scholarly journals Inequalities of Jackson type for functions with values in Hilbert space

2021 ◽  
Vol 16 ◽  
pp. 10
Author(s):  
V.F. Babenko ◽  
S.V. Savela
Keyword(s):  
Hilbert Space ◽  
Periodic Function ◽  
Modulus Of Continuity ◽  
Trigonometric Polynomials ◽  
Jackson Type

We present the generalization of M.I. Chernykh's results about the estimate of the best $L_2$-approximation of periodic function $f$ by trigonometric polynomials by its $L_2$-modulus of continuity, in the case of functions with values in Hilbert space.

scholarly journals Estimation of deviation of continuous $2\pi$-periodic functions from corresponding trigonometric polynomials of S.N. Bernstein's type

2021 ◽  
pp. 43
Author(s):  
N.Ya. Yatsenko
Keyword(s):  
Periodic Function ◽  
Trigonometric Polynomial ◽  
Modulus Of Continuity ◽  
Trigonometric Polynomials ◽  
Periodic Functions

We have established the estimation of deviation of continuous $2\pi$-periodic function $f(x)$ from the trigonometric polynomial of S.N. Bernstein's type that corresponds to it, by the modulus of continuity of the function $f(x)$.

scholarly journals On approximation of continuous functions by piecewise-continuous ones

1987 ◽  
pp. 65
Author(s):  
T.V. Nakonechnaia ◽  
T.A. Grankina
Keyword(s):  
Periodic Function ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Interpolation Spline ◽  
Lagrange Polynomial ◽  
Piecewise Continuous ◽  
Approximation Of Continuous Functions ◽  
Second Degree

Let $f(x) \in W^r H_{\omega} [-\pi, \pi]$ ($r = 0;1$) and $x_k = \frac{k\pi}{n} = h \cdot k$ ($k = 0, \pm 1, \ldots, \pm n$). We call $2\pi$-periodic function $S_2(f, x)$ an interpolation spline of order 2 if, in any segment $[x_k - \frac{h}{2}, x_k + \frac{h}{2}]$, it is the Lagrange polynomial of second degree that interpolates the function $f(x)$ in the points $x_{k-1}$, $x_k$, $x_{k+1}$.We establish that for any concave modulus of continuity $\omega (t)$ the equalities hold:$$\sup\limits_{f \in H_{\omega}[-\pi, \pi]} \| f - S_2(f) \|_{\infty} = \omega(\frac{h}{2}) + \frac{1}{8} \omega (h),$$\sup\limits_{f \in W^1 H_{\omega}[-\pi, \pi]} \| f - S_2(f) \|_{\infty} = \frac{65}{192} \int\limits_0^{\frac{4}{5}h} \omega (t) dt + \frac{5}{48} \int\limits_{\frac{4}{5}h}^{\frac{6}{5}h} \omega (t) dt$$

scholarly journals On approximation by trigonometrical polynomials in $L_2$ space

2016 ◽  
Vol 24 ◽  
pp. 89
Author(s):  
O.V. Polyakov
Keyword(s):  
Modulus Of Continuity ◽  
Best Approximation ◽  
Generalized Modulus Of Continuity ◽  
Jackson Type ◽  
The Best Approximation ◽  
Differentiable Functions

We obtain certain inequalities of Jackson type, connecting the value of the best approximation of periodic differentiable functions and the generalized modulus of continuity of the highest derivative.

Approximation by max-product sampling Kantorovich operators with generalized kernels

2019 ◽  
pp. 1-26 ◽  
Author(s):  
Lucian Coroianu ◽  
Danilo Costarelli ◽  
Sorin G. Gal ◽  
Gianluca Vinti
Keyword(s):  
Uniform Convergence ◽  
Real Axis ◽  
Modulus Of Continuity ◽  
Higher Dimensions ◽  
Jackson Type ◽  
Convergence Properties ◽  
Type Estimate ◽  
Quantitative Estimates ◽  
Uniform Modulus Of Continuity ◽  
Kantorovich Operators

In a recent paper, for max-product sampling operators based on general kernels with bounded generalized absolute moments, we have obtained several pointwise and uniform convergence properties on bounded intervals or on the whole real axis, including a Jackson-type estimate in terms of the first uniform modulus of continuity. In this paper, first, we prove that for the Kantorovich variants of these max-product sampling operators, under the same assumptions on the kernels, these convergence properties remain valid. Here, we also establish the [Formula: see text] convergence, and quantitative estimates with respect to the [Formula: see text] norm, [Formula: see text]-functionals and [Formula: see text]-modulus of continuity as well. The results are tested on several examples of kernels and possible extensions to higher dimensions are suggested.

NOTE ON JACKSON AND BERNSTE N TYPE APPROXIMATION THEOREMS IN THE CASE OF APPROXIMATION BY ALGEBRAIC POLYNOMIALS N THE SPACES L AND C

2000 ◽  
Vol 36 (3-4) ◽  
pp. 353-358 ◽  
Author(s):  
S. Pawelke
Keyword(s):  
Periodic Function ◽  
Best Approximation ◽  
Trigonometric Polynomials ◽  
Algebraic Polynomials ◽  
Cient Condition ◽  
Approximation Theorems ◽  
Type Approximation ◽  
The Best Approximation ◽  
Approximation By Algebraic Polynomials

We con ider the best approximation E (n,f)by algebraic polynomials of degree at most n for function f in L 1 (-1, 1)or C [-1, 1]and give imple necessary and u .cient condition for E (n,f)=O (n-.),n ›.,u ing the well-known results in the ca e of ap- proximation of periodic function by trigonometric polynomials.

scholarly journals Affine Bessel sequences and Nikishin’s example

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 963-966 ◽  
Author(s):  
V.A. Mironov ◽  
A.M. Sarsenbi ◽  
P.A. Terekhin
Keyword(s):  
Hilbert Space ◽  
Periodic Function ◽  
Spectral Theory ◽  
Continuous Periodic Function ◽  
Affine Systems ◽  
Convergence In Measure ◽  
Affine System ◽  
Bessel Sequences

We study affine Bessel sequences in connection with the spectral theory and the multishift structure in Hilbert space. We construct a non-Besselian affine system fun(x)g1 n=0 generated by continuous periodic function u(x). The result is based on Nikishin?s example concerning convergence in measure. We also show that affine systems fun(x)g1 n=0 generated by any Lipchitz function u(x) are Besselian.

On the Jackson-Type Inequality for the Best Sp-Approximations of Functions by Trigonometric Polynomials

2017 ◽  
Vol 8 ◽  
pp. 27-35
Author(s):  
Alexander N. Shchitov
Keyword(s):  
Best Approximation ◽  
Type Inequality ◽  
Trigonometric Polynomials ◽  
Sharp Constant ◽  
Jackson Type ◽  
Moduli Of Continuity ◽  
Approximation Of Functions ◽  
The Best Approximation

We find the sharp constant in the Jackson-type inequality between the value of the best approximation of functions by trigonometric polynomials and moduli of continuity of m-th order in the spaces Sp, 1 ≤ p < ∞. In the particular case we obtain one result which in a certain sense generalizes the result obtained by L.V. Taykov for m = 1 in the space L2 for the arbitrary moduli of continuity of m-th order (m Є N).

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