Best L2 Local Approximation on Two Small Intervals

In this article, we introduce the τ condition, which is weaker than the L² differentiability. If a function satisfies the τ condition on two points of ?, we prove the existence and characterization of the best local polynomial approximation on these points.     

On the jackson theorem for periodic functions in spaces with integral metric

We consider the approximation of periodic functions by trigonometric polynomials in metric (not normed) spaces that are generalizations of the spaces L _p , 0 < p < 1, and L ₀. In particular, we prove the multidimensional Jackson theorem in L _p (T ^m ), 0 < p < 1.     

An operator onL pwithout best compact approximation

We construct, for 1<p<∞,p ≠ 2, an operator onL ^pwhose distance to the space of compact operators onL ^pis not attained. We also show that the identity operator onL ^p,p ≠ 1,2, ∞ has a unique best compact approximation. Research partially supported by NSF grant DMS-8201635.     

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some L^p（R^n） to L^p（R^n） when the m-th order derivatives of A belong to L^p（R^n） for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are （L^p, L^p）-type operators for 1 〈 p 〈 ∞.     

Global smoothness and uniform convergence of smooth Gauss–Weierstrass singular operators

In this article we continue with the study of smooth Gauss–Weierstrass singular integral operators over the real line regarding their simultaneous global smoothness preservation property with respect to the L_p norm, 1≤p≤∞, by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of continuity with respect to the uniform norm. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function.     

Approximation ofk-Monotone Functions

It is shown that an algebraic polynomial of degree k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inL_p, 0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inL_p, 1<p<∞, iso(n^−k/p).     

Best m-ter m trigono metric approxi mation for the classes $ B_{p,{{\uptheta }}}^r $ of functions of low s moothness

We obtain an exact-order estimate for the best m-term trigonometric approximation of the Besov classes B_p,\uptheta ^r B_{p,{{\uptheta }}}^r of periodic functions of many variables of low smoothness in the space L _q , 1 < p ≤ 2 < q < ∞.     

Analytic approximation of matrix functions in

We consider the problem of approximation of matrix functions of class L^p on the unit circle by matrix functions analytic in the unit disk in the norm of L^p, 2≤p<∞. For an m×n matrix function Φ in L^p, we consider the Hankel operator , 1/p+1/q=1/2. It turns out that the space of m×n matrix functions in L^p splits into two subclasses: the set of respectable matrix functions and the set of weird matrix functions. If Φ is respectable, then its distance to the set of analytic matrix functions is equal to the norm of H_Φ. For weird matrix functions, to obtain the distance formula, we consider Hankel operators defined on spaces of matrix functions. We also describe the set of p-badly approximable matrix functions in terms of special factorizations and give a parametrization formula for all best analytic approximants in the norm of L^p. Finally, we introduce the notion of p-superoptimal approximation and prove the uniqueness of a p-superoptimal approximant for rational matrix functions.     

Best Simultaneous L p -Approximation on Small Regions

In this article, we study the behavior of best simultaneous L ^p -approximation by algebraic polynomials on a union of intervals when the measure of them tend to zero. We also get an interpolation result.     

Some Fourier-type operators for functions on unbounded intervals

In order to approximate functions defined on the real line or on the real semiaxis by polynomials, we introduce some new Fourier-type operators, connected to the Fourier sums of generalized Freud or Laguerre orthonormal systems. We prove necessary and sufficient conditions for the boundedness of these operators in suitable weighted L ^p -spaces, with 1 < p < ∞. Moreover, we give error estimates in weighted L ^p and uniform norms.     

A Sharp Jackson Inequality in Lp(ℝd) with Dunkl Weight

A sharp Jackson inequality in the space L_p(?^d), 1 ≤ p < 2, with Dunkl weight is proved. The best approximation is realized by entire functions of exponential spherical type. The modulus of continuity is defined by means of a generalized shift operator bounded on L_p, which was constructed earlier by the authors. In the case of the unit weight, this operator coincides with the mean-value operator on the sphere.

