Best One-Sided Approximation in the Mean of the Characteristic Function of an Interval by Algebraic Polynomials

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 $(a,b),$ $-1<a<b<1,$ 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 $(-h,h),$ $0<h<1,$ for an even weight $\upsilon$.