Average B-width and infinite dimensional G-width of some smooth function classes on the line

In this paper, we get the exact values of average σ-B width and infinite dimensional σ-G width of Sobolev class B^r _p(R) in the metric L_p(R) (1≤p≤∞) and obtain the exact (σ∈N) and strong asymptotic (σ>1) results of infinite dimensional σ-G widths of Sobolev-Wiener class W^r _pq (R) in the metric L_q(R) and its dual case W^r _p(R) in the metric L_qp(R) (1≤q≤p≤∞).     

L p approximation by Bernstein-Kantorovich quasi-interpolants

Bernstein-Kantorovich quasi-interpolants K^（2r-1）n（f, x） are considered and direct, inverse and equivalence theorems with Ditzian-Totik modulus of smoothness ω^2rφ（f, t）p （1 ≤ p ≤＋∞） are obtained.     

THE GLOBAL APPROXIMATION BY LEFT-BERNSTEIN-DURRMEYER QUASI-INTERPOLANTS IN Lp[0, 1]

In this paper,we will use the 2r-th Ditzian-Totik modulus of smoothness wp^2r(f,t)p to discuss the direct and inverse theorem of approximation by Left-Bernstein-Durrmeyer quasi-interpolants Mn^[2r-1]f for functions of the space Lp[0,1](1≤p≤ ∞)。     

Average Widths of Sobolev-Wiener Classes with Mixed Smoothness in L q (ℝ d )

In this paper, the average σ-Κ width of Sobolev-Wiener classes S ^r _pq W with mixed smoothness in L _q (ℝ ^d ) is studied for 1 < q≤p < ∞, and the weak asymptotical behaviour of these quantities is obtained. Received July 21, 1998, Accepted May 31, 1999     

On weighted approximation by Bernstein-Durrmeyer operators

In this paper, we consider weighted approximation by Bernstein-Durrmeyer operators in L_p[0, 1] (1≤p≤∞), where the weight function w(x)=x^α(1−x)^β,−1/p<α, β<1-1/p. We obtain the direct and converse theorems. As an important tool we use appropriate K-functionals. Supported by Zhejiang Provincial Science Foundation.     

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W_p^k(ℝ^s) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W_p^k(ℝ^s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L_p (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ∈L_p(ℝ^s) (φ∈C(ℝ^s) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,L_p(ℝ^s)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167) in the univariate setting to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C20, 41A25, 39B12. Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant G121210654.     

Some estimates related to fractal measures and Laplacians on manifolds

Let Δ be the Laplace-Beltrami operator on ann dimensional completeC ^∞ manifoldM In this paper we establish an estimate ofe ^tΔ (dμ) valid for allt>0 wheredμ is a locally uniformly α dimensional measure onM 0≤α≤n The result is used to study the mapping properties of (I-tΔ)^-β considered as an operator fromL ^p (M dμ) toL ^p (M dx) wheredx is the Riemannian measure onM β>(n−α)/2p′ 1/p+1/p′=1 1≤p≤∞     

Approximative characteristics of the isotropic classes of periodic functions of many variables

Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov (B _p,θ ^r ) and Nukol’skii (H _p ^r ) classes of periodic functions of many variables in the metric of L _q , 1 ≤ p, q ≤ ∞. We also establish the orders of the best approximations of functions from the same classes in the spaces L ₁ and L _∞ by trigonometric polynomials with the corresponding spectrum.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt Foundation.     

Sampling theorem of Hermite type and aliasing error on the Sobolev class of functions

Denote by B _2σ,p (1 < p < ∞) the bandlimited class p-integrable functions whose Fourier transform is supported in the interval [−σ, σ]. It is shown that a function in B _2σ,p can be reconstructed in L _p(ℝ) by its sampling sequences {f (κπ / σ)} _κ∈ℤ and {f’ (κπ / σ)} _κ∈ℤ using the Hermite cardinal interpolation. Moreover, it will be shown that if f belongs to L _p ^r (ℝ), 1 < p < ∞, then the exact order of its aliasing error can be determined. Project supported by the Scientific Research Common Program of Beijing Municipal Commission of Education under grant number KM 200410009010 and by the Natural Science Foundation of China under grant number 10071006     

The Besov and Triebel-Lizorkin spaces with high order on the Lipschitz curves

The Besov spacesB _p ^α,μ (Γ) and Triebel-Lizorkin spacesF _p ^α,μ (Γ) with high order α∈R on a Lipschitz curve Γ are defined. when 1≤p≤∞, 1≤q≤∞. To compare to the classical case a difference characterization of such spaces in the case |α|<1 is given also. The author is supported in part by the Foundation of Zhongshan University Advanced Research Centre and NSF of China.     

On the order of relative approximation of classes of differentiable periodic functions by splines

In the case where n → ∞, we obtain order equalities for the best L _q -approximations of the classes W _p ^r , 1 ≤ q ≤ p ≤ 2, of differentiable periodical functions by splines from these classes.     

Best approximations of the classes B p,θ r of periodic functions of many variables in uniform metric

We obtain estimates exact in order for the best approximations of the classes B _∞,θ ^r of periodic functions of two variables in the metric of L _∞ by trigonometric polynomials whose spectrum belongs to a hyperbolic cross. We also investigate the best approximations of the classes B _p,θ ^r , 1 ≤ p < ∞, of periodic functions of many variables in the metric of L _∞ by trigonometric polynomials whose spectrum belongs to a graded hyperbolic cross. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1395–1406, October, 2006.     

Theorems for generalized Favard-Kantorovich and Favard-Durrmeyer operators in exponential function spaces

We consider the Kantorovich and the Durrmeyer type modifications of the generalized Favard operators and we prove some direct approximation theorems for functions f such that w _σ f ∈ L _p (R), where 1 ≤ p ≤ ∞ and w _σ (x) = exp(−σx ²), σ > 0.     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials II

We present a definition of general Sobolev spaces with respect to arbitrary measures, Wh,p (Ω,μ) for 1 ≤p≤∞. In [RARP] we proved that these spaces are complete under very light conditions. Now we prove that if we consider certain general types of measures, then Cc∞ (R) is dense in thee spaces. As an application to Sobolev orthogonal polynomials, we study the boundedness of the multiplication operator. This gives an estimation of the zeroes of Sobolev orthogonal polynomials.     

Besov spaces andK-functionals inL p, 0<p<1*

We investigate Besov spaces and their connection with trigonometric polynomial approximation inL _p[−π,π], algebraic polynomial approximation inL _p[−1,1], algebraic polynomial approximation inL _p(S), and entire function of exponential type approximation inL _p(R), and characterizeK-functionals for certain pairs of function spaces including (L _p[−π,π],B _s ^a(L _p[−π,π])), (L _p(R),_s ^a(L_p(R))), , and , where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r. This project is supported by the National Science Foundation of China.     

Bernstein and Gelfand widths of certain classes of analytic functions. II

The Gelfand widths of the unit ball of H²(ν) (the weighted Hardy space) with respect to the metric of the space L_∞(T_r) are considered (here T_r is the circle of radius r centered at the origin), as well as the Bernstein widths of the unit ball of H^∞ with respect to the metric of the space L₂(T_r, μ). Asymptotic formulas for the widths in question are established for arbitrary measures ν, μ. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 134–140. Translated by S. V. Kislyakov.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.     

Discrete characterization of the Paley-Wiener space with several variables

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB _{π, p} (R ⁿ ), 1≤p<∞, i.e., for 1<p<∞,B _{π, p} (R ⁿ ) is isomorphic tol _p (Z ⁿ ), and forp=1,B _{π, 1} (R ⁿ ) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z ⁿ ). This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan.