I am trying to approximate the function $(1-x)^\sqrt{2}$ using Legendre polynomials on the interval $[0,1]$. I have been using $P_n(1-2x)$ as my polynomials as I want all the polynomials to satisfy $P_n(0)=1$. However, when I use the formula $$ \lambda_n = \int_0^1 P_n(1-2x)(1-x)^\sqrt{2}\,\mathrm{d}x $$ to get the coefficients for my series $$ (1-x)^\sqrt{2} = \sum_n \lambda_n P_n(1-2x) $$ I have trouble getting the series to converge to the function. I calculated the first twenty coefficients, which are:
0.414214, 0.171573, 0.0160998, -0.0017419, 0.00043065, -0.000150194,
0.0000640064, -0.0000311783, 0.0000167229, -9.64878*10^-6,
5.89596*10^-6, -3.77369*10^-6, 2.50945*10^-6, -1.72295*10^-6,
1.21651*10^-6, -8.79203*10^-7, 6.48666*10^-7, -4.85355*10^-7,
5.9736*10^-7, -2.88178*10^-7, 2.26099*10^-7
However, when plotted against the original function they have incredibly poor convergence. Is this due to to a mistake in my methodology or is $N=20$ insufficient for convergence of this function? I am a little confused as the coefficients rapidly become small and so will need a very large number to converge to the condition that $f(0)=1$.
The norm of Legendre polynomials (for $n > 0$) is not $1$. In this case $$\int_0^1P_n(1-2x)^2 \mathrm dx = \frac1{2n+1}.$$ Therefore your expansion should be $$\sum_{n=0}^{\infty}(2n+1)\lambda_n P_n(1-2x).$$