Question

Let \(f \in E\) and $$ f(x) \sim \sum_{n=-\infty}^{\infty} c_{n} e^{i n x} $$ be the complex Fourier series of \(f\). Determine the complex Fourier series of \(f(\bar{x}), \overline{f(x)}\), and \(f(-x)\).

Short answer

The Fourier series are \(f(\bar{x}) = \sum_{n=-\infty}^{\infty} c_{n} e^{i n \bar{x}}\), \(\overline{f(x)} = \sum_{n=-\infty}^{\infty} \overline{c_{n}} e^{-i n x}\), and \(f(-x) = \sum_{n=-\infty}^{\infty} c_{n} e^{-i n x}\).

Step-by-step solution

Understanding the problem

We need to determine the complex Fourier series of three variations of function \(f\): \(f(\bar{x})\), \(\overline{f(x)}\), and \(f(-x)\). We have the Fourier series of \(f(x)\) given, and we need to transform this series for each case.

Expanding \(f(\bar{x})\)

If \(f(x)\) is represented by \sum_{n=-\infty}^{\infty} c_{n} e^{i n x}\, then we substitute \(\bar{x}\) into it to get \(f(\bar{x}) = \sum_{n=-\infty}^{\infty} c_{n} e^{i n \bar{x}}\). This is the complex Fourier series of \(f(\bar{x})\).

Expanding \(\overline{f(x)}\)

For the complex conjugate \(\overline{f(x)}\), taking the conjugate of the Fourier series gives \(\overline{f(x)} = \overline{\sum_{n=-\infty}^{\infty} c_{n} e^{i n x}} = \sum_{n=-\infty}^{\infty} \overline{c_{n}} e^{-i n x}\). This is achieved by conjugating each term of the series.

Expanding \(f(-x)\)

For \(f(-x)\), we substitute \(-x\) in place of \(x\) in the Fourier series: \(f(-x) = \sum_{n=-\infty}^{\infty} c_{n} e^{-i n x}\). This is the complex Fourier series of \(f(-x)\).

Key concepts

Fourier transform

The Fourier transform is a mathematical technique that translates a function of time (or space) into a function of frequency. It provides a frequency domain representation of the original time domain function. The idea is to express any function as a sum of simple waves, known as sinusoids, which are typically sinusoidal waves. When you apply the Fourier transform, you move from focusing on how a function behaves over time to understanding what frequency components make up that function.
Consider a signal, such as sound or light. This signal can be broken down into a spectrum of its frequencies. The Fourier transform helps in describing these frequencies by turning the complex function into an infinite series of sines and cosines. This transformation is crucial in various fields, including signal processing, physics, and even engineering.

• The output of a Fourier transform, a frequency spectrum, shows the frequency peaks or lines for each component of the signal.
• This helps in identifying dominant frequencies in the function, making it easier to analyze signals.

By understanding the frequency components, you can, for example, filter specific frequencies out or amplify them, manipulating the original signal's characteristics.

harmonic analysis

Harmonic analysis is the study of functions or signals in terms of basic waves, often sines and cosines. Within harmonic analysis, the Fourier transform plays a significant role by breaking down complex signals into their basic oscillating components. The main goal is to find harmonics, or simple wave frequencies, within a more complex function.
Using complex Fourier series, each function can be understood through the various amplitudes and phases of its sinusoidal components. These components are additive and are expressed in terms of their harmonics, or integral multiples of a base frequency. It's especially useful in areas such as music, where sound waves are analyzed through their frequency contents.

• Harmonic analysis helps decompose signals into their constituent frequencies, understanding both fundamental and overtone harmonics.
• This understanding assists in compression, detection, and even prediction of signals in various domains.

The combination of frequency analysis and harmonic component prediction make it a powerful tool for understanding and manipulating signals.

function transformation

Function transformation in the context of complex Fourier series is about manipulating the input function in ways that alter its Fourier representation. Each variation—whether it's the real part, imaginary part, or conjugate of the function—has a distinct representation in terms of its Fourier series.
For instance, if you take a complex function and find its conjugate, it affects all components of its Fourier series. Each coefficient in the Fourier series might need conjugation or might change its sign depending on the transformation you are performing. Key transformations include:

• Complex Conjugate: Changes the sign of the imaginary part of each coefficient in the Fourier series.
• Shift by Complex Argument: Substituting complex values in the argument, such as replacing x with -x or \(\bar{x}\).

Understanding these transformations is crucial for correct interpretation of Fourier analyses and for adjusting the series to suit various analytical needs. Each transformation reveals different aspects of the original function, giving deeper insights into its frequency domain representation.