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

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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. 
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     author = {V. P. Motornyi},
     title = {On the mean convergence of {Fourier} series in {Legendre} polynomials},
     journal = {Izvestiya. Mathematics },
     pages = {131--144},
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     number = {1},
     year = {1973},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1973_7_1_a5/}
}
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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/