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(c) Prove that any polynomial of degree n that is orthogonal to $1,x,x^{2},...,x^{n-1}$ is a constant multiple of $L_{n}.$ $$L_{n}=\frac{d^{n}}{dx^{n}}(x^{2}-1)^{n}$$ Two elements $X$ are $Y$orthogonal if $(X,Y)=0$

for $(X,Y)$ is defined by: $$(X,Y)=z_{1}\bar{w}_{1}+\ ...+z_{d}\bar{w}_{d}.$$ where $X$ and $Y$ are two vectors in $\Bbb {C}^{d}.$

Show that $L_{n}$ is orthogonal to $x^{m}$ whenever $m<n.$ Hence {$L_{n}$}$_{n=0}^{\infty}$

(d) Let $\mathcal{L}_{n}=L_{n}/\Vert{L_{n}}\Vert,$ which are the normalized Legendre polynomials. Prove{$\mathcal{L}_{n}$} is the family obtained by applying the "Gram-Schmidt process" to {$1,x,...,x^{n},...$}, and conclude that every Riemann integrable function $f$ on $[-1,1]$ has Legendre expansion $$\sum_{n=0}^{\infty}\langle f,\mathcal{L}_{n}\rangle\mathcal{L}_{n}$$ which converges to $f$ in the mean-square sense.

Here are some properties of $L_{n}$

If $f$ is indefinitely differentiable on $[-1,1],$ then $$\int_{-1}^{1}{L}_{n}(x)f(x)dx=(-1)^{n}\int_{-1}^{1}(x^{2}-1)^{n}f^{n}(x)dx,$$

$$\Vert\mathcal{L}_{n}\Vert^{2}=\int_{-1}^{1}\vert L_{n}(x)\vert^{2}dx=\frac {(n!)^{2}2^{2n+1}}{2n+1}.$$ I don't know how to use the two properties and Gram_Schmidt process to solve (c), (d). I don't know how to represent the orthogonality of a polynomial of degree n to $1,x,x^{2},...,x^{n-1}.$

John He
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1 Answers1

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For (c), you don't need the Gram-Schmidt process. Any polynomial of degree $n$, let's call it $p(x)$, can be written as a sum $p(x) = \sum_{k=0}^n a_k L_k(x)$. (To see why this is true, start with the highest-degree term $x^n$ and match the coefficient between $p(x)$ and $L_n(x)$, and then do the same with $L_{n-1}(x)$, $L_{n-2}(x)$, etc.)

From the orthogonality between $p(x)$ and 1, we can conclude that $a_0 = 0$. Then from the orthogonality between $p(x)$ and $x$, we can conclude that $a_1 = 0$. This continues until you have only $a_k$ left.

For (d), you need to know Gram-Schmidt process is. You start with $f_0(x) = 1$, then move to a polynomial $f_1(x)$ of degree 1, i.e. something like $ax + b$, which is orthogonal to $f_0$, and has norm 1. Then you move to a polynomial $f_2$ of degree 2, i.e. something like $ax^2 + bx + c$, which is orthogonal to both $f_0$ and $f_1$, and has norm 1. Et cetera.

(d) is thus a natural consequence of (c). There is in fact some ambiguity because any function in the basis can be multiplied by $e^{i\theta}$ and it will still be orthonormal. We simply adopt some arbitrary convention to decide which one we pick.

The final point in (d) can be proved in exactly the same way as in Lemma 1.2 in Section 3.1.2. We may also use the result from Exercise 16 from Chapter 2.

Petra Axolotl
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