# On the Best Approximation in the Mean by Algebraic Polynomials with Weight and the Exact Values of Widths for the Classes of Functions

### Abstract

The exact value of the extremal characteristicis obtained on the class *L* _{2} ^{ r } (*D* _{ ρ })*,* where *r ∈* ℤ_{+}; \( {D}_{\rho} = \sigma (x)\frac{d^2}{d{ x}^2}+\tau (x)\frac{d}{d x} \) *, σ* and τ are polynomials of at most the second and first degrees, respectively, *ρ* is a weight function, 0 *< p* ≤ 2*,* 0 *< h <* 1*, λ* _{ n }(*ρ*)
are eigenvalues of the operator *D* _{ ρ } *, φ*
is a nonnegative measurable and summable function (in the interval (*a, b*)) which is not equivalent to zero, *Ω* _{ k,ρ } is the generalized modulus of continuity of the *k* th order in the space *L* _{2,ρ } (*a, b*)*,* and *E* _{ n } (*f*)_{2,ρ }
is the best polynomial approximation in the mean with weight *ρ* for a function *f ∈ L* _{2,ρ } (*a, b*)*.* The exact values of widths for the classes of functions specified by the characteristic of smoothness *Ω* _{ k,ρ } and the *K*-functional \( \mathbb{K} \) _{m} are also obtained.

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 65, no. 12, Dec. 2013, pp. 1604–1621, http://umj.imath.kiev.ua/index.php/umj/article/view/2540.