Approximation by entire functions on countable unions of segments of the real axis: 2. proof of the main theorem

In this study, we consider an approximation of entire functions of Hölder classes on a countable union of segments by entire functions of exponential type. It is essential that the approximation rate in the neighborhood of segment ends turns out to be higher in the scale that had first appeared in the theory of polynomial approximation by functions of Hölder classes on a segment and made it possible to harmonize the so-called “direct” and “inverse” theorems for that case, i.e., restore the Hölder smoothness by the rate of polynomial approximation in this scale. Approximations by entire functions on a countable union of segments have not been considered earlier. The first section of this paper presents several lemmas and formulates the main theorem. In this study, we prove this theorem on the basis of earlier given lemmas.     

Some problems of approximation theory in the spacesL p on the line with power weight

In the spaces L _p on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the K-functional. We prove a direct and an inverse theorem of Jackson-Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the K-functional.     

Approximation by Entire Functions on a Countable Union of Segments on the Real Axis: 3. Further Generalization

In this paper, an approximation of functions of extensive classes set on a countable unit of segments of a real axis using the entire functions of exponential type is considered. The higher the type of the approximating function is, the higher the rate of approximation near segment ends can be made, compared with their inner points. The general approximation scale, which is nonuniform over its segments, depending on the type of the entire function, is similar to the scale set out for the first time in the study of the approximation of the function by polynomials. For cases with one segment and its approximation by polynomials, this scale has allowed us to connect the so-called direct theorems, which state a possible rate of smooth function approximation by polynomials, and the inverse theorems, which give the smoothness of a function approximated by polynomials at a given rate. The approximations by entire functions on a countable unit of segments for the case of Hölder spaces have been studied by the authors in two preceding papers. This paper significantly expands the class of spaces for the functions, which are used to plot an approximation that engages the entire functions with the required properties.     

d维Bernstein算子加Jacobi权的收敛阶

１．引 言 设Ｓ＝Ｓｄ（ｄ＝１,２,…）是 Ｒｄ中的单纯形,即记ｋ＝（ｋ１,ｋ２,……,ｋｄ）∈Ｒｄ,ｋｉ为非负整数, ,则Ｓ上定义的函数ｆ所对应的ｄ维Ｂｅｒｎｓｔｅｉｎ算子定义为其中 Ｐｎ,ｋ（ｘ）＝是 Ｂｅｒｎｓｔｅｉｎ基函数．引进多维Ｊａｃｏｂｉ权函数, 这里 ．定义Ｂｅｒｎｓｔｅｉｎ权函数 表示微分算子． 记 是单位向量,即第ｉ个分量为１,其余ｄ－１个分量为０, ．定义函数ｆ在方向ｅ上的ｒ阶对称差分为Ｃ（Ｓ）中的加权Ｓｏｂｏｌｅｖ空间为其中Ｓ为Ｓ的内部．定义加权Ｋ－泛函及加权光滑模其中 为加权范数． …     

Best mean square approximations by entire functions of finite degree on a straight line and exact values of mean widths of functional classes

We obtain exact Jackson-type inequalities in the case of the best mean square approximation by entire functions of finite degree ≤ σ on a straight line. For classes of functions defined via majorants of averaged smoothness characteristics Ω₁(f, t ), t > 0, we determine the exact values of the Kolmogorov mean ν-width, linear mean ν-width, and Bernstein mean ν-width, ν > 0.     

Bernstein型算子加Jacobi权逼近

对于Bernstein型算子,证明它在通常的加权范数下是无界的,通过引进新的加权范数,研究其加Jacobi权的逼近性质,得到加权逼近的正逆定理,从而导出加权逼近特征的等价刻画．     

On approximation of functions by algebraic polynomials in Hölder spaces

We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding to the generalized Jacobi translation and the Ditzian–Totik moduli of smoothness. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria of the precise order of decrease of the best approximation in these spaces. Moreover, we obtain strong converse inequalities for some methods of approximation of functions. As an example, we consider approximation by the Durrmeyer–Bernstein polynomial operators.     

