Approximation Of Periodic Functions Research Articles

Constructive sparse trigonometric approximations for the functions with generalized mixed smoothness

The order bounds (in the case of uniform metrics) and exact order bounds (in the case of integral metrics) for the best m-term trigonometric approximation of periodic functions with generalized mixed smoothness from classes close to the Nikol’skii–Besov-type ones are obtained. Every of the upper bounds is realized by a constructive method based on greedy algorithms.

Nearly Comonotone Approximation of Periodic Functions

Nearly Comonotone Approximation of Periodic Functions

On the constant and step in Jackson’s inequality for best approximations by trigonometric polynomials and by Haar polynomials

Two sharp results for best approximations of periodic functions are established in this paper. We prove the sharpness of the step of the modulus of continuity in Jackson’s inequality with least possible constant for approximations by trigonometric polynomials. We also prove the sharpness of the constants in a Jackson-type inequality for approximations by Haar polynomials in several variables.

Approximations of periodic functions by analogue of Zygmund sums in the spaces Lp(•)

We found order estimates for the upper bounds of the deviations of analogue of Zygmund?s sums on the classes of (?;?)-differentiable functions in the metrics of generalized Lebesgue spaces with variable exponent.

Approximation of periodic functions by Zygmund means in Orlicz spaces

Approximation of periodic functions by Zygmund means in Orlicz spaces

On Jackson Type Inequalities for the Best Approximations of Periodic Functions

UDC 517.5 We develop a new approach to the proof of a generalized Jackson theorem providing an estimate for the best approximation by trigonometric polynomials with the help of the moduli of continuity of an arbitrary order. The proposed approach is based on approximation methods constructed by using Fejer and Jackson–Vallee - Poussin classical kernels. Bibliography :1 4titles.

Approximation of Periodic Functions by Modified Steklov Averages in L 2

In the space L 2 of periodic functions, sharp (in the sense of constants) lower estimates for the deviation of the modified Steklov functions of the first and second orders in terms of the modulus of continuity are established. Similar results are also obtained for even continuous periodic functions with nonnegative Fourier coefficients in the space C. Bibliography: 3 titles.

On the Approximation of Periodic Functions in L2 and the Values of the Widths of Certain Classes of Functions

The sharp Jackson–Stechkin inequalities are received, in which a special module of continuity Ωe m (f;t) determined by Steklov’s function is used instead the usual modulus of continuity of mth order ω m (f;t). Such generalized modulus of continuity of mth order were introduced by V.A. Abilov and F.V. Abilova. The introduced modulus of continuity found their application in the theory of polynomial approximation in Hilbert space in the works by M.Sh. Shabozov and G.A. Yusupov, S.B. Vakarchuk and V.I. Zabutnaya and others. While continuing and developing these direction for some classes of functions defined by modulus of continuity, the new values of n-widths in the Hilbert space L₂were found.

On the Constants in Generalized Jackson-Type Inequalities

Some modifications of Jackson-type inequalities are established, including estimates on the best approximations of periodic functions by trigonometric polynomials in terms of the moduli of continuity of higher orders. Special attention is paid to bounding the constants occurring in the inequalities obtained.

Approximation of Periodic Functions of Many Variables in Metric Spaces by Piecewise-Constant Functions

We prove the direct and inverse Jackson- and Bernstein-type theorems for averaged approximations of periodic functions of many variables by piecewise-constant functions with uniform partition of the period torus in metric spaces with integral metric given by a function ψ of the type of modulus of continuity.

Strong approximation of almost periodic functions

We consider summability methods generated by the class GM (2β) . We generalize some related results of P. Pych-Taberska [Studia Math. XCVI (1990), 91–103] on strong approximation of almost periodic functions by their Fourier series and S. M. Mazhar and V.Totik [J. Approx. Theory, 60(1990), 174–182] on approximation of periodic functions by matrix means of their Fourier series. Mathematics subject classification (2010): 42A24.

Estimates for the Best Approximations of Periodic Functions in Terms of Deviations

We estimate the best approximation of a function by trigonometric polynomials in terms of deviations. To illustrate general results, we consider estimates connected with the Vallee–Poussin integral and the generalized Weierstrass integral.

Inequalities of the type of the generalized Jackson theorem for the best approximations

Very simple methods for deriving upper estimates of the type of the generalized Jackson theorem for the best approximations of periodic functions are proposed, The results obtained are similar to the generalized Jackson theorem, providing estimates in terms of the moduli of continuity. Bibliography: 12 titles.

Best approximations of periodic functions in generalized lebesgue spaces

In generalized Lebesgue spaces with variable exponent, we determine the orders of the best approximations in the classes of (ψ; β)-differentiable 2π-periodic functions, deduce an analog of the well-known Bernstein inequality for the (ψ; β)-derivative, and apply this inequality to prove the inverse theorems of approximation theory in these classes.

Asymptotics of series arising from the approximation of periodic functions by Riesz and Cesàro means

Asymptotic expansions in powers of δ as δ → +∞ of the series $$\sum\limits_{k = 0}^\infty {( - 1)^{(\beta + 1)k} \frac{{Q((\delta ^\alpha - (ak + b)^\alpha ) + )}} {{(ak + b)^{r + 1} }}} , $$ where β ∈ ℤ, α, a, b > 0, and r ∈ ℂ, while Q is an algebraic polynomial satisfying the condition Q(0) = 0, are obtained. In special cases, these series arise from the approximation of periodic differentiable functions by the Riesz and Cesaro means.

Approximation of periodic functions in weighted Orlicz spaces

Approximation of periodic functions in weighted Orlicz spaces

Approximations of periodic functions to $$\mathbb R ^n$$ by curvatures of closed curves

We show that for any \(n\) real periodic functions \(f_1,\ldots , f_n\) with the same period, such that \(f_i>0\) for \(i 0\), there is a closed curve in \(\mathbb R ^{n+1}\) with curvatures \(\kappa _1, \ldots , \kappa _n\) such that \(\left| \kappa _{i(t)}-f_{i(t)}\right|<\varepsilon \) for all \(i\) and \(t\). This does not hold for parametric families of closed curves in \(\mathbb R ^{n+1}\).

Exact order of approximation of periodic functions by one nonclassical method of summation of Fourier series

By using an exact estimate for approximation by known trigonometric polynomials, we strengthen a Jackson-type theorem. Moreover, we determine the exact order of approximation of some periodic functions by these polynomials. For this purpose, we introduce a special modulus of smoothness.

Best approximation of periodic functions of several variables from the classes $ MB_{{p,\theta }}^{\omega } $

We obtain exact-order estimates for the best approximation of periodic functions of several variables from the classes $ MB_{{p,\theta }}^{\omega } $ by trigonometric polynomials with the “numbers” of harmonics from graded hyperbolic crosses in the metric of the space L q :

Approximation of periodic functions in the classes H{sub q}{sup {Omega}} by linear methods

The following result is proved: if approximations in the norm of L{sub {infinity}} (of H{sub 1}) of functions in the classes H{sub {infinity}}{sup {Omega}} (in H{sub 1}{sup {Omega}}, respectively) by some linear operators have the same order of magnitude as the best approximations, then the set of norms of these operators is unbounded. Also Bernstein's and the Jackson-Nikol'skii inequalities are proved for trigonometric polynomials with spectra in the sets Q(N) (in {Gamma}(N,{Omega})). Bibliography: 15 titles.