On lower bounds for the approximation of individual functions by local splines with nonfixed nodes

For functions with the integrable βth power, where β = (r + 1 + 1/p)⁻¹, we obtain asymptotically exact lower bounds for the approximation by local splines of degreer and defectk ≥r/2 in the metric ofL _p. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 12, pp. 1628–1637, December, 1999.     

Analytic approximation of matrix functions in

We consider the problem of approximation of matrix functions of class L^p on the unit circle by matrix functions analytic in the unit disk in the norm of L^p, 2≤p<∞. For an m×n matrix function Φ in L^p, we consider the Hankel operator , 1/p+1/q=1/2. It turns out that the space of m×n matrix functions in L^p splits into two subclasses: the set of respectable matrix functions and the set of weird matrix functions. If Φ is respectable, then its distance to the set of analytic matrix functions is equal to the norm of H_Φ. For weird matrix functions, to obtain the distance formula, we consider Hankel operators defined on spaces of matrix functions. We also describe the set of p-badly approximable matrix functions in terms of special factorizations and give a parametrization formula for all best analytic approximants in the norm of L^p. Finally, we introduce the notion of p-superoptimal approximation and prove the uniqueness of a p-superoptimal approximant for rational matrix functions.     

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).     

Comparison of approximation properties of generalized polynomials and splines

We establish that, for p ∈ [2, ∞), q = 1 or p = ∞, q ∈ [ 1, 2], the classes W _p^rof functions of many variables defined by restrictions on the L _p-norms of mixed derivatives of order r = (r ₁, r ₂, ..., r _m) are better approximated in the L _q-metric by periodic generalized splines than by generalized trigonometric polynomials. In these cases, the best approximations of the Sobolev classes of functions of one variable by trigonometric polynomials and by periodic splines coincide. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1011–1020, August, 1998.     

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.     

On piecewise-polynomial approximation of functions with a bounded fractional derivative in an Lp-norm

We study the error in approximating functions with a bounded (r + α)th derivative in an L_p-norm. Here r is a nonnegative integer, α ε [0, 1), and ƒ^{(r + α)} is the classical fractional derivative, i.e., ƒ^{(r + α)}(y) = ∝₀¹, ^α d(ƒ^(r)(t)). We prove that, for any such function ƒ, there exists a piecewise-polynomial of degree s that interpolates ƒ at n equally spaced points and that approximates ƒ with an error (in sup-norm) ƒ^{(r + α)}_p O(n^{−(r+α−1/p}). We also prove that no algorithm based on n function and/or derivative values of ƒ has the error equal ƒ^{(r + α)}_p O(n^{−(r+α−1/p}) for any ƒ. This implies the optimality of piecewise-polynomial interpolation. These two results generalize well-known results on approximating functions with bounded rth derivative (α = 0). We stress that the piecewise-polynomial approximation does not depend on α nor on p. It does not depend on the exact value of r as well; what matters is an upper bound s on r, s r. Hence, even without knowing the actual regularity (r, α, and p) of ƒ, we can approximate the function ƒ with an error equal (modulo a constant) to the minimal worst case error when the regularity were known.     

Diophantinc approximation by linear forms on manifolds

The following Khintchine-type theorem is proved for manifoldsM embedded in ℝ ^k which satisfy some mild curvature conditions. The inequality |q·x| <Ψ(|q|) whereΨ(r) → 0 asr → ∞ has finitely or infinitely many solutionsqεℤ ^k for almost all (in induced measure) points x onM according as the sum Σ _r ^{= 1/∞} Ψ(r)r ^k−2 converges or diverges (the divergent case requires a slightly stronger curvature condition than the convergent case). Also, the Hausdorff dimension is obtained for the set (of induced measure 0) of point inM satisfying the inequality infinitely often whenψ(r) =r ^−t . τ >k − 1.     

One-sided approximation of functions

The best one-sided approximation of periodic functions by trigonometric polynomials of classW _p ⁰ (K) in the metric ofL is obtained. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 136–140, January, 2000.     

Bivariate spline interpolation with optimal approximation order

Let Δ be a triangulation of some polygonal domain Ω ⊂ R² and let S_q^r(Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We develop the first Hermite-type interpolation scheme for S _q ^r (Δ), q ≥ 3r + 2, whose approximation error is bounded above by Kh ^q +1, where h is the maximal diameter of the triangles in Δ, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and near-singular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of S _q ^r (Δ). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [7] and [18].     

Weighted simultaneous Chebyshev approximation

In this paper we discuss the problem of weighted simultaneous Chebyshev approximation to functions f₁,…f_m ε C(X) (1 m ∞), i.e., we wish to minimize the expression {∑^m_{j = 1} λ_j¦f_j − q¦^p}^1/p∞, where λ_j > 0, ∑^m_{j = 1} λ_j = 1, p 1. For this problem we establish the main theorems of the Chebyshev theory, which include the theorems of existence, alternation, de La Vallée Poussin, uniqueness, strong uniqueness, as well as that of continuity of the best approximation operator, etc.     

On the approximation power of bivariate splines

We show how to construct stable quasi-interpolation schemes in the bivariate spline spaces S _d ^r (Δ) with d⩾ 3r + 2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the L _p norms, and show that the methods also approximate derivatives to optimal order. We pay special attention to the approximation constants, and show that they depend only on the smallest angle in the underlying triangulation and the nature of the boundary of the domain. This revised version was published online in August 2006 with corrections to the Cover Date.     

