpoint locally uniformly rotund. This result applies to the separable subspaces of the bidual of a Banach space with a shrinking unconditional Schauder basis and to the bidual of the James space.     

A note on preconditioned AOR method for -matrices

In this note, some errors in a recent article by Li et al. (Improvements of preconditioned AOR iterative methods for L-matrices, J. Comput. Appl. Math. 206 (2007) 656–665) are pointed out and some correct results are presented.     

On the forms of the truncated LU decomposition

Matrix LU decomposition has six ijk forms. Different forms have different computational complexities and storage requirements, particularly on vector and parallel computers. Other factors governing the choice of a particular form are considered. For treating Fredholm integral equations of the first kind, the truncated LU decomposition of the resulting system matrix is recommended. Required modifications to selected known ijk forms are presented.     

Existence of periodic solutions for a fourth-order -Laplacian equation with a deviating argument

In this paper, we study the existence of periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument as follows:

[φ_p(u^″(t))]^″+f(u^″(t))+g(u(t−τ(t)))=e(t).