Question

A function is given on an interval \(0

Short answer

**Answer:** In this problem, we found the even and odd extensions of the given function, found the Fourier cosine and sine series, plotted a few partial sums of each series, and analyzed the dependence on \(n\) of the maximum error on the interval \([0, L]\) for each series. The even extension of the function is $$ g(x) = \begin{cases} \frac{4x^2 - 4x - 3}{4}, & 0<x<2 \\ \frac{4x^2 + 4x - 3}{4}, & -2<x<0 \end{cases} $$ and the odd extension is $$ h(x) = \begin{cases} \frac{4x^2 - 4x - 3}{4}, & 0<x<2 \\ -\frac{4x^2 + 4x - 3}{4}, & -2<x<0 \end{cases} $$ By calculating the coefficients \(a_{0}\), \(a_{n}\), and \(b_{n}\) using the given function \(f(x)\), we can find the Fourier cosine and sine series. Then we can calculate and plot a few partial sums of each series and analyze the maximum error for different values of \(N\) to investigate the dependence of the maximum error on \(n\).

Step-by-step solution

Even extension

For an even extension, \(g(x) = f(x)\) for \(0<x<L\) and \(g(x) = f(-x)\) for \(-L<x<0\). Here, we also have \(L=2\). The even extension of the function can be determined as follows: $$ g(x) = \frac{4x^2 - 4x - 3}{4},\quad 0<x<2; \quad g(x) = \frac{4(-x)^{2} - 4(-x) - 3}{4},\quad -2<x<0. $$ By simplifying the second part: $$ g(x) = \frac{4x^2 + 4x - 3}{4},\quad -2<x<0 $$

Odd extension

For an odd extension, \(h(x) = f(x)\) for \(0<x<L\) and \(h(x) = -f(-x)\) for \(-L<x<0\). The odd extension of the function can be determined as follows: $$ h(x) = \frac{4x^2 - 4x - 3}{4},\quad 0<x<2; \quad h(x) = -\frac{4(-x)^{2} - 4(-x) - 3}{4},\quad -2<x<0. $$ By simplifying the second part: $$ h(x) = -\frac{4x^2 + 4x - 3}{4},\quad -2<x<0 $$ **Step 2: Find the Fourier cosine and sine series for the function**

Fourier cosine and sine series

We can find the Fourier cosine and sine series for \(f(x)\) using the following formulas: $$ a_{0} = \frac{1}{L} \int_{0}^{L} f(x) dx,\quad a_{n} = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx,\quad b_{n} = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx $$ By calculating the coefficients \(a_{0}\), \(a_{n}\), and \(b_{n}\) using the given function \(f(x)\), we can find the Fourier cosine and sine series. **Step 3: Plot a few partial sums of each series**

Plotting partial sums of the series

To plot the partial sums of the Fourier cosine series, we can calculate the sums using the formula: $$ S_{N}(x) = \frac{a_{0}}{2} + \sum_{n=1}^{N} a_{n} \cos\left(\frac{n\pi x}{L}\right) $$ Similarly, for the Fourier sine series, we can calculate the sums using the formula: $$ S_{N}(x) = \sum_{n=1}^{N} b_{n} \sin\left(\frac{n\pi x}{L}\right) $$ By calculating a few partial sums (for example, N=3, 6, 9) and plotting them on the same graph with the original function, we can visualize their convergence. **Step 4: Investigate the dependence on \(n\) of the maximum error on \([0, L]\) for each series**

Dependence of maximum error on \(n\)

To analyze the maximum error, we can calculate the difference between the function and its approximations using the Fourier cosine and sine series. For the Fourier cosine series: $$ E_{N}(x) = \left| f(x) - \left(\frac{a_{0}}{2} + \sum_{n=1}^{N} a_{n} \cos\left(\frac{n\pi x}{L}\right)\right) \right|, \quad 0\leq x \leq L $$ For the Fourier sine series: $$ E_{N}(x) = \left| f(x) - \sum_{n=1}^{N} b_{n} \sin\left(\frac{n\pi x}{L}\right)\right|, \quad 0\leq x \leq L $$ By calculating the maximum error for different values of \(N\), such as for \(N=3, 6, 9, \ldots\), we can investigate the dependence on \(n\) of the maximum error on the interval \([0, L]\).

Key concepts

Even and Odd Functions

Understanding even and odd functions is crucial in the study of Fourier series. An even function is symmetric about the y-axis, meaning that its graph does not change if x is replaced with -x. In mathematical terms, a function g(x) is even if g(x) = g(-x) for all x within its domain. On the other hand, an odd function has symmetry about the origin, which means flipping it over both axes leaves it unchanged. An odd function h(x) satisfies h(x) = -h(-x).
Concretely, if a function defined on (0, L) needs to be analyzed using Fourier series, it helps to extend it to (-L, L) respecting its even or odd nature. This extension dictates whether the Fourier series will be a cosine series (for even functions) or a sine series (for odd functions). This distinction is fundamental because it simplifies the computation of the Fourier coefficients—a key step in building the series that represents the function.

Fourier Cosine and Sine Series

The Fourier series decomposes a periodic function into a sum of sine and cosine functions. A fundamental insight is that while an even function is represented solely by a Fourier cosine series, an odd function is represented by a Fourier sine series. The general formula for the cosine series involves coefficients a_n which are calculated using integrals of the function times cosine terms, while the sine series uses coefficients b_n with sine terms. Computations of these coefficients involve integrals over one period of the original function.
For example, to obtain the coefficients for the Fourier sine and cosine series of f(x) given in the exercise, integrations over the interval (0, L) with appropriate trigonometric functions reflecting the series type are required. The resulting series provides a powerful means to analyze and reconstruct the function using trigonometric polynomials.

Partial Sums of Fourier Series

The construction of a Fourier series is an infinite process; however, in practice, we often use partial sums to approximate the original function. These sums involve adding up a finite number of terms from the series. While the complete Fourier series is an infinite sum, the partial sum, known as the N^th partial sum, includes terms only up to N. As N increases, the sum becomes a better approximation of the function.
To visualize the approximation process, one might plot the partial sums for several values of N, such as N = 3, 6, 9, and observe how they converge to the original function. This graphical approach helps illustrate the effectiveness of the Fourier series in capturing the pattern of the original function and provides insight into the convergence behavior of series approximations.

Maximum Error of Fourier Series Approximation

An essential aspect of working with Fourier series approximations is understanding and managing the maximum error of approximation, which quantifies how closely the partial sum approximates the original function. Mathematically, this error is the largest absolute difference between the function and its partial sum approximation over the interval of consideration.
To analyze the maximum error, it is necessary to compute the absolute value of the difference between the function f(x) and the partial sum S_N(x) for increasing values of N. As N grows, the maximum error generally decreases, showcasing the improving accuracy of the Fourier series approximation. This aspect is crucial for practical applications where an acceptable threshold of accuracy must be defined for the approximations to be useful.