06 - Lebesgue Inequality

Lebesgue Inequality

Let $f$ be continuous with $f\in C([-1,1])$

Let $p_n$ be the polynomial interpolant of $f$ at $\{x_i{\}}_{i=0}^n$
Let ${\Lambda }_n$ be the Lebesgue Constant then

$$||f-p_n|{|}_{\infty }\le ({\Lambda }_n+1)||f-p_n^{\ast }|{|}_{\infty }$$

Proof

$$||f-p_n|{|}_{\infty }\le ||f-p_n^{\ast }|{|}_{\infty }+||p_n^{\ast }-p_n|{|}_{\infty }$$

As

$$p_n^{\ast }-p_n={\mathcal{L}}_n(p_n^{\ast }-p_n)={\mathcal{L}}_n(p_n^{\ast }-f)$$

since $p_n^{\ast }-p_n\in {\Pi }_n$ thus $L_n(p_n^{\ast }-p_n)=p_n^{\ast }-p_n$
Hence

$$||p_n^{\ast }-p_n|{|}_{\infty }\le ||{\mathcal{L}}_n(p_n^{\ast }-f)||\le {\Lambda }_n||f-p_n^{\ast }|{|}_{\infty }$$

Hence result follows