Prove the least squares approximation property of Legendre...

Prove the least squares approximation property of Legendre polynomials [see $(9.5)$ and (9.6)] as follows. Let $f(x)$ be the given function to be approximated. Let the functions $p($ r) be the normalized Legendre polynomials, that is, $$ p_{l}(x)=\sqrt{\frac{2 l+1}{2}} P_{l}(x) \quad \text { so that } \int_{-1}^{1}\left[p_{l}(x)\right]^{2} d x=1 $$ Show that the I.egendre series for $f(x)$ as far as the $p_{2}(x)$ term is $$ f(x)=\varsigma_{0} p_{0}(x)+\iota_{1} p_{1}(x)+i_{2} p_{2}(x) \quad \text { with } \quad c_{l}=\int_{-1}^{1} f(x) p_{1}(x) d x $$. Write the quadratic polynomial satisfying the least squares condition as $b_{0} P_{0}(x)+b_{1} p_{1}(x)$ $+b_{2} p_{2}(x)$ (by Problem $5.14$ any quadratic polynomial can be written in this form). 'The problem is to find $b_{0}, b_{1}, b_{2}$ so that $$ I=\int_{-1}^{1}\left[f(x)-\left(b_{0} p_{0}(x)+b_{1} p_{1}(x)+b_{2} \rho_{2}(x)\right)\right]^{2} d x $$ is a minimum. Square the bracket and write $I$ as a sum of integrals of the individual terms. Show that some of the integrals are zero by orthogonality, some are 1 because the $\rho_{l}$ 's are normalized, and others are equal to the cofficients $c_{1} .$ Add and subtract $t_{0}^{2}+c_{1}^{2}+c_{2}^{2}$ and show that $$ I=\int_{-1}^{1}\left[f^{2}(x)+\left(b_{0}-c_{0}\right)^{2}+\left(b_{1}-c_{1}\right)^{2}+\left(b_{2}-c_{2}\right)^{2}-\iota_{0}^{2}-c_{1}^{2}-c_{2}^{2}\right] d x $$ Now determine the values of the $b$ 's to make $I$ as small as possible. (Hint: The smallest value the square of a real number can have is zero.) Generalize the proof to polynomials of degree $n$.

Prove the least squares approximation property of Legendre polynomials [see $(9.5)$ and (9.6)] as follows. Let $f(x)$ be the given function to be approximated. Let the functions $p($ r) be the normalized Legendre polynomials, that is,
$$
p_{l}(x)=\sqrt{\frac{2 l+1}{2}} P_{l}(x) \quad \text { so that } \int_{-1}^{1}\left[p_{l}(x)\right]^{2} d x=1
$$
Show that the I.egendre series for $f(x)$ as far as the $p_{2}(x)$ term is
$$
f(x)=\varsigma_{0} p_{0}(x)+\iota_{1} p_{1}(x)+i_{2} p_{2}(x) \quad \text { with } \quad c_{l}=\int_{-1}^{1} f(x) p_{1}(x) d x
$$. Write the quadratic polynomial satisfying the least squares condition as $b_{0} P_{0}(x)+b_{1} p_{1}(x)$ $+b_{2} p_{2}(x)$ (by Problem $5.14$ any quadratic polynomial can be written in this form). 'The problem is to find $b_{0}, b_{1}, b_{2}$ so that
$$
I=\int_{-1}^{1}\left[f(x)-\left(b_{0} p_{0}(x)+b_{1} p_{1}(x)+b_{2} \rho_{2}(x)\right)\right]^{2} d x
$$
is a minimum. Square the bracket and write $I$ as a sum of integrals of the individual terms. Show that some of the integrals are zero by orthogonality, some are 1 because the $\rho_{l}$ 's are normalized, and others are equal to the cofficients $c_{1} .$ Add and subtract $t_{0}^{2}+c_{1}^{2}+c_{2}^{2}$ and show that
$$
I=\int_{-1}^{1}\left[f^{2}(x)+\left(b_{0}-c_{0}\right)^{2}+\left(b_{1}-c_{1}\right)^{2}+\left(b_{2}-c_{2}\right)^{2}-\iota_{0}^{2}-c_{1}^{2}-c_{2}^{2}\right] d x
$$
Now determine the values of the $b$ 's to make $I$ as small as possible. (Hint: The smallest value the square of a real number can have is zero.) Generalize the proof to polynomials of degree $n$.

Key Concepts

Projection Coefficients and Error Minimization

The projection of a function onto a subspace spanned by an orthonormal set involves calculating the inner products (or coefficients) of the function with each basis function. The significance of orthogonality is that it causes cross-terms in the error expression to vanish when the error is expanded. Consequently, the squared error separates into individual terms, and it is minimized when each approximation coefficient matches the corresponding projection coefficient.

Least Squares Approximation

Least squares approximation is a method for approximating a given function by another function (often a polynomial) from a certain subspace such that the integral of the square of their difference (the error) is minimized. When using an orthonormal basis, the minimization process leads directly to the coefficients being the inner products of the function with the corresponding basis functions, thereby yielding the best approximation in the L2-norm.

Generalization to Higher Degree Approximation

The approach demonstrated with quadratic polynomials using Legendre polynomials can be generalized to polynomials of any degree n. The fundamental idea is to represent the function as a sum of projections onto an orthonormal basis for the polynomial space, ensuring that the approximation minimizes the L2-norm of the error over the interval. This general technique underpins many approximation methods in numerical analysis and applied mathematics.

Legendre Polynomials

Legendre polynomials are a sequence of orthogonal polynomials defined on the interval [-1, 1] with respect to the weight function 1. Their orthogonality property makes them exceptionally useful for expanding functions in series, and their normalization leads to the creation of an orthonormal basis, which greatly simplifies the computation of the projection coefficients used in function approximation.

Orthonormal Bases in Function Spaces

An orthonormal basis in a function space, such as the space of square integrable functions on [-1, 1], is a set of functions that are both orthogonal and of unit norm. This property ensures that the inner product of any two different basis functions is zero and that the inner product of a basis function with itself is one. This feature is crucial in least squares approximation because it allows for the unique decomposition of any function into components along the basis functions.

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