     

Walsh Multipliers for Dyadic Hardy Spaces

In this article we are interested in conditions on the coefficients of a Walsh multiplier operator that imply the operator is bounded on certain dyadic Hardy spaces H ^p , 0 < p < ∞. In particular, we consider two classical coefficient conditions, originally introduced for the trigonometric case, the Marcinkiewicz and the Hörmander–Mihlin conditions. They are known to be sufficient in the spaces L ^p , 1 < p < ∞. Here we study the corresponding problem on dyadic Hardy spaces, and find the values of p for which these conditions are sufficient. Then, we consider the cases of H ¹ and L ¹ which are of special interest. Finally, based on a recent integrability condition for Walsh series, a new condition is provided that implies that the multiplier operator is bounded from L ¹ to L ¹, and from H ¹ to H ¹. We note that existing multiplier theorems for Hardy spaces give growth conditions on the dyadic blocks of the Walsh series of the kernel, but these growth are not computable directly in terms of the coefficients.     

Marcinkiewicz–Zygmund Inequalities and Polynomial Approximation from Scattered Data on SO(3)

We consider scattered data approximation problems on SO(3). To this end, we construct a new operator for polynomial approximation on the rotation group. This operator reproduces Wigner-D functions up to a given degree and has uniformly bounded L ^p -operator norm for all 1 ≤ p ≤ ∞. The operator provides a polynomial approximation with the same approximation degree of the best polynomial approximation. Moreover, the operator together with a Markov type inequality for Wigner-D functions enables us to derive scattered data L ^p -Marcinkiewicz–Zygmund inequalities for these functions for all 1 ≤ p ≤ ∞. As a major application of such inequalities, we consider the stability of the weighted least squares approximation problem on SO(3).     

On approximation of functions from below by splines of the best approximation with free nodes

Let M be the set of functions integrable to the power β=(r+1+1/p)^-1. We obtain asymptotically exact lower bounds for the approximation of individual functions from the set M by splines of the best approximation of degree rand defect k in the metric of L _p.     

Estimates of Best Approximation and Fourier Transforms in Integral Metrics

Under some assumptions on a function F and its Fourier transform we prove new estimates of best approximation of F by entire functions of exponential type σ in L_p( ), 1 ≤ p < 2. The proof is based on some inequalities for in L₁( ) which may be treated as generalizations of results of Bausov and Telyakovskii. As an application we obtain exact estimates of best approximation of some infinitely differentiable functions.     

Approximation by Faber-Laurent rational functions in the mean of functions of callsL p(Γ) with 1

Çavu§  A.  Israfilov  D. M. 《分析论及其应用》1995,11(1):105-118

On approximation of functions by trigonometric polynomials with incomplete spectrum in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, 0 < <Emphasis Type="Italic">p</Emphasis> < 1

Let B be a set of integers with certain arithmetic properties. We obtain estimates of the best approximation of functions in the space L _p , 0 < p <1, by trigonometric polynomials that are constructed by the system {e^ikx}_{k ? \mathbbZ\B} \{e^{ikx}\}_{k\in \mathbb{Z}\backslash B} . Bibliography: 13 titles.     

Nonlinear approximation by renormalized trigonometric system

We study the convergence of greedy algorithmwith regard to renormalized trigonometric system. Necessary and sufficient conditions are found for system’s normalization to guarantee almost everywhere convergence, and convergence in L ^p (T) for 1 < p < ∞ of the greedy algorithm, where T is the unit torus. Also the non existence is proved for normalization which guarantees convergence almost everywhere for functions from L ¹(T), or uniform convergence for continuous functions.     

Characterization ofL p (R n ) using Gabor frames

We characterize L^p norms of functions onR ⁿ for 1<p<∞ in terms of their Gabor coefficients. Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of L^p functions converge to the functions almost everywhere and in L^p for 1<p<∞. In L¹ we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L¹ in a certain Cesàro sense. Consequently, we are able to establish that a large class of Gabor families generate Banach frames for L^p (R ⁿ) when 1≤p<∞.