= Legendre polynomial evaluation representer
{c}
{title2=$q_n(t)=\sum_{j=0}^n\overline{p_j(0)}p_j(t)$}
For <orthonormal> <Legendre polynomials> on $[-1,1]$, the <polynomial> $q_n$ represents evaluation at zero on polynomials of degree at most $n$: $\langle r,q_n\rangle=r(0)$. This follows by expanding $r$ in the <orthonormal basis>, or from the <Riesz representation theorem>. With $M=\lfloor n/2\rfloor$ and $c_m=\binom{2m}{m}/4^m$, orthonormality gives
$$
\|q_n\|_2^2=\frac12\sum_{m=0}^M(4m+1)c_m^2=\frac{(2M+1)^2c_M^2}{2}.
$$
The last identity telescopes using $c_m/c_{m-1}=(2m-1)/(2m)$. The norms are unbounded: $c_m\geq1/(2m+1)$ makes the sum dominate a divergent <harmonic series>. Thus the evaluation identities hold eventually for every fixed <polynomial> but cannot extend to convergence for every $L^2$ vector, by the <Uniform boundedness principle>. Point evaluation is also not defined on $L^2$ equivalence classes.
Back to article page
