关于Baskakov—Durrmeyer算子的一致逼近

本文首先给出了Ｂａｓｋａｋｏｖ－Ｄｕｒｒｍｅｙｅｒ算子一致逼近意义下的正定理，并把它推广到一类线性组合的情形，然后讨论了它的导数与光滑模的等价关系，最后给出了二元Ｂａｓｋａｋｏｖ－Ｄｕｒｒｍｅｙｅｒ算子逼近阶的特殊刻画。     

关于Baskakov—Kurrmeyer算子的局部逼近

本文主要讨论了用Ｈｏｌｄｅｒ连续函数表示Ｂａｓｋａｋｏｖ－Ｄｕｒｒｍｅｙｅｒ算子局部逼近阶的特征刻划问题。     

关于Baskakov-Durrmeyer算子的局部逼近

本文主要讨论了用Ｈｏｌｄｅｒ连续函数表示Ｂａｓｋａｋｏｖ－Ｄｕｒｒｍｅｙｅｒ算子局部逼近阶的特征刻划问题。     

Baskakov-Durrmeyer算子的点态逼近

本文应用Ｄｉｔｚｉａｎ－Ｔｏｔｉｋ模得到Ｂａｓｋａｋｏｖ－Ｄｕｒｒｍｅｙｅｒ算子线性组合的点态逼近的等 价定理．     

Baskakov算子加权逼近的收敛阶

本文讨论了Ｂａｓｋａｋｏｖ算子加Ｊａｃｏｂｉ权逼近的收敛性，首先指出了按通常的加权范数，Ｂａｓｋａｋｏｖ算子是无界的。然后引入一种新的范数，在此范数下Ｂａｓｋａｋｏｖ算子具有压缩性，最后借助于Ｋ－泛函，我们着重讨论了它的特征刻划问题。     

一类Durrmeyer算子线性组合的加权逼近

本文主要给出了一类Ｂｅｒｎｓｔｅｉｎ－Ｄｕｒｒｍｅｙｅｒ算子的线性组合在Ｌｐ逼近意义下加Ｊａｃｏｂｉ权逼近时的特征刻划．     

某些多元线性正算子的加权逼近

本文首先给出了在Ｌｐ逼近意义下某些线性正算子加Ｊａｃｏｂｉ权逼近时的特征定理，作为应用，我们给出了多元Ｂａｓｋａｋｏｖ型算子、多元Ｓｚａｓｚ－Ｍｉｒａｋｊａｎ型算子和多元Ｂｅｔａ算子加权逼近时的特征刻划．     

关于Szász-Mirakjan—Durrmeyer算子在Besov空间上的逼近阶

本文主要给出了Ｓｚａｓｚ－Ｍｉｒａｋｊａｎ－Ｄｕｒｒｍｅｙｅｒ算子在Ｂｅｓｏｖ空间上逼近阶的特征刻划     

关于多元Baskakov算子的加权逼近

本文首先指出一类多元Ｂａｓｋａｋｏｖ算子在通常的加权范数下是无界的．然后给出了一类新的加权范数,在此范数下它是压缩的．最后利用多元分解技巧,解决了多元Ｂａｓｋａｋｏｖ算子加权逼近的特征刻划文问题．     

Baskakov-Durrmeyer型算子同时逼近的强逆不等式

本文对Ｂａｓｋａｋｏｖ－Ｄｕｒｒｍｅｙｅｒ型算子Ｍｎ（ｆ，ｘ）证明了，当１＜ｐ∞时，存在某一正数ｍ，使得ω２φｆ（２ｒ），１ｎｐＭ（‖Ｍ（２ｒ）ｎｆ－ｆ（２ｒ）‖ｐ＋‖Ｍ（２ｒ）ｍｎｆ－ｆ（２ｒ）‖ｐ＋１ｎ‖ｆ（２ｒ）‖ｐ，φ２（ｘ）＝ｘ（１＋ｃｘ）     

Direct and converse results for operators of Baskakov-Durrmeyer type

We consder the n-th so-called operators of Baskakov-Durrmeyer type, which result from the classical Baskakov-type operators with weights p_nk, if the discrete values f(k/n) are replaced by the integral terms (n-c∫ ₀ ^∞ p _n k(t)f(t)dt. The main differences between these operators and their classical and Kantorovicvariants respectively is that they commute. We prove direct and converse theorems also for linear combinations of the operators and results of Zamansky-Sunouchi type. As an important tool for measuring the smootheness of a function we use the Ditzian-Totik modulus of smoothness and its equivalence to appropriate K-functionals. This paper is part of the author's dissertation.     

