Geodesic
Rational functions of best approximation in integral metrics
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (1982), pp. 43-48

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We prove that any rational function of degree $\le n$ with real coefficients best approximating a function $f\in L_q(a,b)$, $1$, in the metric of $L_q(a,b)$ is of degree exactly $n$. We find the signs of the polynomials best approximating in the metric of $L_q([a,b])$, $1\le q\le\infty$ for some classes of differentiate functions. 
@article{VMUMM_1982_5_a11,
     author = {A. K. Ramazanov},
     title = {Rational functions of best approximation in integral metrics},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {43--48},
     publisher = {mathdoc},
     number = {5},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_1982_5_a11/}
}
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A. K. Ramazanov. Rational functions of best approximation in integral metrics. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (1982), pp. 43-48. http://geodesic.mathdoc.fr/item/VMUMM_1982_5_a11/