In Which the Riesz Representation Theorem is Very Mysterious
Define \[\langle p, q \rangle = \int_{-1}^1 p(x)q(x)\,dx\] to be an inner product on \( \mathscr{P}_n(\mathbb{R}) \), the space of polynomials of degree at most \(n\) with real coefficients. Note that \(\varphi(p)=p(0)\) is a linear functional on \( \mathscr{P}_n(\mathbb{R}) \). Therefore, by the Riesz representation theorem, there must exist a unique polynomial \(q_n(x)\) such that \[\int_{-1}^1 p(x)q(x)\,dx = p(0)\] for all \(p\) with degree at most \(n\).
To compute these polynomials, define the matrix \(G\) via \[G_{i,j} = \int_{-1}^1 x^i x^j\,dx = \frac{1-(-1)^{i+j+1}}{i+j+1}.\] (We zero-index.) Then, if \(a\) is the vector giving the coefficients of \(p\) and \(b\) the coefficients of \(q\), we know that \[b’Ga = a_0 = (1, 0, \dots, 0) a,\] or, rearranging, \[(b’G-(1, 0, \dots, 0))a = 0.\] Since this must be true for all \(a\), \[b’G-(1, 0, \dots, 0) = 0,\] which lets us directly compute \[b = G^{-1}(1, 0, \dots, 0)’.\]
Seeing as these polynomials are finite-degree analogues of the Dirac delta – although, experimentally, they do not appear to converge to it even pointwise – it is interesting to ask how well they approximate the infinite peak. Specifically, we want to know their value at zero. This can be calculated as \[(1, 0, \dots, 0)G^{-1}(1, 0, \dots, 0)’.\]
Experimentally, we find \[q_n(0) \sim \frac{n}{\pi}.\]