Three-monotone spline approximation

For r≥3, n∈N and each 3-monotone continuous function f on [a,b] (i.e., f is such that its third divided differences [x₀,x₁,x₂,x₃]f are nonnegative for all choices of distinct points x₀,…,x₃ in [a,b]), we construct a spline s of degree r and of minimal defect (i.e., s∈C^r−1[a,b]) with n−1 equidistant knots in (a,b), which is also 3-monotone and satisfies ‖f−s‖_L∞[a,b]≤cω₄(f,n⁻¹,[a,b])_∞, where ω₄(f,t,[a,b])_∞ is the (usual) fourth modulus of smoothness of f in the uniform norm. This answers in the affirmative the question raised in [8, Remark 3], which was the only remaining unproved Jackson-type estimate for uniform 3-monotone approximation by piecewise polynomial functions (ppfs) with uniformly spaced fixed knots.Moreover, we also prove a similar estimate in terms of the Ditzian–Totik fourth modulus of smoothness for splines with Chebyshev knots, and show that these estimates are no longer valid in the case of 3-monotone spline approximation in the L_p norm with p<∞. At the same time, positive results in the L_p case with p<∞ are still valid if one allows the knots of the approximating ppf to depend on f while still being controlled.These results confirm that 3-monotone approximation is the transition case between monotone and convex approximation (where most of the results are “positive”) and k-monotone approximation with k≥4 (where just about everything is “negative”).     

Sets of rigged paths with Virasoro characters

Let {M _r,s ^(p,p′)}_{1≤r≤p−1,1≤s≤p′−1} be the irreducible Virasoro modules in the (p,p′)-minimal series. In our previous paper, we have constructed a monomial basis of ⊕ _r=1 ^p−1 M _r,s ^(p,p′) in the case 1<p′/p<2. By ‘monomials’ we mean vectors of the form , where φ _−n ^(r′,r):M _r,s ^(p,p′)→M _r′,s ^(p,p′) are the Fourier components of the (2,1)-primary field and |r ₀,s〉 is the highest weight vector of . In this article, we introduce for all p<p′ with p≥3 and s=1 a subset of such monomials as a conjectural basis of ⊕ _r=1 ^p−1 M _r,1^(p,p′). We prove that the character of the combinatorial set labeling these monomials coincides with the character of the corresponding Virasoro module. We also verify the conjecture in the case p=3.      

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≤ ∞)。     

Minimax approximation by rational fractions of the inverse polynomial type

Rational fractions of the formR(x)/(c ₁ +c ₂ x +c ₃ x ² + ...) ^r are useful for approximating decay type functions over infinite and semi-infinite domains. A procedure is given which produces the optimal coefficients with no more effort than for linear approximations. No initial guess is needed for the values of the coefficients nor for the maximum error of approximation.     

Shape-preserving approximation inL p

We prove a direct theorem for shape preservingL _p -approximation, 0p, in terms of the classical modulus of smoothnessw ₂(f, t _p ¹ ). This theorem may be regarded as an extension toL _p of the well-known pointwise estimates of the Timan type and their shape-preserving variants of R. DeVore, D. Leviatan, and X. M. Yu. It leads to a characterization of monotone and convex functions in Lipschitz classes (and more general Besov spaces) in terms of their approximation by algebraic polynomials.Communicated by Ron DeVore.     

Whitney type inequalities for local anisotropic polynomial approximation

We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in L_p(Q) with 1≤p≤∞. Here Q is a d-parallelepiped in R^d with sides parallel to the coordinate axes. We consider the error of best approximation of a function f by algebraic polynomials of fixed degree at most r_i−1 in variable , and relate it to a so-called total mixed modulus of smoothness appropriate to characterizing the convergence rate of the approximation error. This theorem is derived from a Johnen type theorem on equivalence between a certain K-functional and the total mixed modulus of smoothness which is proved in the present paper.     

Raising approximation order of refinable vector by increasing multiplicity

An algorithm is presented for raising an approximation order of any given orthogonal multiscaling function with the dilation factor a. Let φ(x) = [φ1(x),φ2(x),…,φr(x)]T be an orthogonal multiscaling function with the dilation factor a and the approximation order m. We can construct a new orthogonal multiscaling function φnew(x) = [ φT(x). f3r 1(x),φr 2(x),…,φr s(x)}T with the approximation order m L(L ∈ Z ). In other words, we raise the approximation order of multiscaling function φ(x) by increasing its multiplicity. In addition, we discuss an especial setting. That is, if given an orthogonal multiscaling function φ(x) = [φ1 (x), φ2(x), …, φr(x)]T is symmetric, then the new orthogonal multiscaling function φnew(x) not only raise the approximation order but also preserve symmetry. Finally, some examples are given.     

Smooth approximation of mather sets of monotone twist mappings

In the theory of monotone twist mappings of a cylinder one constructs for every rotation number α invariant minimal sets M_α. In this paper an approximation of these Mather sets by smooth invariant curves M^v_α is devised, which for v → 0 converge to M_α almost everywhere. The main point of the construction is that the approximating curves M^v_α form for v > 0 a smooth foliation. The approximation is achieved with the help of a regularized version of the Percival variational problem. © 1994 John Wiley & Sons, Inc.     

The approximation of reachable sets of control systems with integral constraint on controls

In this paper an approximation method for the construction of reachable sets of control systems with integral constraints on the control is considered. It is assumed that the control system is non-linear with respect to the phase state vector and is linear with respect to the control vector. The admissible control functions are chosen from the ball centered at the origin with radius μ₀ in L_p, p > 1. The reachable set is replaced by the set which consists of finite number of points. The estimated accuracy of the Hausdorff distance between the reachable set and the set which is approximately constructed is obtained.