Nikolskii type inequality for cardinal splines

The Nikolskii type inequality for cardinal splines

Estimate of fourier transforms with respect to the system of generalized eigenfunctions of the Schrödinger operator with Stummel-type potential

Suppose thatА is a nonnegative self-adjoint extension to { } of the formal differential operator−Δu+q(x)u with potentialq(x) satisfying the condition {

Integrals involving products of bessel functions

Summary The integrals and , where n is any positive integer, are evaluated in terms ofMacRobert E-functions and generalized hypergeometric functions.     

On the existence of components of the Noether-Lefschetz locus with given codimension

It is known that the codimension c of a component of the Noether-Lefschetz locus NL(d) satisfies . We prove that ford≥47 and for every integer there exists a component of NL(d) with codimension c. This is done with families of surfaces of degree d in ℙ³ containing a curve lying on a cubic or on a quartic surface or a curve with general moduli. Moreover we produce an explicit example, for everyd≥4, of components of maximum codimension , thus giving a new proof of the fact that these components are dense in the locus of smooth surfaces (density theorem).     

Monotonicity of Average Power Means

A new numerical inequality for average power means is presented. Let and let be a sequence of positive numbers. Consider the operator . We denote by the superposition of these operators. The following assertion is proved: if . Bibliography: 2 titles.     

On the dimension of the cartesian product of relations and orders

It is known that for incidence structures and , max , wheref dim stands for Ferrers relation. We shall show that under additional assumptions on and , both bounds can be improved. Especially it will be shown that the square of a three-dimensional ordered set is at least four-dimensional.     

Asymptotic behaviour of the phase in non-smooth obstacle scattering

In this paper, we study the asymptotic behaviour of the scattering phases(λ) of the Dirichlet Laplacian associated with obstacle , where Ω is a bounded open subset of ℝ ⁿ (n≥2) with non-smooth boundary ∂Ω and connected complement Ω _e =ℝ ⁿ . We can prove that if Ω satisfies a certain geometrical condition, then

On representations of Ψ* andC *-algebras ofb-pseudo-differential operators on manifolds with corners

The author computes the Jacobson topology and the corresponding Borel structure on the spectrum of . Here denotes the C^*-algebra generated by the algebra of b-pseudo-differential operators of orderO on a compact manifold with corners (in the sense of Melrose). The author also reduces the problem of classification of representations to the problem of classification of positive Borel measures on certain well-known Borel spaces. Bibliography:29 titles. Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 85–117.     

The true order of magnitude of Lamé rounding error sums

Let (z ∈ ℝ). Further let λ denote a large real parameter. We show that for arbitrary real numbersk and α withk>=2.7013 and 0<α≦1,

Quantitative aspect of correction theorems. II

Assume that 0<ε≤1, F ∈ C( ), E={≠0}, δ>0. Then there exists a function G with uniformly convergent Fourier series such that |G|+|F−G|≤(1+δ)|F|, m{F≠G}≤εmE, and . Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 83–91.     

Resonant nonlinear singular problems in the limit circle case

Existence results are presented for the resonant singular boundary value problem a.e. on [0, 1] with lim _t0+py=y(1)=0. Here we donot assume but only that .     

On the range of a generalized derivation

A generalized derivation , is defined by the formula , where and is the Banach algebra of bounded linear operators in a Hilbert space . Sufficient conditions under which and are given. Bibliography: 8 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 111–119.     

A criterion for the existence of bounded nonoscillatory solutions for nonlinear retarded differential equatory

Summary For the equation (⋆) below we give a condition which guarantees the existence of a bounded nonoscillatory solution x with Entrata in Redazione il 19 luglio 1977.     

On the Lipschitz Constant for a Nonisometric Bi-Lipschitz Transformation of a Cantor Set

A bi-Lipschitz continuous mapping of a space X is a bijection such that , where . We write if f is a Lipschitz (bi-Lipschitz) mapping of X into itself and denote by the set of all bi-Lipschitz mappings of X that are not isometry. Thus, if and blip . For X we consider a standard Cantor set K on the real line (with standard metric). The main result of this paper is formulated as follows: where Bibliography: 2 titles.     

The solutions of the<Emphasis Type="Italic">n</Emphasis>-dimensional Bessel diamond operator and the Fourier-Bessel transform of their convolution

In this article, the operator is introduced and named as the Bessel diamond operator iteratedk times and is defined by

Iteration theory in hyperbolic domains

We introduce the notion of pointwise regularity ( ) of Colombeau’s generalized functions and give comparison theorems between regularity, - andC ^∞-regularity. We also define the notion of a pointwise wave front set and establish a theorem concerning the effect of a linear generalized partial differential operator on such a wave front.     

Sequence Spaces l p,q in Probabilistic Characterizations of Weak Type Operators

We study operators (not necessarily linear) defined on a quasi-Bahach space X and taking values in the space of real-valued Lebesgue-measurable functions. Factorization theorems for linear and superlinear operators with values in the space are proved with the help of the Lorentz sequence spaces . Sequences of functions belonging to fixed bounded sets in the spaces are characterized for and . The possibility of distinguishing weak type operators (bounded in the space ) from operators factorizable through is obtained in terms of sequences of independent random variables. A criterion under which an operator is symmetrically bounded in order in , is established. Some refinements of the above-mentioned results are obtained for translation shift-invariant sets and operators. Bibliography: 30 titles.     

Extension to R + n+1 of singular integrals and fractional integrals

In this paper, a natural R ₊ ⁿ⁺¹ extension of singular integrals, i.e.,T _κ:f→K*φ _t *t with K a standard C-Z kernel and ϕ usual one, is investigated. One of the main results is: Let (dμ, udx) ∈C₁ and u-Mw, w∈A_∞, then T_k is of type (L^p(udx), L^p(dμ)). As a related topic, a maximal operator is proved to be of type , where , provided (dμ, udx) ∈C₁ and u∈ A_∞. Supported by National Science Foundation of China     

Three primes theorem in a short interval (V)

In this paper, we shall prove that for a sufficiently large odd numberN, the equation has solutions. The Project Supported by National Natural Science Foundation of China