Kolmogorov and linear widths of weighted Besov classes

We study the Kolmogorov m-widths and the linear m-widths of the weighted Besov classes on [−1,1], where Lq,μ, 1?q?∞, denotes the Lq space on [−1,1] with respect to the measure , μ>0. Optimal asymptotic orders of and as m→∞ are obtained for all 1?p,τ?∞. It turns out that in many cases, the orders of are significantly smaller than the corresponding orders of the best m-term approximation by ultraspherical polynomials, which is somewhat surprising.     

Positive integral operators in unbounded domains

We study positive integral operators in with continuous kernel k(x,y). We show that if the operator is compact and Hilbert-Schmidt. If in addition k(x,x)→0 as |x|→∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and is trace class. Replacing the first assumption by the stronger then and the bilinear series converges also in L¹. Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.     

A note on maximality of analytic crossed products

Let G be a compact abelian group with the totally ordered dual group which admits the positive semigroup . Let N be a von Neumann algebra and be an automorphism group of on N. We denote to the analytic crossed product determined by N and α. We show that if is a maximal σ-weakly closed subalgebra of , then induces an archimedean order in .     

Wavelet frames with irregular matrix dilations and their stability

Let , B and A_j () be real nonsingular n×n matrices, λ_k () be real numbers. In this paper we present a sufficient condition for the system to be a frame for . This sufficient condition also shows the stability of the system with respect to the perturbation of matrix dilation parameters and the perturbation of translation parameters .     

On powers of the Heaviside function for negative integers

The distributions and were defined as the neutrix limit of the sequences and respectively for , see [J.D. Nicholos, B. Fisher, The distribution composition , J. Math. Anal. Appl. 258 (2001) 131-145; B. Fisher, On defining the distribution , Univ. u Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 15 (1985) 119-129]. We here consider these distributions when r=0. In other words, we define the sth powers of the Heaviside function H(x) in the distributional sense for negative integers. Further compositions are also considered.     

Majorization of regular measures and weights with finite and positive critical exponent

For the sets , 1?p<∞, of positive finite Borel measures μ on the real axis with the set of algebraic polynomials P dense in Lp(R,dμ), we establish a majorization principle of their “boundaries,” i.e. for every there exists such that dμ/dν?1. A corresponding principle holds for the sets , p>0, of non-negative upper semi-continuous on R functions (weights) w such that P is dense in the space : For every there exists such that w?ω.     

Convergence in mean of some random Fourier series

For a symmetric stable process X(t,ω) with index α∈(1,2], f∈Lp[0,2π], p?α, and , we establish that the random Fourier-Stieltjes (RFS) series converges in the mean to the stochastic integral , where fβ is the fractional integral of order β of the function f for . Further it is proved that the RFS series is Abel summable to . Also we define fractional derivative of the sum of order β for an, An(ω) as above and . We have shown that the formal fractional derivative of the series of order β exists in the sense of mean.     

Various types of stochastic integrals with respect to fractional Brownian sheet and their applications

In this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let be a fractional Brownian sheet with Hurst parameters H=(H₁,H₂), and (²[0,1],B(²[0,1]),μ) a measure space. By using the techniques of stochastic calculus of variations, we introduce stochastic line integrals along all sufficiently smooth curves γ in ²[0,1], and four types of stochastic surface integrals: , i=1,2, , , , . As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters H₁,H₂∈(1/4,1). Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process.     

Reflexivity and hyperreflexivity of operator spaces

In this paper, we show that if is an n-dimensional subspace of L(V) such that every nonzero transformation of has rank greater than or equal to 2n−1 then is algebraically reflexive. If is an n-dimensional subspace of B(H) such that every nonzero transformation of  has rank greater than or equal to 2n−1 then is hyperreflexive. We also consider how to construct some new hyperreflexive subspaces.     

Continuous dependence results for inhomogeneous ill-posed problems in Banach space

We apply semigroup theory and other operator-theoretic methods to prove Hölder-continuous dependence on modeling for the inhomogeneous ill-posed Cauchy problem in Banach space. The inhomogeneous ill-posed Cauchy problem is given by , u(0)=χ, 0?t<T; where −A is the infinitesimal generator of a holomorphic semigroup on a Banach space X, χ∈X, and . For a suitable function f, the approximate problem is given by , v(0)=χ. Under certain stabilizing conditions, we prove that for a related norm, where and M are computable constants independent of β, 0<β<1, and ω(t) is a harmonic function. These results extend earlier work of Ames and Hughes on the homogeneous ill-posed problem.     

Asymptotics for self-normalized random products of sums of i.i.d. random variables

Let be a sequence of independent and identically distributed positive random variables, which is in the domain of attraction of the normal law, and tn be a positive, integer random variable. Denote , , where denotes the sample mean. Then we show that the self-normalized random product of the partial sums, , is still asymptotically lognormal under a suitable condition about tn.     

Continuity and Schatten-von Neumann p-class membership of Hankel operators with anti-holomorphic symbols on (generalized) Fock spaces

In this paper we investigate Hankel operators with anti-holomorphic symbols ∈L²(C,m|z|), where are general Fock spaces. We will show that is not continuous if the corresponding symbol is not a polynomial . For polynomial symbols we will give necessary and sufficient conditions for continuity and compactness in terms of N and m. For monomials we will give a complete characterization of the Schatten-von Neumann p-class membership for p>0. Namely in case 2k<m the Hankel operators are in the Schatten-von Neumann p-class iff p>2m/(m−2k); and in case 2k?m they are not in the Schatten-von Neumann p-class.     

On the asymptotic stability of equilibrium

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a positive definite function V which has infinitesimal upper bound such that is negative definite. In this paper we prove that if is bounded then the condition that is negative definite can be weakened and replaced by that and is negative definite.     

Mean inequalities in certain Banach function spaces

Boundedness of the Hardy operator and its adjoint is characterized between Banach function spaces Xq and Lp. By applying a limiting procedure, corresponding boundedness of the geometric mean operator is also derived.