Approximations of almost periodic functions by entire ones

In this paper we propose a new proof of the well-known theorem by S. N. Bernstein, according to which among entire functions which give on (−∞,∞) the best uniform approximation of order σ of periodic functions there exists a trigonometric polynomial whose order does not exceed σ. We also prove an analog of this Bernstein theorem and an analog of the Jackson theorem for uniform almost periodic functions with an arbitrary spectrum.     

Bernstein型算子同时逼近误差

该文证明了C[0,1]空间中的函数及其导数可以用Bernstein算子的线性组合同时逼近,得到逼近的正定理与逆定理.同时,也证明了Bernstein算子导数与函数光滑性之间的一个等价关系.该文所获结果沟通了Bernstein算子同时逼近的整体结果与经典的点态结果之间的关系.     

Bernstein型算子线性组合加Jacobi权逼近及高阶导数的等价定理

本文利用加权Ditzian-Totik光滑模证明Bernstein型算子的线性组合加权逼近阶估计和等价定理;同时,研究加Jacobi权下Benstein型算子的高阶导数与所逼近函数光滑性之间的关系.     

Some approximation properties of a Durrmeyer variant of q‐Bernstein–Schurer operators

In this paper, we will propose a Durrmeyer variant of q‐Bernstein–Schurer operators. A Bohman–Korovkin‐type approximation theorem of these operators is considered. The rate of convergence by using the first modulus of smoothness is computed. The statistical approximation of these operators is also studied. Copyright © 2016 John Wiley & Sons, Ltd.     

Qp空间中的de la Vallée Poussin平均

本文研究了单位球上的Q_p空间中的de la Vallée Poussin平均算子,并通过高阶光滑模来建立Jackson逼近定理.此外,我们还得到了Bernstein不等式,K-泛函和光滑模的等价刻画等结果.     

THE SMOOTHNESS AND DIMENSION OF FRACTAL INTERPOLATION FUNCTIONS

In this paper, we investigate the smoothness of non-equidistant fractal interpolation functions We obtain the Holder exponents of such fractal interpolation functions by using the technique of operator approximation. At last, We discuss the series expressiong of these functions and give a Box-counting dimension estimation of “critical” fractal interpohltion functions by using our smoothness results.     

Bernstein-Kantorovich算子的迭代布尔和的逼近性质

本文利用光滑模及最佳逼近多项式的性质,研究了Bernstein-Kantorovich算子的迭代布尔和对Lp[0,1]中的函数的逼近性质,得到了逼近正定理,弱逆不等式及等价定理.     

Approximation of Multivariate Functions with a Certain Mixed Smoothness by Entire Functions

ＡｐｐｒｏｘｉｍａｔｉｏｎｏｆＭｕｌｔｉｖａｒｉａｔｅＦｕｎｃｔｉｏｎｓｗｉｔｈ　ａＣｅｒｔａｉｎＭｉｘｅｄＳｍｏｏｔｈｎｅｓｓｂｙＥｎｔｉｒｅＦｕｎｃｔｉｏｎｓＷａｎｇＨｅｐｉｎｇ（汪和平）；ＳｕｎＹｏｎｇｓｈｅｎｇ（孙永生）（Ｄｅｐａｒｔｍｅｎｔ...     

关于Bernstein算子的收敛速度

本文对一类函数建立了Bernstein算子的一致逼近定理,而且给出了其逆定理的一个简短证明。     

Bézier variant of the Jakimovski–Leviatan–Păltănea operators based on Appell polynomials

In this paper, we introduce the Bézier variant of the Jakimovski–Leviatan–P?lt?nea operators based on Appell polynomials. We establish some local results, a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness and also study the rate of convergence for the functions having a derivative of bounded variation for these operators.     

用指数型整函数的最佳限制逼近

我们研究了由仅有实零点的代数多项式导出的微分算子确定的广义Sobolev类利用指数型整函数作为逼近工具的最佳限制逼近问题.利用Fourier变换和周期化等方法,得到在L_2(R)范数下的广义Sobolev光滑函数类的相对平均宽度和最佳限制逼近的精确常数,以及当0是这个代数多项式的一个至多2重的零点时,得到最佳限制逼近在L_1(R)范数和一致范数下的广义Sobolev类的精确到阶的结果.