A REMARK ON KOLMOGOROV'S COMPARISON THEOREM

In approximation theory the theorem of Kolmogorov concerning the comparison of derivatives of differentiable functions defined on the real line is well-known. It plays an important r le in establishing sharp inequalities between the norms of derivatives of a function. In this note we establish a comparison theorem of Kolmogorov type on a class of functions which are defined on the real line and can be continuated analytically in a stripped region containing the real llne. As a consequence we have derived an inequality of Landau-Kolmogorov type on this function class, and moreover, we have applied it to get the exact estimation for the Kolmogorov's N-widths of the analytic function class.     

A REMARK ON KOLMOGOROVS COMPARISON THEOREM

In approximation theory the theorem of Kolmogorov concerning the comparison of derivatives of differentiable functions defined on the real line is well—known. It plays an important role in establishing sharp inequalities between the norms of derivatives of a function. In this note we establish a comparison theorem of Kolmogorov type on a class of functions which are defined on the real line and can be contlnuated analytically in a stripped region containing the real line. As a consequence we have derived an inequality of Landau-Kolmogorov type on this function class, and moreover, we have applied it to get the exact estimation for the Kolmogorov''s N-widths of the analytic function class.     

Lower bound for quantum integration error on anisotropic Sobolev classes

We study the approximation of the integration of multivariate functions in the quantum model of computation. Using a new reduction approach we obtain a lower bound of the n-th minimal query error on anisotropic Sobolev class R（Wpr（[0, 1]d）） （r R＋d）. Then combining this result with our previous one we determine the optimal bound of n-th minimal query error for anisotropic Hblder- Nikolskii class R（H∞r（[0,1]d）） and Sobolev class R（W∞r（[0,1]d））. The results show that for these two types of classes the quantum algorithms give significant speed up over classical deterministic and randomized algorithms.     

On a Class of Generalized Sampling Functions

In this note, we discuss a class of so-called generalized sampling functions. These functions are defined to be the inverse Fourier transform of a family of piecewise constant functions that are either square integrable or Lebegue integrable on the real number line. They are in fact the generalization of the classic sinc function. Two approaches of constructing the generalized sampling functions are reviewed. Their properties such as cardinality, orthogonality, and decaying properties are discussed. The interactions of those functions and Hilbert transformer are also discussed.     

ESTIMATES OF LINEAR RELATIVE n-WIDTHS IN L~p [0,1]

In this paper we will show that if an approximation process { L n }n ∈ N is shapepreserving relative to the cone of all k-times differentiable functions with non-negative k-th derivative on [0,1],and the operators L n are assumed to be of finite rank n,then the order of convergence of D k L n f to D k f cannot be better than n 2 even for the functions x k,x k+1,x k+2 on any subset of [0,1] with positive measure.Taking into account this fact,we will be able to find some asymptotic estimates of linear relative n-width of sets of differentiable functions in the space L p [0,1],p ∈ N.     

ON BERNSTEIN POLYNOMIALS OF FUNCTIONS OF BOUNDED VARIATION OF ORDER p

1.IntroductionLet f be a function defined on[0,1],then the Bernstein poly-nomial B_n(f,x)of f is T.Popoviciu(see[1])and G.G.Lorentz studied the operator(1.1)and estimated the rate of convergence of B_n(f,x).F.Herzogand J.D. Hill,and others studied the convergence of(1.1)fordiscontinuous functions,they proved that if f is bounded on[0,1]and x is a discontinuity point of the first kind,then     

An Extension of the Turan Inequality in L_p-space for 0

周颂平 《数学研究与评论》1986,(2)

ON N-WIDTH OF GENERALIZED BERNOULLI KERNEL

Let P(x)=x~r+a_lx~(r-1)+…+a_r be a polynomial,zeros of whichare all real,and D=d/(dx).We denote L~r the class of 2π-periodiccontinuous functions such that f~(r-1)are absolutely continuous on[0,2π]and f~(i)(o)=f~(i)(2π),i=1,…,r-1.Put     

On the Bounded Variation of Operator-Valued Functions

In[1],the Kothe nuclear sequence spaces are described and in[2]the characte-rization of the semi-nuclear and the barreled locally convex spaces by the boundedvariation of functions respectively.In this paper several kinds of bounded variationof operator-valued functions are studied. Let X and Y are Banach spaces,T(t):[0,1]→L(X,Y)be a operator-valued fun-ction.We called T(t)be uniformly bounded variation,if there exist a constantM>0,such that for any partition △:0=t_0相似文献   

Some radius problems related to a certain subclass of analytic functions

For real parameters α and β such that 0≤α1β,we denote by S(α,β) the class of normalized analytic functions which satisfy the following two-sided inequality:αR(zf′(z)/f(z))β,z∈U,where U denotes the open unit disk.We find a sufficient condition for functions to be in the class S(α,β) and solve several radius problems related to other well-known function classes.     

