Question

A function is given on an interval \(0

Short answer

Question: Sketch the even and odd extension functions of the given function \(f(x) = x^3 - 5x^2 + 5x + 1\) over three periods and find the Fourier cosine and sine series for the given function. Plot a few partial sums of each series and investigate the dependence on \(n\) of the maximum error on \([0, L]\) for each series.

Step-by-step solution

1. Sketch the even and odd extension functions

First, we need to extend the given function to be even and odd functions. The even extension function \(g(x)\) can be found by reflecting \(f(x)\) about the y-axis for \(-L<x<0\). Similarly, the odd extension function \(h(x)\) can be found by reflecting \(f(x)\) about the origin for \(-L<x<0\). To find the period of these functions, we note that \(L=3\); hence, the period of both \(g(x)\) and \(h(x)\) is \(2L = 6\). Then we can sketch the graphs of the even and odd extension functions over three periods, i.e., for \(-18 < x < 18\).

2. Find the Fourier cosine and sine series

To find the Fourier cosine series for the given function, we need to find the coefficients \(a_0\), \(a_n\) using the formula: $$ a_0 = \frac{2}{L}\int_{0}^{L} f(x) dx, \quad a_n = \frac{2}{L}\int_{0}^{L} f(x) \cos(\frac{n\pi x}{L}) dx $$ And the Fourier sine series can be found by using the coefficients \(b_n\): $$ b_n = \frac{2}{L}\int_{0}^{L} f(x) \sin(\frac{n\pi x}{L}) dx $$

3. Plot a few partial sums of each series

By considering a few terms in the Fourier series expansion, we can find a few partial sums of Fourier cosine and Fourier sine series. Plot these partial sums to visualize the approximations of \(f(x)\) using the Fourier series.

4. Investigate the dependence on \(n\) of the maximum error on \([0, L]\)

The maximum error between the original function \(f(x)\) and its Fourier series approximation depends on the number of terms (\(n\)) included in the series. To investigate this dependence, we can calculate the maximum absolute difference between \(f(x)\) and its approximations using the Fourier cosine and sine series for different values of n on the interval \([0, L]\). This way, we can study how the maximum error varies as we increase the number of terms in the series.

Key concepts

Even and Odd Function Extensions

In the world of Fourier analysis, extending a function to be even or odd can significantly simplify the process of finding its Fourier series. An even function, noted \(g(x)\), is symmetric around the y-axis, meaning that \(g(-x) = g(x)\). So, when you want to create an even extension of a function defined on \(0 < x < L\), you 'mirror' the function across the y-axis for the segment \( -L < x < 0\).

On the other hand, an odd function, denoted \(h(x)\), has rotational symmetry about the origin, satisfying \(h(-x) = -h(x)\). To extend a function oddly, you reflect it across the origin to the interval \( -L < x < 0\), resulting in a function that continues to display its odd characteristics.

The importance of these extensions lies in their influence on the terms of the Fourier series; even functions will solely require cosine terms, whereas odd functions will rely purely on sine terms. This characteristic ultimately streamlines the calculation process, making the Fourier series less complex to compute.

Fourier Cosine and Sine Series

When we delve into the realm of the Fourier series, we come across two special types of series expansions: the Fourier cosine series and the Fourier sine series. These are particularly useful when working with functions that have been extended to be even or odd, respectively.

The Fourier cosine series is used to approximate even functions and consists only of cosine terms (thus the name). The coefficients of these terms, \(a_0\) and \(a_n\), capture the function's 'shape' and are calculated with specific integrals over the function's domain. Similarly, the Fourier sine series approximate odd functions with series that contain only sine terms, with coefficients \(b_n\) capturing the odd symmetry of the function.

These series play a crucial role because they allow us to decompose complex periodic signals into simpler, sinusoidal components. This decomposition is not just mathematically elegant but also underpins many practical applications, from signal processing to solving differential equations.

Partial Sums of Fourier Series

In a practical sense, the Fourier series is an infinite sum of sine and cosine terms. However, when we calculate these series, we generally truncate them to a finite number of terms, creating what are known as the partial sums. These partial sums provide an approximation of the original function.

By plotting these sums, we can visualize how the approximation improves as we add more terms. Initially, with only a few terms, the approximation may appear rough and quite distant from the original function. As we include more terms, the approximation becomes smoother and more accurate, capturing more of the finer details in the function's behavior.

These partial sums are not just theoretical constructs; they are crucial in computational applications where infinite series can not be processed. By investigating these sums, students can gain insights into the convergence of the Fourier series and its effectiveness in representing the original function.

Fourier Series Approximation Error

A central question in the study of Fourier series is how well the series approximates a given function. The answer to this lies in the approximation error, which is defined as the difference between the original function and its Fourier series representation. A key aspect of this error is its dependence on the number of terms, \(n\), in the series.

As \(n\) increases, the approximation typically becomes better, and the error decreases. This relationship is particularly important as it can inform us how many terms are needed to achieve a desired level of accuracy. For instance, if a high level of precision is required for a particular application, we need to consider including a larger number of terms.

To comprehend the behavior of the approximation error, one can calculate the maximum absolute difference between the function and its Fourier series for different values of \(n\). This practice fosters a deeper understanding of the rate of convergence and the factors that can affect the efficiency of approximation in varying contexts.