(a) Use the polynomials and weighted norm from Exercise...

(a) Use the polynomials and weighted norm from Exercise 4.5.12 to find the quadratic least squares approximation to $f(t)=1 / t$. In what sense is your quadratic approximation "best"? (b) Now find the best approximating cubic polynomial. (c) Compare the graphs of the quadratic and cubic approximants with the original function and discuss what you observe.

Key Concepts

Least Squares Approximation

Least squares approximation is a method used to determine the best fit of a function by minimizing the error, typically measured as the integral of the square of the difference between the target function and its approximate representation. In this context, it involves finding the polynomial coefficients that minimize this squared error, which ensures that the error is evenly distributed in a mean square sense.

Weighted Norm

A weighted norm is a specific way of measuring the size or error of a function where different parts of the domain can be given different levels of importance through a weight function. This modified norm is integral to the least squares process where the error is not simply the straight L2 norm but is weighted to better capture the function's behavior on various intervals or domains.

Orthogonal Projection

Orthogonal projection in a function space is the process of projecting a function onto a subspace (such as the span of a set of polynomials) so that the error between the function and its projection is orthogonal to that subspace. This concept is fundamental in least squares approximation as it provides a means to identify the coefficients of the approximating polynomial by solving for when the residual is orthogonal to the basis functions.

Polynomial Approximation

Polynomial approximation involves representing a given function by a polynomial of a certain degree such that the approximation is as close as possible to the function in the chosen norm. Both quadratic (degree 2) and cubic (degree 3) approximations are used to capture the behavior of complex functions, with higher degree polynomials generally providing a better fit over a range of values, albeit sometimes at the risk of overfitting in regions where the function behaves irregularly.

Graphical Analysis

Graphical analysis involves plotting the original function along with its polynomial approximations to visually assess their accuracy and behavior. Through this comparison, one can observe how well the approximants trace the original function's trend. Differences between the quadratic and cubic fits highlight the trade-off between simplicity and accuracy, as the cubic approximation may capture nuances that the quadratic does not while potentially introducing additional oscillatory behavior.

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