Geodesic
Best approximations of functions in the $L_p$ metric by Haar and Walsh polynomials
Sbornik. Mathematics, Tome 16 (1972) no. 2, pp. 265-285 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 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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     title = {Best approximations of functions in the $L_p$ metric by {Haar} and {Walsh} polynomials},
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B. I. Golubov. Best approximations of functions in the $L_p$ metric by Haar and Walsh polynomials. Sbornik. Mathematics, Tome 16 (1972) no. 2, pp. 265-285. http://geodesic.mathdoc.fr/item/SM_1972_16_2_a8/

[1] A. Haar, “Zur Theorie der orthogonaien Funktionensysteme”, Math. Ann., 69 (1910), 331–371 | DOI | MR | Zbl

[2] G. Aleksich, Problemy skhodimosti ortogonalnykh ryadov, IL, Moskva, 1963 | MR

[3] J. L. Walsh, “A closed set of normal orthogonal functions”, Amer. J. Math., 45 (1923), 5–24 | DOI | MR | Zbl

[4] S. Kachmazh, G. Shteingauz, Teoriya ortogonalnykh ryadov, Fizmatgiz, Moskva, 1958

[5] B. Sz-Nagy, “Approximation properties of orthogonal expansions”, Acta Sci. Math. Szeged, 15 (1953/54), 31–57 | MR

[6] P. L. Ulyanov, “O ryadakh po sisteme Khaara”, Matem. sb., 63(105) (1964), 356–391 | Zbl

[7] B. I. Golubov, “O ryadakh Fure nepreryvnykh funktsii po sisteme Khaara”, Izv. AN SSSR, seriya matem., 28 (1964), 1271–1296 | MR | Zbl

[8] Z. Ciesielski, “Properties of the orthonormal Franclin system”, Studia Math., 27:3 (1966), 289–323 | MR | Zbl

[9] Ch. Watari, “Best approximation by Walsh polynomials”, Tohoku Math. J., 15:1 (1963), 1–5 | DOI | MR

[10] N. J. Fine, “On Walsh functions”, Trans. Amer. Math. Soc., 65 (1949), 372–414 | DOI | MR | Zbl

[11] A. F. Timan, Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, 1960

[12] E. P. Dolzhenko, Ravnomernye approksimatsii ratsionalnymi funktsiyami (algebraicheskimi i trigonometricheskimi) i globalnye funktsionalnye svoistva, 166:3 (1966), 526–529 | Zbl

[13] A. A. Konyushkov, “Nailuchshie priblizheniya trigonometricheskimi polinomami i koeffitsienty Fure”, Matem. sb., 44(86) (1958), 53–84

[14] P. L. Ulyanov, “Teoremy vlozheniya i sootnosheniya mezhdu nailuchshimi priblizheniyami (modulyami nepreryvnosti) v raznykh metrikakh”, Matem. sb., 81(123) (1970), 104–131 | Zbl

[15] P. L. Ulyanov, “Vlozhenie nekotorykh klassov funktsii”, Izv. AN SSSR, seriya matem., 32 (1968), 649–686 | Zbl

[16] M. Z. Solomyak, “Ob ortogonalnom bazise v prostranstve Banakha”, Vestnik LGU, seriya matem. mekh., 1957, no. 1, 27–36 | Zbl

[17] G. G. Xardi, D. E. Littlvud, G. Polia, Neravenstva, IL, Moskva, 1948

[18] J. Marcinkiewicz, “Quelques théorèmes sur les series orthogonales”, Ann. Soc. Polon. Math., 16 (1937), 85–96

[19] B. I. Golubov, “O $p$-variatsii funktsii”, Matem. zametki, 5:2 (1969), 195–204 | MR | Zbl

[20] A. I. Rubinshtein, “O ryadakh s lakunami”, Izv. VUZov, Matematika, 1963, no. 3, 137–148 | MR | Zbl

[21] J. Schauder, “Eine Eigenschaft des Haarschen Orthogonalsystems”, Math. Z., 28 (1928), 317–320 | DOI | MR | Zbl

[22] G. H. Hardy, J. E. Littlwood, “A convergence criterien for Fourier series”, Math. Z., 28:4 (1928), 612–634 | DOI | MR | Zbl

[23] V. K. Dzyadyk, “O prodolzhenii funktsii, udovletvoryayuschikh usloviyu Lipshitsa v metrike $L_p$”, Matem. sb., 40(82) (1956), 239–242 | MR

[24] P. L. Ulyanov, “Ob absolyutnoi i ravnomernoi skhodimosti ryadov Fure”, Matem. sb., 72(114) (1967), 193–225 | Zbl

[25] S. M. Nikolskii, “Ryad Fure funktsii s dannym modulem nepreryvnosti”, DAN SSSR, 52:3 (1946), 191–194

[26] S. B. Stechkin, “O poryadke nailuchshego priblizheniya nepreryvnykh funktsii”, Izv. AN SSSR, seriya matem., 15 (1951), 219–242 | Zbl