Jackson and Bernstein theorems for the weight exp(−|x|) on ℝ

In 1978, Freud, Giroux and Rahman established a weightedL ₁ Jackson theorem for the weight exp(−|x|) on the real line, using methods that work only inL ₁. This weight is somewhat exceptional, for it sits on the boundary between weights like exp(-|x|^α), α≥1, where weighted polynomials are dense, and the case α<1, where weighted polynomials are not dense. We obtain the firstL _p Jackson theorem for exp(−|x|), valid in allL _p , 0<p≤∞, as well as for higher order moduli of continuity. We also establish a converse Bernstein type theorem, characterizing rates of approximation in terms of smoothness of the approximated function. Research supported by NSF grant DMS 0400446.     

Converse and Smoothness Theorems for Erdo&#x030B;s Weights inL_(0<p?∞)

We prove converse and smoothness theorems of polynomial approximation in weightedL_pspaces with norm ‖fW‖_Lp()(0<p?∞) for Erdo&#x030B;s weights on the real line. In particular we prove characterization theorems involving realization functionals and thereby establish some interesting properties of our weighted modulus of continuity.     

On Hille-Tamarkin operators and Schatten classes

We determine the smallest Schatten class containing all integral operators with kernels inL _p(L_{p', q})^symm, where 2 <p∞ and 1≦q≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1].     

On the sharpness of Jackson’s inequality in the spaces L p on the half-line with power weight

In the space L _p , 1 ≤ p < 2, on the half-line with power weight, Jackson’s inequality between the value of the best approximation of a function by even entire functions of exponential type and its modulus of continuity defined by means of a generalized shift operator is well known. The question of the sharpness of the inequality remained open. For the constant in Jackson’s inequality, we obtain a lower bound, which proves its sharpness.     

A note on the sunouchi operator with respect to the walsh—kaczmarz system

An analogue of the so—called Sunouchi operator with respect to the Walsh—Kaczmarz system will be investigated. We show the boundedness of this operator if we take it as a map from the dyadic Hardy space H ^p to L ^p for all 0<p≤1.. For the proof we consider a multiplier operator and prove its (H ^p H ^p)—boundedness for 0<p≤1. Since the multiplier is obviously bounded from L ² to L ², a theorem on interpolation of operators can be applied to show that our multiplier is of weak type (1,1) and of type (q q) for all 1<q<∞. The same statements follow also for the Sunouchi operator.     

On the Order of Decrease of Uniform Moduli of Smoothness for the Classes of Functions E p,m [ɛ]

In this paper, we solve the problem of the exact order of decrease of uniform moduli of smoothness for the classes of 2π-periodic functions of several variables with a given majorant of the sequence of total best approximations in the metric of L _p , 1 ≤ p < ∞.     

On Jodeit’s multiplier extension theorems

Let Δ(x) = max {1 - ¦x¦, 0} for all x ∈ ?, and let ξ_[0,1) be the characteristic function of the interval 0 ≤x < 1. Two seminal theorems of M. Jodeit assert that A and ξ_[0,1) act as summability kernels convertingp-multipliers for Fourier series to multipliers forL ^P (?). The summability process corresponding to Δ extendsL ^P (T)-multipliers from ? to ? by linearity over the intervals [n, n + 1],n ∈ ?, when 1 ≤p < ∞, while the summability process corresponding to ξ_[0,1) extends L^P(T)-multipliers by constancy on the intervals [n, n + 1),n ∈ ?, when 1 <p < ∞. We describe how both these results have the following complete generalization: for 1 ≤p < ∞, an arbitrary compactly supported multiplier forL ^P (?) will act as a summability kernel forL ^P (T)-multipliers, transferring maximal estimates from L^P(T) to L^P(?). In particular, specialization of this maximal theorem to Jodeit’s summability kernel ξ_{[0, 1)} provides a quick structural way to recover the fact that the maximal partial sum operator on L^P(?), 1 <p < ∞, inherits strong type (p,p)-boundedness from the Carleson-Hunt Theorem for Fourier series. Another result of Jodeit treats summability kernels lacking compact support, and we show that this aspect of multiplier theory sets up a lively interplay with entire functions of exponential type and sampling methods for band limited distributions.     

The Choquet theorem and stochastic equations

где,у- произвольное положительное число,В —абсолютная положительная постоянная, — произв ольная последовател ьность знаков ±1. Из этого неравенства следует, что если 1<P<∞ иg∈L _p (0, 1), то. Отсюда сразу вытекае т, что система Франкли на является безусловным базисом в пространствахL _p (0, 1) для 1 <р<∞.     

Direct and converse theorems of the theory of approximation in the metric of Lp for 0 < p < 1

A direct theorem of the Jackson type and several converse theorems are established for the approximation of periodic functions of period 2Π by trigonometrical polynomials in the metric of L_p, 0 < p < 1.     

