Mean approximation of functions by Fourier-Gegenbauer sums

Necessary and sufficient conditions for best approximations of functions in the $L^2_{(1-x^2)^\alpha}(-1,1)$ metric, $-1/2\le\alpha<1/2$ to zero at a certain rate are established (for $\alpha=?1/2$ known results are obtained). Inequalities for algebraic polynomials are used in the reasoning.