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 the Fourier series of \(f\). Prove that there exist \(\left\\{A_{n}\right\\}_{n=0}^{\infty}\) and \(\left\\{\alpha_{n}\right\\}_{n=0}^{\infty}\), where \(-\frac{\pi}{2}<\alpha_{n} \leq \frac{\pi}{2}\), such that $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right]=A_{0}+\sum_{n=1}^{\infty} A_{n} \cos \left(n x-\alpha_{n}\right) . $$ In a similar way, prove that there exist \(\left\\{B_{n}\right\\}_{n=0}^{\infty}\) and \(\left\\{\beta_{n}\right\\}_{n=1}^{\infty}\), where \(-\frac{\pi}{2}<\) \(\beta_{n} \leq \frac{\pi}{2}\), such that $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right]=B_{0}+\sum_{n=1}^{\infty} B_{n} \sin \left(n x+\beta_{n}\right) . $$

Short answer

The amplitudes are given by \( A_n = B_n = \sqrt{a_n^2 + b_n^2} \), with corresponding phases \( \alpha_n = \tan^{-1}(b_n/a_n) \) and \( \beta_n = \tan^{-1}(-a_n/b_n) \).

Step-by-step solution

Represent Terms as Amplitude-Phase Form

Express each term from the Fourier series in the amplitude-phase form:\[ a_n \cos(nx) + b_n \sin(nx) = A_n \cos(nx - \alpha_n) \]where the amplitude \( A_n \) and phase \( \alpha_n \) need to be determined.

Calculate Amplitude A_n

The amplitude \( A_n \) is given by the Euclidean norm of coefficients \( a_n \) and \( b_n \):\[ A_n = \sqrt{a_n^2 + b_n^2} \]

Determine Phase Angle α_n

The phase \( \alpha_n \) is determined using the trigonometric relation:\[ \tan(\alpha_n) = \frac{b_n}{a_n} \]This gives:\[ \alpha_n = \tan^{-1}\left(\frac{b_n}{a_n}\right) \]Ensure \(-\frac{\pi}{2} < \alpha_n \leq \frac{\pi}{2}\) by adjusting the quadrant based on the signs of \( a_n \) and \( b_n \).

Confirm with Trigonometric Identity

Verify the identity by expanding:\[ A_n \cos(nx - \alpha_n) = A_n (\cos(nx)\cos(\alpha_n) + \sin(nx)\sin(\alpha_n)) \]This simplifies to:\[ = a_n \cos(nx) + b_n \sin(nx) \]Confirm alignment with the original \( a_n \cos(nx) + b_n \sin(nx) \).

Represent in Sinusoidal Form with Phase Shift

Next, express each term using sinusoidal form with a phase shift:\[ a_n \cos(nx) + b_n \sin(nx) = B_n \sin(nx + \beta_n) \]Again, find \( B_n \) and \( \beta_n \).

Calculate Amplitude B_n

As before, \( B_n \) is determined by:\[ B_n = \sqrt{a_n^2 + b_n^2} \]Notice that the amplitude remains the same.

Determine Phase Angle β_n

The phase \( \beta_n \) can be found using:\[ \tan(\beta_n) = -\frac{a_n}{b_n} \]Thus:\[ \beta_n = \tan^{-1}\left(-\frac{a_n}{b_n}\right) \]Adjust \( \beta_n \) within \( -\frac{\pi}{2} < \beta_n \leq \frac{\pi}{2} \).

Verify Sinusoidal Identity

Verify by expanding the sinusoidal form:\[ B_n \sin(nx + \beta_n) = B_n (\sin(nx)\cos(\beta_n) + \cos(nx)\sin(\beta_n)) \]Thus:\[ = b_n \sin(nx) + a_n \cos(nx) \]Matching the original format confirms correctness.

Key concepts

Amplitude and Phase

When working with Fourier series, it's important to understand how signals can be represented using amplitude and phase. This involves converting the expression \(a_n \cos(nx) + b_n \sin(nx)\) into a form that utilizes both amplitude \(A_n\) and phase \(\alpha_n\). This transformation allows for a more intuitive understanding of wave behavior.

• Amplitude \(A_n\): This measures the signal's strength. Mathematically, it is determined by the Euclidean norm, \(A_n = \sqrt{a_n^2 + b_n^2}\).
• Phase \(\alpha_n\): This indicates the wave's shift along the x-axis. It's calculated using the trigonometric identity, \(\tan(\alpha_n) = \frac{b_n}{a_n}\).

Transitioning the representation to amplitude and phase form not only aids in analysis but also simplifies many calculations.

Trigonometric Identities

Trigonometric identities are mathematical tools that simplify expressions involving trig functions like sine and cosine. In the context of Fourier series, they allow us to express combinations of sine and cosine in a more usable form.A key identity used here is the cosine addition formula:\(\cos(nx - \alpha_n) = \cos(nx)\cos(\alpha_n) + \sin(nx)\sin(\alpha_n)\)Using this identity enables us to demonstrate that the amplitude-phase form of a Fourier component aligns with its original form. This is important because:

• It helps verify that the conversion to the amplitude-phase form was done correctly.
• It simplifies the understanding and calculation for multiple terms involving trigonometric functions.

Euclidean Norm

The Euclidean norm is a fundamental concept in vector mathematics, often depicted as the vector's length or magnitude. In Fourier series, it helps in defining the amplitude of a signal. Here's why it's useful in calculating amplitudes:

• Magnitude Calculation: It sums the squares of all components, giving a robust measure of magnitude with \(A_n = \sqrt{a_n^2 + b_n^2}\).
• Simplicity: Provides a straightforward way to aggregate multiple components into a single scalar value that represents the 'size' of the vector.

Using the Euclidean norm thus streamlines the process of finding amplitudes for complex waveforms within the Fourier series.

Phase Angle

The phase angle \(\alpha_n\) is crucial in understanding how waves shift over the cycle. It is determined by trigonometric relationships derived from the coefficients of sine and cosine in the Fourier series.To find \(\alpha_n\), use the identity:\(\tan(\alpha_n) = \frac{b_n}{a_n}\).The steps to ensure correctness include:

• Inverse Tangent Calculation: Calculate \(\alpha_n\) using \(\tan^{-1}(\frac{b_n}{a_n})\).
• Range Adherence: Ensure \(-\frac{\pi}{2} < \alpha_n \leq \frac{\pi}{2}\) by adjusting for the correct quadrant.
• Sine and Cosine Check: Verify with a trigonometric identity to ensure that \(\alpha_n\) correctly shifts the wave.

This ensures the phase angle correctly reflects the actual position of the waveform within its cycle.