Geodesic
On the mean convergence of Fourier series in Legendre polynomials
Izvestiya. Mathematics , Tome 7 (1973) no. 1, pp. 131-144.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we study the convergence of Fourier series in Legendre polynomials in the space $L_p$, if $1\leqslant p\leqslant4/3$ or $4\leqslant p\infty$ (i.e. in the case when the Lebesgue constants are unbounded). The fundamental result consists in the fact that with the improvement of the differential-difference properties of the function, the convergence is less affected by the growth of the Lebesgue constant ($1\leqslant p\leqslant4/3$). For functions with sufficiently good differential-difference properties the partial sums of the Fourier–Legendre series give an approximation in the $L_p$ ($1$) metric of an order as good as the best. 
@article{IM2_1973_7_1_a5,
     author = {V. P. Motornyi},
     title = {On the mean convergence of {Fourier} series in {Legendre} polynomials},
     journal = {Izvestiya. Mathematics },
     pages = {131--144},
     publisher = {mathdoc},
     volume = {7},
     number = {1},
     year = {1973},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1973_7_1_a5/}
}
TY  - JOUR
AU  - V. P. Motornyi
TI  - On the mean convergence of Fourier series in Legendre polynomials
JO  - Izvestiya. Mathematics 
PY  - 1973
SP  - 131
EP  - 144
VL  - 7
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1973_7_1_a5/
LA  - en
ID  - IM2_1973_7_1_a5
ER  - 
%0 Journal Article
%A V. P. Motornyi
%T On the mean convergence of Fourier series in Legendre polynomials
%J Izvestiya. Mathematics 
%D 1973
%P 131-144
%V 7
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1973_7_1_a5/
%G en
%F IM2_1973_7_1_a5
V. P. Motornyi. On the mean convergence of Fourier series in Legendre polynomials. Izvestiya. Mathematics , Tome 7 (1973) no. 1, pp. 131-144. http://geodesic.mathdoc.fr/item/IM2_1973_7_1_a5/

[1] Pollard H., “The mean convergence of orthogonal series, I”, Trans. Amer. Math. Soc., 62 (1947), 387–403 ; “II, Ibedem”, 63 (1948), 355–367 | DOI | MR | Zbl | DOI | MR | Zbl

[2] Pollard H., “The mean convergence of orthogonal series”, Duke Math. J., 16:1 (1949), 189–191 | DOI | MR | Zbl

[3] Neuman J., Rudin W., “Mean convergence of orthogonal series”, Proc. Amer. Math. Soc., 3 (1952), 219–222 | DOI | MR

[4] Muckenhoupt B., “Mean convergence of Jacobi series”, Proc. Amer. Math. Soc., 23:2 (1969), 306–310 | DOI | MR | Zbl

[5] Suetin P. K., “O predstavlenii nepreryvnykh i differentsiruemykh funktsii ryadami Fure po mnogochlenam Lezhandra”, Dokl. AN SSSR, 158:6 (1964), 1275–1277 | MR | Zbl

[6] Agakhanov S. A., Natanson G. I., “Priblizhenie funktsii summami Fure–Yakobi”, Dokl. AN SSSR, 166:1 (1966), 9–10 | Zbl

[7] Motornyi V. P., “Nekotorye voprosy priblizheniya funktsii algebraicheskimi mnogochlenami v integralnoi metrike”, Dokl. AN SSSR, 172:3 (1967), 537–540 | MR | Zbl

[8] Motornyi V. P., “Priblizhenie funktsii algebraicheskimi polinomami v metrike $L_p$”, Izv. AN SSSR. Ser. matem., 35:4 (1971), 874–899 | MR | Zbl

[9] Telyakovskii S. A., “Priblizhenie differentsiruemykh funktsii chastnymi summami ikh ryadov Fure”, Matem. zametki, 4:3 (1968), 291–300 | Zbl

[10] Segë G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962

[11] Muckenhoupt B., “Mean convergence of Hermite and Laguerre series, II”, Trans. Amer. Math. Soc., 147:2 (1970), 433–460 | DOI | MR

[12] Hardy G. H., Littlewood J. E., “A convergence criterion for Fourier series”, Math. Z., 28 (1928), 612–634 | DOI | MR | Zbl

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

[14] Potapov M. K., “Nekotorye neravenstva dlya polinomov i ikh proizvodnykh”, Vestnik MGU, 1960, no. 2, 10–19 | MR

[15] Ibragimov I. P., “Nekotorye neravenstva dlya algebraicheskikh mnogochlenov”, Issledovaniya po sovremennym problemam konstruktivnoi teorii funktsii, Fizmatgiz, M., 1961, 139–143 | MR

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

[17] Zigmund A., Trigonometricheskie ryady, t. I, Mir, M., 1965 | MR

[18] Wing G. M., “The mean convergence of orthogonal series”, Amer. J. Math., 72 (1950), 792–807 | DOI | MR