A trigonometric polynomial of degree n is a function of the...

A trigonometric polynomial of degree $n$ is a function of the form $$ p(t)=a_0+a_1 \cos t+b_1 \sin t+a_2 \cos 2 t+b_2 \sin 2 t+\cdots+a_n \cos n t+b_n \sin n t, $$ where $a_0, a_1, b_1, \ldots, a_n, b_n$ are the coefficients. Find the trigonometric polynomial of degree $n$ that is the least squares approximation to the function $f(t)=1 /\left(1+t^2\right)$ on the interval $[-\pi, \pi]$ based on the $k$ equally spaced data points $t_j=-\pi+\frac{2 \pi j}{k}, j=0, \ldots, k-1$, (omitting the right-hand endpoint), when (a) $n=1, k=4$, (b) $n=2, k=8$, (c) $n=2, k=16,(d) n=3, k=16$. Compare the graphs of the trigonometric approximant and the function, and discuss. (e) Why do we not include the right-hand endpoint $t_k=\pi$ ?

A trigonometric polynomial of degree $n$ is a function of the form
$$
p(t)=a_0+a_1 \cos t+b_1 \sin t+a_2 \cos 2 t+b_2 \sin 2 t+\cdots+a_n \cos n t+b_n \sin n t,
$$
where $a_0, a_1, b_1, \ldots, a_n, b_n$ are the coefficients. Find the trigonometric polynomial of degree $n$ that is the least squares approximation to the function $f(t)=1 /\left(1+t^2\right)$ on the interval $[-\pi, \pi]$ based on the $k$ equally spaced data points $t_j=-\pi+\frac{2 \pi j}{k}, j=0, \ldots, k-1$, (omitting the right-hand endpoint), when (a) $n=1, k=4$, (b) $n=2, k=8$, (c) $n=2, k=16,(d) n=3, k=16$. Compare the graphs of the trigonometric approximant and the function, and discuss. (e) Why do we not include the right-hand endpoint $t_k=\pi$ ?

Key Concepts

Exclusion of the Right-Hand Endpoint

The right-hand endpoint in a periodic interval is typically omitted because it coincides with the left-hand endpoint, leading to redundancy. Including both endpoints would result in double-counting the same value, potentially skewing the approximation and violating the assumption of distinct, uniformly distributed sample points over one complete period.

Equally Spaced Data Points

Utilizing equally spaced data points ensures a uniform sampling over the interval of interest, which is especially important when approximating periodic functions. Uniform spacing simplifies computations and facilitates the use of discrete Fourier techniques, allowing for accurate recovery and representation of the function’s behavior over its period.

Discrete Fourier Analysis

Discrete Fourier analysis involves decomposing functions into sums of sine and cosine terms based on sampled data points. This approach is fundamental when fitting trigonometric polynomials to data, as it leverages the orthogonality properties of trigonometric functions on uniformly sampled intervals to simplify the computation of the best-fit coefficients.

Trigonometric Polynomial

A trigonometric polynomial is a finite linear combination of sine and cosine functions, typically used to approximate periodic functions. Its degree indicates the highest frequency component present in the approximation, similar to the partial sums of a Fourier series, and it offers a flexible tool for representing periodic phenomena by adjusting the number of harmonics included.

Least Squares Approximation

Least squares approximation is a method to determine the coefficients of the approximating function by minimizing the sum of the squared differences (errors) between the function's values and the data points. This technique ensures that the approximation is, in a specific sense, optimally close to the given data, making it a highly effective tool in function approximation and data fitting.

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