Question

Let \(f: \mathbb{R} \rightarrow \mathbb{C}\) be a piecewise continuous function which is \(\pi\)-periodic. Let $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos 2 n x+b_{n} \sin 2 n x\right] $$ be the Fourier series of \(f\) on \([0, \pi]\), and $$ f(x) \sim \frac{A_{0}}{2}+\sum_{n=1}^{\infty}\left[A_{n} \cos n x+B_{n} \sin n x\right] $$ the Fourier series of \(f\) on \([-\pi, \pi]\). Express the \(A_{n}\) and \(B_{n}\) in terms of the \(a_{n}\) and \(b_{n}\).

Short answer

For even \(n\), \(A_{2n} = a_{n}\) and \(B_{2n} = -b_{n}\); for odd \(n\), \(A_{n} = B_{n} = 0\).

Step-by-step solution

Fourier Coefficients on [0, \pi]

The Fourier series on \([0, \pi]\) is given by \[ f(x) \sim \frac{a_{0}}{2} + \sum_{n=1}^{\infty} \left[ a_{n} \cos 2nx + b_{n} \sin 2nx \right]. \] This series uses frequencies of \(2n\) only because the function is defined only over half a period \([0, \pi]\).

Fourier Coefficients on [-\pi, \pi]

The Fourier series on \([-\pi, \pi]\) is \[ f(x) \sim \frac{A_{0}}{2} + \sum_{n=1}^{\infty} \left[ A_{n} \cos nx + B_{n} \sin nx \right]. \] This represents the standard Fourier series for a function defined over a full period. Here, \(n\) ranges over all positive integers, not just even integers.

Relationship Between Coefficients

To relate these, recall that the series on \([0,\pi]\) uses even frequencies, \(2n\). This means each term in the \([0, \pi]\) series relates to every second term in the \([-\pi, \pi]\) series for cosine terms, i.e., \(A_{2n} = a_{n}\), and for sine terms, \(-B_{2n} = b_{n}\).

Express Odd Terms in Terms of Zero

For odd \(n\) in \([-\pi, \pi]\), no corresponding term exists in the \([0, \pi]\) series, which covers only even frequencies. Thus, \(A_{n} = 0\) and \(B_{n} = 0\) for odd \(n\).

Final Result

Combining all, the relationships are given by: \( A_{2n} = a_{n} \), \( B_{2n} = -b_{n} \). For odd \(n\), \( A_{n} = 0 \) and \( B_{n} = 0 \). This maintains the periodicity and agreements between defining ranges.

Key concepts

Piecewise Continuous Function

A piecewise continuous function is a type of function that can be broken down into segments, each of which is continuous on its own. Essentially, this function might seem erratic at first, but when analyzed in specific intervals, it behaves predictably without any surprises like sudden jumps or breaks.
For the function to be piecewise continuous, it should meet several key conditions:

• Each segment of the function must be defined and continuous within a specific interval.
• The points where the function switches from one kind of behavior to another (like switching from one equation to another) should not have discontinuities that are too severe (like infinite jumps).
• Technically speaking, as long as the number of discontinuities is finite and simple, the function qualifies.

Understanding this type of function is essential when dealing with Fourier series, as they allow us to bend the usual rules of continuity to include more real-world scenarios.

Periodic Function

A periodic function is one that repeats its values at regular intervals or periods. This is a key idea in the study of waveforms and signals, as these functions model repetitive phenomena found in nature and various technologies.
The fundamental characteristics of a periodic function are:

• The function repeats its behavior after a certain period, denoted as \(T\).
• For any point \(x\) on the function, \(f(x) = f(x + T)\) holds true for all \(x\).
• This periodicity is crucial in Fourier series as it allows us to analyze the function in terms of simpler repetitive components, aiding in simplifying complex waveforms.

In the context of the given exercise, the function is \(\pi\)-periodic, meaning it repeats every \(\pi\) units. This property is essential for converting a function into its Fourier series representation.

Fourier Coefficients

Fourier coefficients are the building blocks of the Fourier series, breaking down complex functions into sums of sines and cosines. Understanding these coefficients is pivotal, as they effectively give weights to these trigonometric functions, summarizing the original function's essential features.
The Fourier series decomposition involves:

• A constant term (denoted here as \(\frac{a_0}{2}\) or \(\frac{A_0}{2}\)) which represents the average value over a period.
• Cosine terms with coefficients \(a_n\) or \(A_n\), capturing the even symmetry components of the function.
• Sine terms with coefficients \(b_n\) or \(B_n\), accounting for the odd symmetry parts.
• Each coefficient tells us how much of a particular frequency component is present in the overall function.

This decomposition is particularly helpful in the case of piecewise continuous and periodic functions to handle regions separately without losing the periodicity of the original function.

Frequency Components

In the context of Fourier series, frequency components refer to the individual sine or cosine functions that make up the complete waveform. Each of these components represents a specific frequency, elucidating the concept of decomposing complex signals into simpler waves.
Key takeaways about frequency components include:

• Each term in the Fourier series corresponds to a frequency that is a multiple of the fundamental frequency.
• The function's period determines these frequencies, sometimes expressed in terms of cycles per period.
• In the context of the exercise, we see that terms like \(2n\) align with the frequencies within a partial period, while \(n\) relates to a full period, explaining the presence of even or odd frequencies.
• Understanding these components is essential for dissecting complex waveforms into manageable parts, giving insights into the behavior and composition of signals.

By breaking down a function into its frequency components, we can analyze and synthesize signals more efficiently, essential for tasks in signal processing and communications.