On the geometric mean operator with variable limits of integration

A new criterion for the weighted L _p ?L _q boundedness of the Hardy operator with two variable limits of integration is obtained for 0 < q < q + 1 ≤ p < ∞. This criterion is applied to the characterization of the weighted L _p ?L _q boundedness of the corresponding geometric mean operator for 0 < q < p < ∞.     

Approximation by Nörlund means of quadratical partial sums of double Walsh-Fourier series

In this article we discuss the Nörlund means of cubical partial sums of Walsh-Fourier series of a function in L ^p (1 ≤ p ≤ ∞). We investigate the rate of the approximation by this means, in particular, in Lip(α, p), where α > 0 and 1 ≤ p ≤ ∞. In case p = ∞ by L ^p we mean C _W , the collection of the uniformly W-continuous functions. Our main theorems state that the approximation behavior of the two-dimensional Walsh- Nörlund means is so good as the approximation behavior of the one-dimensional Walsh- Nörlund means. As special cases, we get the Nörlund logarithmic means of cubical partial sums of Walsh-Fourier series discussed recently by Gát and Goginava [5] in 2004 and the (C, β)-means of Marcinkiewicz type with respect to double Walsh-Fourier series discussed by Goginava [10]. Earlier results on one-dimensional Nörlund means of the Walsh-Fourier series was given by Móricz and Siddiqi [14].     

Boundary properties of functions harmonic in a strip

We study in the L_p-norm, 1≤p≤∞, the boundary properties of the solution to the Dirichlet problem for the stripA ={(x, y):?∞0} and its dependence on the structural properties of the given boundary values (symmetric, antisymmetric). In particular, for the case of symmetric boundary values we obtain direct and inverse theorems on approximation in terms of the general modulus of continuity of second order.     

Sufficient conditions for the Lebesgue integrability of Fourier transforms

We give sufficient conditions for the Lebesgue integrability of the Fourier transform of a function f ∈ L ^p (?) for some 1 < p ≤ 2. These sufficient conditions are in terms of the L ^p integral modulus of continuity of f; in particular, they apply for functions in the integral Lipschitz class Lip(α, p) and for functions of bounded s-variation for some 0 < s < p. Our theorems are nonperiodic versions of the classical theorems of Bernstein, Szász, Zygmund and Salem, and recent theorems of Gogoladze and Meskhia on the absolute convergence of Fourier series.     

Inductive limits of spaces of vector- valued integrable functions

For an order-continuous Banach function space Λ and a separated inductive limit E:= ind_n E _n, we prove that ind_n A {E_n} is a topological subspace of Λ {E}; moreover, both spaces coincide if the inductive limit is hyperstrict. As a consequence, we deduce that if E is an LF-space, then L ^p {E} is barrelled for 1 ≤ p ≤ ∞.     

Global weak solutions of the Cauchy problem for semilinear pseudo-hyperbolic equations

The Cauchy problem for a class of semilinear pseudo-hyperbolic equations is considered. For the corresponding linear problems, we obtain L _p → L _q estimates. By using these estimates, we prove global solvability theorems. We also establish the behavior of solutions as t → + ∞.     

Optimal arguments in jackson’s inequality in the power-weighted space L 2(ℝ d )

This paper is devoted to the determination of the optimal arguments in the exact Jackson inequality in the space L ₂ on the Euclidean space with power weight equal to the product of the moduli of the coordinates with nonnegative powers. The optimal arguments are studied depending on the geometry of the spectrum of the approximating entire functions and the neighborhood of zero in the definition of the modulus of continuity. The optimal arguments are obtained in the case where the first skew field is a l _p ^d -ball for 1 ≤p≤ 2, and the second is a parallelepiped.     

Smoothness and Best Approximation with Respect to Freud-Type Weights,a New Approach

A new measure of smoothness is defined and related to best approximation by polynomials with respect to weighted L _p (R) with Freud-type weights. Other related norms are also discussed. Comparisons with the known measure of smoothness on weighted L _p spaces are obtained. Related K-functionals and realization functionals are introduced. The new measure of smoothness allows us to consider a more general class of function spaces, to achieve Marchaud, Jackson and Bernstein-type inequalities, and to relate it to expressions involving the coefficients of the expansion by orthogonal polynomials with respect to Freud-type weights. Some of the results are new for approximation by Hermite polynomials in the weighted L _p space with the weight \({e^{-x^{2}}}\) .     

Converse theorems for approximation by bernstein polynomials in Lp[0,1] (1<p<∞)

The class of all continuous functions possessing n^?α(1/p<α≤1) order of approximation by Bernstein polynomials inL _p[0, 1] is characterized.     

Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces W _p ^k (? ⁿ , E) with k ∈ ?₀, 1 ≤ p ≤ ∞. Here E is an arbitrary Banach space. We also apply the theory to solve non-autonomous parabolic pseudodifferential equations in Sobolev spaces.