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.     

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.     

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 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 < ∞.     

Best m-ter m approxi mation of the classes$$ B_{\infty, \theta }^r $$ of functions of many variables by polyno mials in the haar syste m

We obtain an exact-order estimate for the best m-term approximation of the classes B_{￥, q}^r B_{\infty, \theta }^r of periodic functions of many variables by polynomials in the Haar system in the metric of the space L _q , 1 < q < ∞.     

Trigonometric and orthoprojection widths of classes of periodic functions of many variables

We obtain exact order estimates for trigonometric and orthoprojection widths of the Besov classes B ^r _p,θ and Nikol’skii classes Hr p of periodic functions of many variables in the space L _q for certain relations between the parameters p and q.     

Asymptotic estimates for the best trigonometric and bilinear approximations of classes of functions of several variables

We obtain exact order estimates for the best M-term trigonometric approximations of the Besov classes B _∞,θ ^r in the space L _q . We also determine the exact orders of the best bilinear approximations of the classes of functions of 2d variables generated by functions of d variables from the classes B _p,θ ^r with the use of translation of arguments.     

On the bestm-term trigonometric and orthogonal trigonometric approximations of functions from the classesL Ψ β,ρ

We obtain estimates exact in order for the best trigonometric and orthogonal trigonometric approximations of the classesL _Ψ^β,ρ of functions of one variable in the spaceL _q in the case 2<p <q < ∞.     

Trigonometric widths of the classes B p,θ r of functions of many variables in the space L q

We obtain estimates exact in order for the trigonometric widths of the Besov classes B _p,θ^r of periodic functions of many variables in the space L _q for 1 ≤ p ≤ 2 < q < p/(p - 1). Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1089–1097, August, 1998.     

Completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight

We study the problem on the completeness of orthogonal systems in asymmetric spaces with sign-sensitive weight. Theorems of general form are obtained. In particular, the necessary and sufficient conditions on α, β, q ₁, and q ₂ for which the known orthogonal systems are everywhere dense in asymmetric spaces L _(α,β);q ([0, 1]) are found. Theorem. Let α, β, q ₁, q ₂ ∈ [1,+∞]. The following orthogonal systems are dense in asymmetric spaces L _(α,β);q ([0, 1]) if and only if either max{α, β, q ₁, q ₂} < + ∞ or max {α, β} < +∞, q ₁ = q ₂ = +∞: trigonometric, algebraic, Haar’s system, Meyer’s system of wavelets, Shannon-Kotel’nikov wavelets, Stromberg and Lemarie-Battle wavelets, the Walsh system, and the Franklin system. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Best orthogonal trigonometric approximations of the classes <Emphasis Type="Italic">B</Emphasis><Superscript>Ω</Superscript><Subscript><Emphasis Type="Italic">p</Emphasis>,θ</Subscript> of periodic functions of many variables

We obtain exact-order estimates for the best orthogonal trigonometric approximations of the classes B ^Ω _p,θ of periodic functions of many variables in the space L _q .     

Inequalities for upper bounds of functionals on the classes w r H ω and their applications

We show that the well-known results on estimates of upper bounds of functionals on the classes W ^r H ^ω of periodic functions can be regarded as a special case of Kolmogorov-type inequalities for support functions of convex sets. This enables us to prove numerous new statements concerning the approximation of the classes W ^r H ^ω , establish the equivalence of these statements, and obtain new exact inequalities of the Bernstein-Nikol’skii type that estimate the value of the support function of the class H ^ω on the derivatives of trigonometric polynomials or polynomial splines in terms of the L ^ϱ -norms of these polynomials and splines.     

The Wirtinger-Steklov inequality between the norm of a periodic function and the norm of the positive cutoff of its derivative

We study the sharp constant in the inequality between the L _p -mean (p ≥ 0) of a 2π-periodic function with zero mean value and the L _q -norm (q ≥ 1) of the positive cutoff of its derivative. We obtain estimates of the constant from below for 0 ≤ p ≤ ∞ and from above for 1 ≤ p ≤ ∞ for an arbitrary 1 ≤ q ≤ ∞. We write out the values of the sharp constant in the cases p = 2, 1 ≤ q ≤ ∞ and p = ∞, 1 ≤ q ≤ ∞.     

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≤∞).     

Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials

In the spaceL _q, 1<q<∞ we establish estimates for the orders of the best approximations of the classes of functions of many variablesB _1,θ ^r andB _p,α ^r by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials. It is shown that, in many cases, the estimates obtained in the present work are better in order than in the case of approximation by polynomials with harmonics from the hyperbolic cross.     

Recovery of band limited functions via cardinal splines

The main result of this paper asserts that if a function f is in the class B_π,p, 1 <p < ∞; that is, those p-integrable functions whose Fourier transforms are supported in the interval [ - π, π], then f and its derivatives f^(j) j = 1, 2, …, can be recovered from its sampling sequence{f(k)} via the cardinal interpolating spline of degree m in the metric ofL _q(ℝ)), 1 <p=q < ∞, or 11 <p=q < ⩽ ∞.     

Optimal query error of quantum approximation on some Sobolev classes

We study the approximation of the imbedding of functions from anisotropic and general-ized Sobolev classes into Lq([0,1]d) space in the quantum model of computation. Based on the quantum algorithms for approximation of finite imbedding from LpN to LNq , we develop quantum algorithms for approximating the imbedding from anisotropic Sobolev classes B(Wpr ([0,1]d)) to Lq([0,1]d) space for all 1 q,p ∞ and prove their optimality. Our results show that for p < q the quantum model of computation can bring a speedup roughly up to a squaring of the rate in the classical deterministic and randomized settings.     

CONVERGENCE RATE FOR ITERATES OF q-BERNSTEIN POLYNOMIALS

Recently, q-Bernstein polynomials have been intensively investigated by a number of authors. Their results show that for q ≠ 1, q-Bernstein polynomials possess of many interesting properties. In this paper, the convergence rate for iterates of both q-Bernstein polynomials and their Boolean sum are estimated. Moreover, the saturation of {Bn（., qn）} when n → ∞ and convergence rate of Bn（f,q;x） when f ∈ C^n-1 [0, 1], q → ∞ are also presented.     

Kolmogorov and linear widths of classes of <Emphasis Type="Italic">s</Emphasis>-monotone integrable functions

Let s ∈ ℕ and let Δ ₊ ^s be the set of functions x: I ↦ ℝ on a finite interval I such that the divided differences [x; t ₀, ..., t _s ] of order s of these functions are nonnegative for all collections of s + 1 different points t ₀, ..., t _s ∈ I. For the classes Δ ₊ ^s B _p : = Δ ₊ ^s ∩ B _p , where B _p is the unit ball in L _p , we determine the orders of Kolmogorov and linear widths in the spaces _Lq for 1 ≤ q > p ≤ ∞. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1633–1652, December, 2005.     

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≤∞). Supported by the National Natural Science Foundation of China (No. 19671012) and by the Doctoral Programme Foundation of Institution of Higher Education of National Education Committee of China.