Projection onto the constant polynomials requires the residual to be orthogonal to

1

. For the original inner product,

0 = \int_{- 1}^{1} (x^{2} - p_{0}) d x = \frac{2}{3} - 2 p_{0}, p_{0} = \frac{1}{3} .

(1)

For the weighted inner product the corresponding normal equation is

0 = \int_{- 1}^{1} x^{2 n} (x^{2} - \overset{p}{ˉ}_{0} (n)) d x = \frac{2}{2 n + 3} - \overset{p}{ˉ}_{0} (n) \frac{2}{2 n + 1} .

(2)

Hence

\overset{p}{ˉ}_{0} (n) = \frac{2 n + 1}{2 n + 3} = 1 - \frac{2}{2 n + 3} ⟶ 1.

(3)

It increases monotonically towards one. Although the weight vanishes at zero, it is positive elsewhere, and a nonzero continuous function is nonzero on an interval of positive weight. Thus the weighted expression really is an inner product and the constant minimizer is unique.

The normalized weight

(2 n + 1) x^{2 n} /2

has total mass one and assigns mass

r^{2 n + 1}

[- r, r]

for each

0 < r < 1

. Its mass therefore concentrates near

- 1

and

1

, where

x^{2} = 1

. This explains why the weighted best constant tends to one rather than to the unweighted spatial average. It is endpoint concentration in weighted constant approximation.