# The inequality of Bernstein for even j - polynomial of Schlomilch.

Санина Е. Л. The inequality of Bernstein for even j - polynomial of Schlomilch. // Taurida Journal of Computer Science Theory and Mathematics, – 2016. – T.15. – №2. – P. 104-
https://doi.org/10.37279/1729-3901-2016-15-2-104-111

The role of the Bernstein inequality in various problems of approximation theory in problems of differential equations well known books from MS Nikolsky [1], NI Bari [2] (p. 895), Sigmund [3] (p. 41), and others. Bernstein inequality in the simplest case, it follows from the Riesz interpolation

$T'_{n}(t) = \frac{1}{4\pi}\sum\limits_{k=1}^{2n}(−1)^{k+1} \frac{1}{sin^{2} \frac{θ_{k}}{2}}T_{n}(t + θ_{k}), θ_{k} =\frac{2k − 1}{2n}\pi$,

for trigonometric polynomials

$T_{n}(t) = \frac{a_{0}}{2}+\sum\limits^{n}_{k=1}(a_{k} sin kt + b_{k} cos kt)$.

and has the form

$||T'_{n}(t)||_{C[−\pi,\pi]} \le n||T_{n}(t)||_{C[\pi,\pi]}$.

$j$-polynomials Schlomilch introduced in [4]. Properties of the even and odd $j$-polynomials are described in [5]. The need to study this kind of polynomials associated with many problems of the theory of functions of weighted spaces, singular differential equations [6].
Classic series Schlomilch have the form

$\frac{a_{0}}{2\Gamma(\nu + 1)} +\sum\limits^{\infty}_{m=1}\left( a_{m}\frac{J_{ν}(mx)}{(mx/2)^{ν}}+ b_{m}\frac{H_{ν}(mx)}{(mx/2)^{ν}}\right)$,

where $J_{ν}$ — Bessel function of the first kind, and $H_{ν}$ — Struve function. How $\frac{J_{ν}(mx)}{(mx/2)^{ν}}$ — even and $\frac{H_{ν}(mx)}{(mx/2)^{ν}}$ — odd.

Among the Fourier-Bessel series, Dini series Schlomilch series most resemble trigonometric Fourier series, as the series Schlomlich generated by trigonometric series using integrals Schlomilch or Sonin. Therefore, the properties Schlomilch series can be explored using the properties of trigonometric series. In [7] noted that the application for the full series Schlomilch (with even and odd components), there are physical limitations of the use of Struve functions.

General view even $j$-polynomials Schlomilch $n$ has the form

$S_{n}(x) = \frac{a_{0}}{2}+\sum\limits^{n}_{m=1}a_{m} j_{ν}(mx)$.

Received interpolation formula for $B$-derivative $j$-polynomials Schlomlich, which is a consequence of the Riesz interpolation formula for trigonometric polynomials:

$B_{ν} (S_{n})(x) = \frac{1}{4n^{2}}\sum\limits^{2n}_{k, m=1}\frac{(−1)^{k+m}}{\left(cos\frac{\Theta_{k−m}}{2} − cos\frac{\Theta_{k+m}}{2}\right)^{2}}\Pi^{\nu}_{x}T_{n}(x + \Theta_{k+m})$,

where $B_{ν} = \frac{d^{2}}{dx^{2}} + \frac{2ν+1}{x} \frac{d}{dx}$ — singular differential operator of Bessel, $ν > −1/2$.

On the basis of this formula is an analogue of the Bernstein inequality, where the role of thederivative of the Bessel operator executes and performs the role of a trigonometric polynomial of even $j$-polynomial Schlomlich:

$|B^{k}_{ν} S_{n}(x)| \le n^{2k} \sup\limits_{x\in [0,\pi]}|S_{n}(x)| \le n^{2k} M, x \in [0, \pi], k \in N$.

This is the main result of this work. Using this inequality will allow to explore the direct and inverse theorems approximation theory, embedding theorem in weighted classes Nikolsky functions Besov, Sobolev-Kipriyanova.

Keywords: Bessel operator, Poisson operator interpolation formula, j-polynomial Schlomilch, Bernstein’s inequality

UDC:
517.9