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 .     

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.     

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, , .     

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.     

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

     

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 .     

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.     

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 number of positive solutions of nonlinear systems

We prove that appropriate combinations of superlinearity and sublinearity of with respect to at zero and infinity guarantee the existence, multiplicity, and nonexistence of positive solutions to boundary value problems for the n-dimensional system 0<t<1. The vector-valued function is defined by where and ? covers the two important cases ?(u′)=u′ and ?(u′)=|u′|p−2u′, p>1. Our methods employ fixed point theorems in a cone.     

Eventually norm continuous semigroups on Hilbert space and perturbations

The eventually norm continuous semigroups on Hilbert space and perturbation are studied in this paper. By resolvent of infinitesimal generator, the sufficient and necessary conditions for eventually norm continuous semigroups are given. Using the result obtained, it is proved that if is infinitesimal generator of an eventually norm continuous semigroup T(t), then there is a subspace Ξ_A of such that, for any , the semigroup S(t) generated by preserves the property of T(t).     

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.     

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.     

Bounded Toeplitz products on the weighted Bergman spaces of the unit ball

Let p>1 and let q denote the number such that (1/p)+(1/q)=1. We give a necessary condition for the product of Toeplitz operators to be bounded on the weighted Bergman space of the unit ball (α>−1), where and , as well as a sufficient condition for to be bounded on . We use techniques different from those in [K. Stroethoff, D. Zheng, Bounded Toeplitz products on Bergman spaces of the unit ball, J. Math. Anal. Appl. 325 (2007) 114-129], in which the case p=2 was proved.