Let g1(t), …, gn(t) be prescribed, linearly independent...

Let $g_1(t), \ldots, g_n(t)$ be prescribed, linearly independent functions. Explain how to best approximate a function $f(t)$ by a linear combination $c_1 g_1(t)+\cdots+c_n g_n(t)$ when the least squares error is measured in a weighted $\mathrm{L}^2$ norm $\|f\|_w^2=\int_a^b f(t)^2 w(t) d t$ with weight function $w(t)>0$.

Key Concepts

Orthogonal Projection

Orthogonal projection in the context of function approximation is the process of decomposing the target function into two parts: one that lies within the subspace spanned by the chosen basis functions and one that is orthogonal to that subspace. Finding the best approximation involves projecting the target function onto the subspace such that the error is orthogonal to every basis function with respect to the weighted inner product.

Linear Independence

Linear independence refers to a set of functions (or vectors) where no function can be written as a linear combination of the others. This property ensures that each function in the set adds a unique dimension to the space of approximants, which is essential for obtaining a unique solution when performing least squares approximation.

Normal Equations

The normal equations are derived from the condition that the residual (the difference between the function and its approximation) must be orthogonal to the span of the basis functions. They provide a set of linear equations in the coefficients of the approximation, typically involving the Gram matrix, and solving them yields the least squares best approximation in the weighted L² norm.

Weighted L² Norm

The weighted L² norm is a measure of a function's size that incorporates a weight function into the standard L² norm. It defines the norm of a function f(t) by integrating the square of the function times a positive weight, w(t), over a specified interval. This allows different parts of the domain to be emphasized or de-emphasized when measuring the error or distance between functions.

Least Squares Approximation

Least squares approximation is a method for finding the best fit of a model function to a set of data or another function in terms of minimizing the error. In the context of function approximation, it involves choosing coefficients for a linear combination of basis functions such that the weighted L² norm of the difference (error) between the target function and the approximant is minimized.

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