Geodesic
Convergence in the mean of the Fourier series in orthogonal polynomials
Matematičeskie zametki, Tome 14 (1973) no. 2, pp. 161-172.

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For weights $p(t)$ and $q(t)$ with a finite number of power-law-type singularities we obtain necessary and sufficient conditions for the inequality $$\|s_n^{(p)}(f)q\|_{L^\eta(-1,1)}\le C\|fq\|_{L^\eta(-1,1)},$$ to hold, where $s_n^{(p)}(f)$ is a partial sum of the Fourier series of the function $f$ in terms of polynomials orthogonal on $[-1,1]$ with weight $p(t)$. This inequality is used to solve the problem concerning convergence in the mean and also convergence almost everywhere of the partial sum $s_n^{(p)}(f)$. 
@article{MZM_1973_14_2_a0,
     author = {V. M. Badkov},
     title = {Convergence in the mean of the {Fourier} series in orthogonal polynomials},
     journal = {Matemati\v{c}eskie zametki},
     pages = {161--172},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_14_2_a0/}
}
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V. M. Badkov. Convergence in the mean of the Fourier series in orthogonal polynomials. Matematičeskie zametki, Tome 14 (1973) no. 2, pp. 161-172. http://geodesic.mathdoc.fr/item/MZM_1973_14_2_a0/