A Remark on Adaptive Decomp osition for Nonlinear Time-frequency Analysis

In recent study the bank of real square integrable functions that have nonlinear phases and admit a well-behaved Hilbert transform has been constructed for adaptive representation of nonlinear signals. We first show in this paper that the available basic functions are adequate for establishing an ideal adaptive decomposi-tion algorithm. However, we also point out that the best approximation algorithm, which is a common strategy in decomposing a function into a sum of functions in a prescribed class of basis functions, should not be considered as a candidate for the ideal algorithm.     

关于Reissner厚板模型的解——对“关于简支厚板与薄板的关系”的讨论

In this paper, the method proposed in [2] is extended to the case of thickplate, the problem in the solution of Reissner model of thick plate is reduced to a solution oftwo displacement functions ω and f, and the general form of the solution of this model isderived. For a simply supported polygonal thick plate, f vanishes identically, and hence therelation between the solution of thick plate and that of similar thin plate can be established.According to this relation, the solutions of a class of thick plates may be derived from thecorresponding solutions of thin plates.     

The Pointwise Multipliers From Space F（p, q, s） to Bloch Type Space in C^n

Let B be the unit ball in C^n. By H(B) we denote the class of all holomorphic functions on B and H^∞ denotes the class of all bounded holomorphic functions on B.     

Construction and Dimension Analysis for a Class of Fractal Functions

In this paper, we construct a class of nowhere differentiable continuous functions by means of the Cantor series expression of real numbers. The constructed functions include some known nondifferentiable functions, such as Bush type functions. These functions are fractal functions since their graphs are in general fractal sets. Under certain conditions, we investigate the fractal dimensions of the graphs of these functions, compute the precise values of Box and Packing dimensions, and evaluate the Hausdorff dimension. Meanwhile, the Holder continuity of such functions is also discussed.     

Third Hankel Determinant for the Inverse of Starlike and Convex Functions

Denote S to be the class of functions which are analytic, normalized and univalent in the open unit disk U = {z : |z| 1}. The important subclasses of S are the class of starlike and convex functions, which we denote by S~* and C.In this paper, we obtain the third Hankel determinant for the inverse of functions ■ elonging to S~* and C.     

指数函数系的不完备性和极小性

Necessary and sufficient conditions are obtained for the incompleteness and the minimality of the exponential system E（Λ,M） = {z l e λ n z ： l = 0,1,...,m n-1;n = 1,2,...} in the Banach space E 2 [σ] consisting of some analytic functions in a half strip.If the incompleteness holds,each function in the closure of the linear span of exponential system E（Λ,M） can be extended to an analytic function represented by a Taylor-Dirichlet series.Moreover,by the conformal mapping ζ = φ（z） = e z ,the similar results hold for the incompleteness and the minimality of the power function system F （Λ,M） = {（log ζ） l ζ λ n ： l = 0,1,...,m n-1;n = 1,2,...} in the Banach space F 2 [σ] consisting of some analytic functions in a sector.     

NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS

§1.IntroductionLet(?,F)be a measurable space and let Lb(F)be the space of F-measurable andbounded real functions.A nonlinear expectation is a continuous functionalE[·]:Lb(F)?→Rthat is order preserving(i.e.,E[X1]≥E[X2],if X1≥X2)and constant preserving     

Shannon-type sampling for multivariate non-bandlimited signals

In this paper, starting from a function analytic in a neighborhood of the unit disk and based on Bessel functions, we construct a family of generalized multivariate sinc functions, which are radial and named radial Bessel-sinc (RBS) functions being time-frequency atoms with nonlinear phase. We obtain a recursive formula for the RBS functions in R d with d being odd. Based on the RBS function, a corresponding sampling theorem for a class of non-bandlimited signals is established. We investigate a class of radial functions and prove that each of these functions can be extended to become a monogenic function between two parallel planes, where the monogencity is taken to be of the Clifford analysis sense.     

Average Width and Optimal Recovery of Some Anisotropic Classes of Smooth Functions Defined on the Euclidean Space R~d

§ 1.Introduction and Main Results　 In[1 ] ,[2 ] ,the authors studied some problems of optimal recovery of functions de-fined on a cube,for a class of functions with partial derivatives of a fixed order havingmoduli of continuity not exceeding a given modules of continuity,and for the unit ballsSHαp in the spaces Hαp satisfying the mixed Holder conditionα,respectively.They ob-tained some weak asymptotic results.　 In[3 ] ,[4 ] and[5] ,Magarill-Il' yaev,Liu and Sun studied some proble…     

On the Approximation of Continuous Function by Harmonic Means

Let f(x) be a continuous and periodic function with period 2π and [f]=a_o/2+sum from n=1 to ∞(a_ncos nx+b_nsin nx) be its Fourier series. Denote by s_n(f,x) the n-th partial sums of [f] and by ω(f,δ) the moduls of continuity of f(x). When ω(δ) is a modul of continuity, we denote by H[ω]。the class of all functions for which ω(f,t)≤ω(t). It is well known that the n-th harmonic means and the n-th Cesro means of f∈C_2, are defined as##属性不符