Exact asymptotic behavior of singular values of a class of integral operators

We find an exact asymptotic formula for the singular values of the integral operator of the form , a Jordan measurable set) where and L is slowly varying function with some additional properties. The formula is an explicit expression in terms of L and T.     

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.     

Spectral Estimations for the Laplace Operator on the Discrete Heisenberg Group

We estimate the spectral measure of the Laplace operator for the discrete Heisenberg group with generators x and y in the vicinity of the unity. Bibliography: 7 titles.     

The Limit of Lebesgue Constants for Summation Methods of the Fourier―Legendre Series Defined by a Multiplier Function

Let C[-1,1] be the space of continuous functions f:[-1,1] with the uniform norm, let _Pk be the Legendre polynomials such that _Pk (1)=1, and let _J0 be the Bessel function of zero index. We consider sequences of linear operators (summation methods) _Un:C [-1,1] C[-1,1] defined by a multiplier function as follows:

A note on normal varieties of monounary algebras

A variety is called normal if no laws of the form s = t are valid in it where s is a variable and t is not a variable. Let L denote the lattice of all varieties of monounary algebras (A,f) and let V be a non-trivial non-normal element of L. Then V is of the form with some n > 0. It is shown that the smallest normal variety containing V is contained in for every m > 1 where C denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of L consisting of all normal elements of L is isomorphic to L.     

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.     

The impact of unbounded swings of the forcing term on the asymptotic behavior of functional equations

Necessary and sufficient conditions have been found to force all solutions of the equation

On the Sunouchi Operator with Respect to the Walsh-Kaczmarz System

Define the operator of Sunouchi (f L ¹) on the Walsh group with respect to the Walsh-Kaczmarz system. In this paper we prove that the operator T is of weak type (1, 1), of type (H ¹, L ¹) and of type (p, p) for all 1 < p 2.     

Hardy and Bellman Transformations in Spaces Close to L ∞ and L 1

The replacement of coefficients of a trigonometric series by their arithmetic averages gives rise to the Hardy operator. The Bellman operator is its adjoint. The spaces _Lp with p[1,) are invariant under the Hardy transformation. This result was proved by Hardy. On the other hand, the space L is not invariant under the Hardy transformation, and the space _L1 is not invariant under the Bellman transformation. Golubov proved that the space BMO is not invariant under the Hardy transformation and is not invariant under the Bellman operator. In the present paper, the exact ``shift' of the domain under the action of these operators is described for certain Orlicz, Lorenz, and Marcinkiewicz spaces and the spaces BMO and . For the Hardy operator, this shift occurs if the domain is close to L , and for the Bellman operator the same happens if the domain is close to L ₁. Bibliography: 15 titles.     

L p Boundedness for the Multilinear Singular Integral Operator

L^p (\mathbbRⁿ )L^{p} (\mathbb{R}^{n} ) boundedness is considered for the maximal multilinear singular integral operator which is defined by

Discrete spectrum and principal functions of non-selfadjoint differential operator

In this article, we consider the operator L defined by the differential expression in L ₂(–, ), where q is a complex valued function. Discussing the spectrum, we prove that L has a finite number of eigenvalues and spectral singularities, if the condition holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.     

Wolff's Inequality for Radially Nonincreasing Kernels and Applications to Trace Inequalities

We extend Th. Wolff's inequality to a general class of radially decreasing convolution kernels. As an application we obtain characterizations of nonnegative Borel measures on R ⁿ such that the trace inequality holds for every f in L ^p (dx).     

On a System of Step Functions

It is well known that the Riemann hypothesis is equivalent to the following statement: the identity function belongs to the linear span in L ²(0,1) of the family

Explicit Strong Solutions of SPDE’s with Applications to Non-Linear Filtering

A differential calculus for random fields is developed and combined with the S-transform to obtain an explicit strong solution of the Cauchy problem

Gaussian radial‐basis functions: Cardinal interpolation of ℓp and power‐growth data

Suppose λ is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal function \(L_\lambda (x) = \sum\nolimits_{k \in \mathbb{Z}} {c_k \exp ( - \lambda (x - k)^2 ),x \in \mathbb{R}} ,\) satisfying the interpolatory conditions \(L_\lambda (j) = \delta _{0j} ,j \in \mathbb{Z}.\) The paper considers the Gaussian cardinal interpolation operator

$(\mathcal{L}_\lambda {\text{y}})(x): = \sum\limits_{k \in \mathbb{Z}} {y_k L_\lambda (x - k),{\text{ y}} = (y_k )_{k \in \mathbb{Z}} ,{\text{ }}x \in \mathbb{R}} ,$

as a linear mapping from ℓ^p(ℤ) into L ^p(ℝ), 1≤ p ∞, and in particular, its behaviour as λ→0⁺. It is shown that \(\left\| {\mathcal{L}_\lambda } \right\|_p \) is uniformly bounded (in λ) for 1 < p < ∞, and that \(\left\| {\mathcal{L}_\lambda } \right\|_1 \asymp \log (1/\lambda )\) as λ→0⁺. The limiting behaviour is seen to be that of the classical Whittaker operator

$\mathcal{W}:{\text{y}} \mapsto \sum\limits_{k \in \mathbb{Z}} {y_k \frac{{\sin \pi (x - k)}}{{\pi (x - k)}}} ,$

in that \(\lim _{\lambda \to 0^ + } \left\| {\mathcal{L}_\lambda {\text{y}} - \mathcal{W}{\text{y}}} \right\|_p = 0,\) for every \({\text{y}} \in \ell ^p (\mathbb{Z}){\text{ and }}1 < p < \infty .\) It is further shown that the Gaussian cardinal interpolants to a function f which is the Fourier transform of a tempered distribution supported in (-π,π) converge locally uniformly to f as λ→0⁺. Multidimensional extensions of these results are also discussed.

     

Correction to: ‘The UCT, the Milnor Sequence, and a Canonical Decomposition of the Kasparov Groups’

In this note we correct a mistake in K-Theory 10 (1996), 49–72. In that paper we asserted that under bootstrap hypotheses the short exact sequence

Second-Order Subexponential Behavior of Subordinated Sequences

Suppose that , , and are three discrete probability distributions related by the equation (E): , where denotes the k-fold convolution of In this paper, we investigate the relation between the asymptotic behaviors of and . It turns out that, for wide classes of sequences and , relation (E) implies that , where is the mean of . The main object of this paper is to discuss the rate of convergence in this result. In our main results, we obtain O-estimates and exact asymptotic estimates for the difference .     

Uniform Approximation of Nonperiodic Functions Defined on the Entire Axis

Using the following notation: C is the space of continuous bounded functions f equipped with the norm , V is the set of functions f such that , the set E consists of fCV and possesses the following property:

Efficient Estimation in a Semiparametric Autoregressive Model

This paper constructs efficient estimates of the parameter in the semiparametric auto-regression model ,with a smooth function and independent and identically distributed innovations _t with zero means and finite variances. This will be done under the assumptions that and that the errors have a density with finite Fisher information for location. The former condition guarantees that the process can be chosen to be stationary and ergodic.     

Sharp Estimates for Deviations of Linear Approximation Methods for Periodic Functions by Linear Combinations of Moduli of Continuity of Different Order

Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that is odd, . Then