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 Mat. Sb. (N.S.), 1972, Volume 87(129), Number 2, Pages 254–274 (Mi msb3048)

Best approximations of functions in the $L_p$ metric by Haar and Walsh polynomials

B. I. Golubov

Abstract: In this work the modulus of continuity of functions in the $L_p$ metric $(1\leqslant p<\nobreak\infty)$ is estimated through its best approximations in this metric by Haar and Walsh polynomials. Besides, estimates of best approximations of functions by Haar and Walsh polynomials in the $L_q$ metric are obtained by the same approximations in the $L_p$ metric $(1\leqslant p<q\leqslant\infty)$. In the last case, the results are analogous to those which were proved for approximations by trigonometric polynomials by P. L. Ul'yanov and also by S. B. Stechkin and A. A. Konyushkov.
Bibliography: 26 titles.

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English version:
Mathematics of the USSR-Sbornik, 1972, 16:2, 265–285

Bibliographic databases:

UDC: 517.5
MSC: Primary 41A30; Secondary 41A10

Citation: B. I. Golubov, “Best approximations of functions in the $L_p$ metric by Haar and Walsh polynomials”, Mat. Sb. (N.S.), 87(129):2 (1972), 254–274; Math. USSR-Sb., 16:2 (1972), 265–285

Citation in format AMSBIB
\Bibitem{Gol72} \by B.~I.~Golubov \paper Best approximations of functions in the $L_p$ metric by Haar and Walsh polynomials \jour Mat. Sb. (N.S.) \yr 1972 \vol 87(129) \issue 2 \pages 254--274 \mathnet{http://mi.mathnet.ru/msb3048} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=293315} \zmath{https://zbmath.org/?q=an:0235.42012|0249.42015} \transl \jour Math. USSR-Sb. \yr 1972 \vol 16 \issue 2 \pages 265--285 \crossref{https://doi.org/10.1070/SM1972v016n02ABEH001425} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. È. A. Storozhenko, V. G. Krotov, P. Oswald, “Direct and converse theorems of Jackson type in $L^p$ spaces, $0<p<1$”, Math. USSR-Sb., 27:3 (1975), 355–374
2. Oswald P., “Spline Approximation in the Lp-Metric, 0 Less-Than-Or-Equal-to P Less-Than-Or-Equal-to 1”, Math. Nachr., 94 (1980), 69–96
3. V. I. Ivanov, “Approximation in $L_p$ by polynomials in the Walsh system”, Math. USSR-Sb., 62:2 (1989), 385–402
4. G. A. Akishev, “Obobschennaya sistema Khaara i teoremy vlozheniya v simmetrichnye prostranstva”, Fundament. i prikl. matem., 8:2 (2002), 319–334
5. P. A. Terekhin, “Best approximation of functions in $L_p$ by polynomials on affine system”, Sb. Math., 202:2 (2011), 279–306
6. S. S. Volosivets, “Teoremy vlozheniya dlya $\mathbf{P}$-ichnykh prostranstv Khardi i $VMO$”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:4(2) (2014), 518–525
7. S. B. Vakarchuk, A. N. Shchitov, “Estimates for the error of approximation of functions in $L_p^1$ by polynomials and partial sums of series in the Haar and Faber–Schauder systems”, Izv. Math., 79:2 (2015), 257–287
8. S. A. Stasyuk, “Priblizhenie nekotorykh gladkostnykh klassov periodicheskikh funktsii mnogikh peremennykh polinomami po tenzornoi sisteme Khaara”, Tr. IMM UrO RAN, 21, no. 4, 2015, 251–260
9. S. S. Volosivets, B. I. Golubov, “Generalized absolute convergence of series from Fourier coeficients by systems of Haar type”, Russian Math. (Iz. VUZ), 62:1 (2018), 7–16
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