Best Polynomial Approximations and Widths of Certain Classes of Functions in the Space \( L_2 \)

Abstract

In the paper exact values of the n-widths are found for the class of differentiable periodic functions in the space L2[0, 2π], satisfying the condition $$\left(\int_0^t \tau \Omega_{2/m}^m(f(r),\tau)d\tau\right)^{m/2} \leq \Phi(t)$$, where $$0 < t \leq \pi/n$$, $$m, n, r \in \mathbb{N}$$, $$\Omega_m(f(r),\tau)$$ is the generalized modulus of continuity of order $$m$$ of the derivative $$f(r) \in L^2[0,2\pi]$$, and $$\Phi(t)$$ is a continuous non-decreasing function, such that $$\Phi(0) = 0$$ and $$\Phi(t) > 0$$ for $$t > 0$$.