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.     

The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems

By constructing the comparison functions and the perturbed method, it is showed that any solution u∈C²(Ω) to the semilinear elliptic problems Δu=k(x)g(u), x∈Ω, u_|∂Ω=+∞ satisfies , where Ω is a bounded domain with smooth boundary in RN; , −2<σ, c₀>0, ; g∈C¹[0,∞), g?0 and is increasing on (0,∞), there exists ρ>0 such that , ∀ξ>0, , .     

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 .     

Strong convergence of the composite Halpern iteration

Let C be a closed convex subset of a uniformly smooth Banach space E and let T:C→C be a nonexpansive mapping with a nonempty fixed points set. Given a point u∈C, the initial guess x₀∈C is chosen arbitrarily and given sequences , and in (0,1), the following conditions are satisfied:

(i)

;

(ii)

αn→0, βn→0 and 0<a?γn, for some a∈(0,1);

(iii)

, and . Let be a composite iteration process defined by

     

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.     

On some iterated weighted spaces

It is proved that the Hörmander and spaces (Ω₁⊂Rn, Ω₂⊂Rm open sets, 1?p<∞, ki Beurling-Björck weights, k=k₁⊗k₂) are isomorphic whereas the iterated spaces and are not if 1<p≠q<∞. A similar result for weighted Lp-spaces of entire analytic functions is also obtained. Finally a result on iterated Besov spaces is given: and are not isomorphic when 1<q≠2<∞.     

q-Differential operator identities and applications

In this paper, we construct a new q-exponential operator and obtain some operator identities. Using these operator identities, we give a formal extension of Jackson's transformation formula. A formal extension of Bailey's summation and an extension of the Sears terminating balanced transformation formula are also derived by our operator method. In addition, we also derive several interesting a formal extensions involving multiple sum about three terms of Sears transformation formula and Heine's transformation formula.     

Trace representation of weights on partial O-algebras

The purpose of this paper is to investigate when a weight ? on a partial O^∗-algebra is a trace weighted by a positive self-adjoint operator Ω, that is, whenever s.t. X†∈L(X) and ?(X†□X)<∞. It is shown that if contains the inverse N of a positive compact operator such that the weak multiplication N□N is defined, then every weight ? on satisfying ?(N□N)<∞ is a trace weighted by some positive trace operator.     

Generalized vector valued almost periodic and ergodic distributions

Schwartz's almost periodic distributions are generalized to the case of Banach space valued distributions , and furthermore for a given arbitrary class A to for φ∈ test functions D(R,C)}. It is shown that this extension process is iteration complete, i.e. . Moreover the T from are characterized in various ways, also tempered distributions with P={X-valued functions of polynomial growth} are shown. Under suitable assumptions , , where for all h>0}, is defined with the corresponding extension of Mh. With an extension of the indefinite integral from to D^′(R,X) a distribution analogue to the Bohl-Bohr-Amerio-Kadets theorem on the almost periodicity of bounded indefinite integrals of almost periodic functions is obtained, also for almost automorphic, Levitan almost periodic and recurrent functions, similar for a result of Levitan concerning ergodic indefinite integrals. For many of the above results a new (Δ)-condition is needed, we show that it holds for most of the A needed in applications. Also an application to the study of asymptotic behavior of distribution solutions of neutral integro-differential-difference systems is given.     

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.     

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.     

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.     

Multipliers between Sobolev spaces and fractional differentiation

We characterize the pointwise multipliers which maps a Sobolev space to a Sobolev space in the case |s|<r<d/2.     

Weighted norm inequalities of Bochner-Riesz means

Let w be a Muckenhoupt weight and be the weighted Hardy spaces. We use the atomic decomposition of and their molecular characters to show that the Bochner-Riesz means are bounded on for 0<p?1 and δ>max{n/p−(n+1)/2,[n/p]rw⁻¹(rw−1)−(n+1)/2}, where rw is the critical index of w for the reverse Hölder condition. We also prove the boundedness of the maximal Bochner-Riesz means for 0<p?1 and δ>n/p−(n+1)/2.     

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.     

Systems of orthogonal polynomials arising from the modular j-function

Let be the polynomial whose zeros are the j-invariants of supersingular elliptic curves over . Generalizing a construction of Atkin described in a recent paper by Kaneko and Zagier (Computational Perspectives on Number Theory (Chicago, IL, 1995), AMS/IP 7 (1998) 97-126), we define an inner product on for every . Suppose a system of orthogonal polynomials {P_n,ψ(x)}_n=0^∞ with respect to exists. We prove that if n is sufficiently large and ψ(x)P_n,ψ(x) is p-integral, then over . Further, we obtain an interpretation of these orthogonal polynomials as a p-adic limit of polynomials associated to p-adic modular forms.