Geodesic
Best One-Sided Approximation in the Mean of the Characteristic Function of an Interval by Algebraic Polynomials
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 4, pp. 110-125 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\upsilon$ be a weight on $(-1,1),$ i.e., a measurable integrable nonnegative function nonzero almost everywhere on $(-1,1)$. Denote by $L^\upsilon(-1,1)$ the space of real-valued functions $f$ integrable with weight $\upsilon$ on $(-1,1)$ with the norm $\|f\|=\int_{-1}^{1}|f(x)|\upsilon(x)\,dx$. We consider the problems of the best one-sided approximation (from below and from above) in the space $L^\upsilon(-1,1)$ to the characteristic function of an interval $(a,b),$ $-1$ by the set of algebraic polynomials of degree not exceeding a given number. We solve the problems in the case where $a$ and $b$ are nodes of a positive quadrature formula under some conditions on the degree of its precision as well as in the case of a symmetric interval $(-h,h),$ $0$ for an even weight $\upsilon$.
Keywords: one-sided approximation, characteristic function of an interval, algebraic polynomials. 
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     title = {Best {One-Sided} {Approximation} in the {Mean} of the {Characteristic} {Function} of an {Interval} by {Algebraic} {Polynomials}},
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M. V. Deikalova; A. Yu. Torgashova. Best One-Sided Approximation in the Mean of the Characteristic Function of an Interval by Algebraic Polynomials. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 4, pp. 110-125. http://geodesic.mathdoc.fr/item/TIMM_2018_24_4_a8/

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