Question

Let \(f \in E\) and $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ denote its Fourier series. Define the two functions $$ g(x)=\frac{f(x)+f(-x)}{2}, \quad h(x)=\frac{f(x)-f(-x)}{2} . $$ Find the Fourier series of \(g\) and of \(h\).

Short answer

The Fourier series of \( g(x) \) is \( \frac{a_{0}}{2} + \sum_{n=1}^{\infty} a_{n} \cos(nx) \) and of \( h(x) \) is \( \sum_{n=1}^{\infty} b_{n} \sin(nx) \).

Step-by-step solution

Understand the Functions g(x) and h(x)

The function \( g(x) = \frac{f(x) + f(-x)}{2} \) is the even part of the function \( f(x) \). It averages the values at \( x \) and \( -x \). Conversely, the function \( h(x) = \frac{f(x) - f(-x)}{2} \) is the odd part of \( f(x) \), highlighting the asymmetry of \( f(x) \) about the y-axis.

Decompose f(x) into Even and Odd Parts

The even function \( g(x) \) will only include cosine terms from the Fourier series of \( f(x) \) because cosine is an even function. The odd function \( h(x) \) will only include sine terms because sine is odd.

Find Fourier Series of g(x)

For \( g(x) = \frac{f(x) + f(-x)}{2} \), plug in the Fourier series definition: \[ g(x) = \frac{1}{2} \left( \frac{a_{0}}{2} + \sum_{n=1}^{\infty} \left[ a_{n} \cos(nx) + b_{n} \sin(nx) \right] + \frac{a_{0}}{2} + \sum_{n=1}^{\infty} \left[ a_{n} \cos(nx) - b_{n} \sin(nx) \right] \right) \] Simplifying, the sine terms cancel out (since they are odd): \[ g(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty} a_{n} \cos(nx) \]

Find Fourier Series of h(x)

For \( h(x) = \frac{f(x) - f(-x)}{2} \), substitute the Fourier series: \[ h(x) = \frac{1}{2} \left( \frac{a_{0}}{2} + \sum_{n=1}^{\infty} \left[ a_{n} \cos(nx) + b_{n} \sin(nx) \right] - \frac{a_{0}}{2} - \sum_{n=1}^{\infty} \left[ a_{n} \cos(nx) - b_{n} \sin(nx) \right] \right) \] Simplifying, the cosine terms cancel out (since they are even): \[ h(x) = \sum_{n=1}^{\infty} b_{n} \sin(nx) \]

Write Final Fourier Series for g(x) and h(x)

From the simplifications, the Fourier series for \( g(x) \) is: \( g(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty} a_{n} \cos(nx) \). The Fourier series for \( h(x) \) is: \( h(x) = \sum_{n=1}^{\infty} b_{n} \sin(nx) \).

Key concepts

Even and Odd Functions

In mathematics, even and odd functions play crucial roles, especially in Fourier analysis. A function is called **even** if it satisfies the condition \( f(x) = f(-x) \) for all \( x \). This means the function is symmetric about the y-axis. Examples of even functions include \( \cos(x) \) and \( x^2 \). On the other hand, a function is labeled **odd** if it meets the criterion \( f(-x) = -f(x) \) for every \( x \). These functions have rotational symmetry about the origin. Examples of odd functions include \( \sin(x) \) and \( x^3 \).

When decomposing a function \( f(x) \) into its even part, \( g(x) = \frac{f(x) + f(-x)}{2} \), and its odd part \( h(x) = \frac{f(x) - f(-x)}{2} \), you can fully capture its symmetry properties. This decomposition is essential for simplifying complex analyses involving Fourier series.

Understanding even and odd functions helps simplify the calculation of Fourier coefficients, as each type of function contributes only its kind of terms (either cosine or sine) to the overall series.

Cosine and Sine Functions

The cosine and sine functions are foundational in trigonometry, deeply intertwined with the Fourier series. Cosine is an **even function**, defined by the equation \( \cos(-x) = \cos(x) \). This symmetry about the y-axis makes cosine functions indispensable for representing even parts of periodic functions in a Fourier series.

On the opposite side, we have sine, an **odd function**. Its defining property is \( \sin(-x) = -\sin(x) \), demonstrating symmetry about the origin. In Fourier series, the sine components capture the odd characteristics of functions. Together, these two functions form the building blocks for modeling complex, periodic behaviors.

In practical applications, these functions help to decompose a complex periodic function \( f(x) \) into simpler sinusoidal components, making them easy to analyze and interpret. This decomposition allows systems to be studied more conveniently in terms of their harmonic oscillations.

Function Decomposition

Function decomposition in the context of Fourier series is an incredibly powerful tool that simplifies the study of complex periodic signals. Essentially, decomposition involves breaking down a complicated function into simpler, more manageable pieces.

These pieces, often the even part \( g(x) \) and the odd part \( h(x) \), correspond to distinct types of symmetries in the original function. The even part consists solely of cosine (even) terms, while the odd part comprises sine (odd) terms, as seen in the final formulas:

• \( g(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty} a_{n} \cos(nx) \)
• \( h(x) = \sum_{n=1}^{\infty} b_{n} \sin(nx) \)

This decomposition is critical because it allows the function to be expressed as a sum of these simpler functions which inherently have well-understood properties. Beyond easing calculations, it provides insights into how different parts of the function contribute to its overall behavior, which is especially useful in fields like engineering and physics.

By isolating these components, one can investigate the even and odd features of the function separately, thus gaining deeper insights into its structural nature and behavior over its domain.