Formula_5375

\[\begin{gathered}\left|\hat{P}_{n}(x)\right| \leqslant \sqrt{\frac{2 n+1}{2}}, \quad x \in[-1,1], \\\left(1-x^{2}\right)^{1 / 4}\left|\hat{P}_{n}(x)\right| \leqslant \sqrt{\frac{2 n+1}{\pi n}}, \quad x \in[-1,1] .\end{gathered}\]