Approximation of trigonometric functions by trigonometric polynomials with interpolation

The paper studies the approximation order of periodic functions by trigonometric polynomials with interpolation in arbitrary set of nodes. A method of construction of Hermite interpolation polynomials is pointed out.     

Approximation of periodic functions by trigonometric polynomials in the mean

For arbitrary summation methods we obtain inequalities between upper bounds of deviations in the L metric and corresponding upper bounds in the C metric with respect to a certain class of functions. These inequalities constitute a generalization of known relationships due to S. M. Nikol'skii. We consider the cases wherein these inequalities become exact or asymptotic equalities.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 15–26, July, 1974     

Approximation of periodic functions by interpolation polynomials in L1

Asymptotically precise estimates are obtained for the deviation, in the L₁-norm, of interpolation polynomials with equally-spaced nodes from certain classes of functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 781–786, June, 1990.     

Classification of infinitely differentiable periodic functions

The set of infinitely differentiable periodic functions is studied in terms of generalized -derivatives defined by a pair of sequences ψ ₁ and ψ ₂. In particular, we establish that every function f from the set has at least one derivative whose parameters ψ ₁ and ψ ₂ decrease faster than any power function. At the same time, for an arbitrary function f ∈ different from a trigonometric polynomial, there exists a pair ψ whose parameters ψ ₁ and ψ ₂ have the same rate of decrease and for which the -derivative no longer exists. We also obtain new criteria for 2π-periodic functions real-valued on the real axis to belong to the set of functions analytic on the axis and to the set of entire functions. Deceased. (A. I. Stepanets) Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1686–1708, December, 2008.     

Approximation of bandlimited functions by trigonometric polynomials

Let σ > 0. For 1 ≦ p ≦ ∞, the Bernstein space B _σ ^p is a Banach space of all f ∈ L ^p (?) such that f is bandlimited to σ; that is, the distributional Fourier transform of f is supported in [?σ,σ]. We study the approximation of f ∈ B _σ ^p by finite trigonometric sums $$ P_\tau (x) = \chi _\tau (x) \cdot \sum\limits_{|k| \leqq \sigma \tau /\pi } {c_{k,\tau } e^{i\frac{\pi } {\tau }kx} } $$ in L ^p norm on ? as τ → ∞, where χ _τ denotes the indicator function of [?τ, τ].     

Concerning the order of approximation of periodic continuous functions by trigonometric interpolation polynomials

Letx _kn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetR _n(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:R _n(x_kn)=f(x_kn),R _n ^(j)(x_kn)=0,j=1,2,4,k=0,1…,n−1. We setw ₂(t,f) as the second modulus of continuity off(x). Then we prove that |R _n(x)-f(x)|=0(nw₂(1/nf)). We also examine the question of lower estimate of ‖R _n-f‖. This generalizes an earlier work of the author.     

Approximation of infinitely differentiable multivariate functions is intractable

We prove that L_∞-approximation of C^∞-functions defined on [0,1]^d is intractable and suffers from the curse of dimensionality. This is done by showing that the minimal number of linear functionals needed to obtain an algorithm with worst case error at most ε(0,1) is exponential in d. This holds despite the fact that the rate of convergence is infinite.     

Approximation of several dimensional functions by trigonometric polynomials

Let f(x, y) be a periodic function defined on the region D

Approximation of weakly differentiable periodic functions

An asymptotic equality is found for the lower bounds of the best approximations of the classes C , under a condition of slow growth of (·).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 406–412, March, 1990.     

The exact error of trigonometric interpolation for differentiable functions

In [1], G. Halász gives some properties of the order of trigonometric approximation as a function of the Lipschitz parameter. Here we show that these properties completely characterize this function.     

Approximation of continuous functions by trigonometric polynomials almost everywhere

We consider the problem of the rate of approximation of continuous 2π-periodic functions of class W^rH[ω]C by trigonometric polynomials of order n on sets of total measure. We prove that when r≥0,ω(δ)δ ^?1 → ∞ (δ → 0) there exists a function f ε W^rH[ω]C such thatf ε W^rH[ω]C and for any sequence {t_{n n=1} ^∞ we have almost everywhere on [0, 2π] $\begin{array}{l} \overline {\mathop {\lim }\limits_{n \to \infty } } \left| {f(x) - t_n (x)} \right|n^r \omega ^{ - 1} (1/n) > C_x > 0, \\ \overline {\mathop {\lim }\limits_{n \to \infty } } \left| {\tilde f(x) - t_n (x)} \right|n^r \omega ^{ - 1} (1/n) > C_x > 0. \\ \end{array}$      

Approximation of nonnegative functions by means of exponentiated trigonometric polynomials

We consider the problem of approximating a nonnegative function from the knowledge of its first Fourier coefficients. Here, we analyze a method introduced heuristically in a paper by Borwein and Huang (SIAM J. Opt. 5 (1995) 68–99), where it is shown how to construct cheaply a trigonometric or algebraic polynomial whose exponential is close in some sense to the considered function. In this note, we prove that approximations given by Borwein and Huang's method, in the trigonometric case, can be related to a nonlinear constrained optimization problem, and their convergence can be easily proved under mild hypotheses as a consequence of known results in approximation theory and spectral properties of Toeplitz matrices. Moreover, they allow to obtain an improved convergence theorem for best entropy approximations.     

Approximation of infinitely differentiable multivariate functions is not strongly tractable

This article studies tractability and strong tractability for multivariate approximation of infinitely differentiable functions, using either standard information or continuous linear information. We prove that this approximation problem is not strongly tractable.     

Approximation of differentiable functions by polynomials in domains with outward cusps

For Isotropic Sobolev spaces defined on domains with outward cusps of power type, a method of approximation of functions and their derivatives by algebraic polynomials is indicated, and the corresponding approximation estimates are given.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 534–540, April, 1992.