Direct and Converse Results on Weighted Simultaneous Approximation by the Method of Operators of Baskakov-Durrmeyer Type

We consider the n- th. operators of Baskakov-Durrmeyer type, which result from the classical Baskakov-type operators with weights p _nk, if the values f(k/n) are replaced by the integral terms \((n - c)\int_{I} p_{nk} (t) f(t) dt \) , where I denotes the corresponding interval which depends on the parameter c. For integrable functions the rate of weighted simultaneous approximation will be related to the Ditzian-Totik modulus of smoothness.     

Almost sure convergence and bounded entropy

It is shown that the almost sure convergence property for certain sequences of operators {S _n{ implies a uniform bound on the metrical entropy of the sets {S _nf|n=1, 2, ...{, wheref is taken in theL ²-unit ball. This criterion permits one to unify certain counterexamples due to W. Rudin [Ru] and J.M. Marstrand [Mar] and has further applications. The theory of Gaussian processes is crucial in our approach.     

Certain positive linear operators

Properties (including the approximating ones) are investigated of positive linear operators L_n(f; x) for which the relation $$L_n \left( {\left( {t - x} \right)f\left( t \right); x} \right) = \frac{{\varphi \left( x \right)}}{n}L'_n \left( {f\left( t \right); x} \right)$$ is fulfilled, as well as the properties of operators L_n ^(m)(f;x). The results are applicable, in particular, to Bernstein polynomials, to the operators of Mirak'yan, Baskakov, and others.     

A family of Bernstein quasi-interpolants on [0,1]

Suppose that we want to approximate f∈C[0,1] by polynomials inP, using only its values on X_n={i/n, 0≤i≤n}. This can be done by the Lagrange interpolant L_n f or the classical Bernstein polynomial B_n f. But, when n tends to infinity, L_n f does not converge to f in general and the convergence of B_n f to f is very slow. We define a family of operators B _n ^(k) , n≥k, which are intermediate ones between B _n ⁽⁰⁾ =B _n ⁽¹⁾ =B_n and B _n ⁽ⁿ⁾ =L_n, and we study some of their properties. In particular, we prove a Voronovskaja-type theorem which asserts that B _n ^(k) f−f=O(n^−[(k+2)/2]) for f sufficiently regular. Moreover, B _n ^(k) f uses only values of B_n f and its derivaties and can be computed by De Casteljau or subdivision algorithms.     

Approximation of functions by certain nonlinear integral operators

We study approximation properties of certain nonlinear integral operators L _n * obtained by a modification of given operators L _n . The operators L _n;r and L _n;r * of r-times differentiable functions are also studied. We give theorems on approximation orders of functions by these operators in polynomial weight spaces.     

Spectral theory for commutative algebras of differential operators on Lie groups

The joint spectral theory of a system of pairwise commuting self-adjoint left-invariant differential operators L₁,…,Ln on a connected Lie group G is studied, under the hypothesis that the algebra generated by them contains a “weighted subcoercive operator” of ter Elst and Robinson (1998) [52]. The joint spectrum of L₁,…,Ln in every unitary representation of G is characterized as the set of the eigenvalues corresponding to a particular class of (generalized) joint eigenfunctions of positive type of L₁,…,Ln. Connections with the theory of Gelfand pairs are established in the case L₁,…,Ln generate the algebra of K-invariant left-invariant differential operators on G for some compact subgroup K of Aut(G).     

Wavelet-Like Transforms for Admissible Semi-Groups; Inversion Formulas for Potentials and Radon Transforms

We introduce a family of wavelet-like transforms associated to certain admissible semigroups of operators acting on L_p-spaces, and prove the corresponding reproducing formula of Calderòn’s type. The new transforms constitute a unified approach to inversion of a wide class of integral operators in analysis and applications. We illustrate the general theory by considering Riesz and Bessel potentials (associated to the ordinary and the generalized translation), and the k-plane Radon transform on ℝⁿ.     

关于多元Baskakov—Durrmeyer算子的Lp逼近

本文给出一类Baskakov—Durrmeyer算子及其线性组合,利用Ditzian-Totik光滑模及相应的K-泛函,得到了这类算子L_p-逼近的特征刻划。     

A note on convergent polynomials of interpolation

In this note we define a sequence {L_n(f;x)} of interpolatory polynomials based on a system x_n={x_kn, k=1,2,…n} of nodes to be a sequence of QLIP if for every f(x)∈C[−1,1], L_n(f; x) tends uniformly to f(x) and ρ_n=1+o(1) as n→∞, where ρ_n is the ratio of the number of points in x_n, at which L_n(f;x) coincides with f(x), and the degree of L_n(f;x). Two sequences of QLIP are constructed, one of which is based on a Bernstein process and the other the Freud-Sharma's construction.