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This article is cited in 10 scientific papers (total in 10 papers)
An exact Jackson–Stechkin inequality for $L^2$-approximation on the interval with the Jacobi weight and on projective spaces
A. G. Babenko Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
Abstract:
Let $L^2_{\alpha,\beta}$ be the Hilbert space of real-valued functions on $[0,\pi]$ with scalar product
$$
(F,G)=\int_{0}^{\pi}F(x)G(x)(\sin\dfrac{x}{2})^{2\alpha+1}
(\cos\dfrac{x}{2})^{2\beta+1} dx,\qquad \alpha>-1,\quad \beta>-1,
$$
and norm $\|F\|=(F,F)^{1/2}$. We prove in the case when $\alpha>\beta\geqslant-1/2$ the following exact Jackson–Stechkin inequality
$$
E_{n-1} (F)\leqslant\omega_r(F,2x_{n}^{\alpha,\beta}),\quad
F\in L^2_{\alpha,\beta},
$$
between the best of $F$ by cosine-polynomials of order $n-1$ and its generalized modulus of continuity of (real) order $r\geqslant 1$: $n\geqslant\max\{2,1+
\frac{\alpha-\beta}{2}\}$ if $\beta>-\frac12$ , $n\geqslant 1$ if $\beta=-\frac12$ , where $x_{n}^{\alpha,\beta}$ is the first positive zero of the Jacobi cosine-polynomial
$P_{n}^{(\alpha,\beta)}(\cos x)$. We deduce from this inequality similar inequalities for mean-square approximations of functions of several variables given on projective spaces.
DOI:
https://doi.org/10.4213/im219
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English version:
Izvestiya: Mathematics, 1998, 62:6, 1095–1119
Bibliographic databases:
MSC: 41A50, 41A10, 42A10, 41A25, 41A17 Received: 30.09.1997
Citation:
A. G. Babenko, “An exact Jackson–Stechkin inequality for $L^2$-approximation on the interval with the Jacobi weight and on projective spaces”, Izv. RAN. Ser. Mat., 62:6 (1998), 27–52; Izv. Math., 62:6 (1998), 1095–1119
Citation in format AMSBIB
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http://mi.mathnet.ru/eng/izv219https://doi.org/10.4213/im219 http://mi.mathnet.ru/eng/izv/v62/i6/p27
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