Question

Approximate a function by a sum of sines. We consider the piecewise constant function $$ f(t)= \begin{cases}1, & 0

Short answer

The approximation improves significantly with larger \( n \) and depends on \( \alpha \), with least error seen away from the jump discontinuity.

Step-by-step solution

Write the Function Definitions

First, define the piecewise function \( f(t, T) \) and the approximation function \( S(t, n, T) \) using Python's computational capabilities. Use loops and mathematical operations to implement these functions.

Python Function for f(t, T)

Implement the function \( f(t, T) \) in Python. This function checks the value of \( t \) and returns 1, 0, or -1 according to the specified conditions:```pythondef f(t, T): if 0 < t < T/2: return 1 elif t == T/2: return 0 else: return -1```

Python Function for S(t, n, T)

Implement the approximation function \( S(t, n, T) \) which uses a sum of sine functions:```pythondef s(t, n, T): result = 0 for i in range(1, n + 1): term = (4 / np.pi) * (1 / (2 * i - 1)) * np.sin((2 * (2 * i - 1) * np.pi * t) / T) result += term return result```

Tabulate Error Data

Use the defined functions to calculate \( f(t) - S(t, n) \) for specified values of \( n \) and \( t \) (as fractions of \( T \)). Compile these results into a table. Use Python's numpy and tabular display capabilities:```pythonimport numpy as npT = 2 * np.pin_values = [1, 3, 5, 10, 30, 100]alpha_values = [0.01, 0.25, 0.49]print("t (α*T) n = 1 n = 3 n = 5 n = 10 n = 30 n = 100")for alpha in alpha_values: t = alpha * T errors = [f(t, T) - s(t, n, T) for n in n_values] print(f"{alpha:.2f}T\t" + "\t".join(f"{error:.4f}" for error in errors))```

Comment on Quality of Approximation

Observe the table to see how approximation improves as \( n \) increases and analyze how the error values decrease, indicating that the approximation is better. Note how errors behave differently depending on the value of \( t \) relative to \( T \).

Key concepts

Sine Functions

Sine functions are mathematical functions that describe smooth, periodic oscillations. They appear frequently in trigonometry and are fundamental in harmonic analysis, which is concerned with representing functions as the superposition of basic waves. This exercise showcases sine functions used in the context of creating a Fourier series, a powerful method for approximating more complex periodic functions.

The general formula for a sine function is given by: \( \sin(x) = a \sin(bx + c) + d \).

• Here, \(a\) affects the amplitude.
• Parameter \(b\) affects the period.
• Parameter \(c\) shifts the phase.
• Finally, \(d\) shifts the entire function vertically.

These functions are significant in this exercise because they are components of the Fourier series used to approximate the function \(f(t)\). Each term in the series is a sine wave, and adding enough terms can closely mimic the behavior of complex functions.

Python Programming

Python is an excellent language for computational tasks because of its simplicity and vast ecosystem of libraries. It allows you to easily create and manipulate mathematical functions, making it ideal for this exercise.

In this exercise, two primary Python functions are crafted. The first is \(\text{def f(t, T)}\), representing a piecewise function. It uses conditional statements to return values at different intervals of \(t\). The second is \(\text{def s(t, n, T)}\), representing the summation of sine functions to form the Fourier series approximation.

• The use of \(\text{np.sin}\) from the NumPy library facilitates the computation of sine values.
• A loop construct iteratively sums the series components, closely approximating \(f(t)\) as the number of terms \(n\) increases.

Using Python to define and calculate these functions allows us to quickly model and analyze the approximation error across various settings.

Approximation Error

Approximation error is critical in evaluating how well a Fourier series models a function. In this context, it is the difference \( f(t) - S(t; n)\), where \( S(t; n)\) is our series approximation.

Errors are calculated for several values of \(n\) and \(t\), allowing analysis of how the error decreases as \(n\) increases. Observations from the exercise suggest:

• Increasing \(n\) reduces the error, implying a more accurate approximation to \(f(t)\). This is because more terms in the series better capture the function's intricacies.
• The approximation error might behave differently depending on \(t\), with certain values of \(\alpha\) leading to more significant errors due to discontinuities in the piecewise function at points like \( t = T/2 \).

Understanding and minimizing approximation error are vital when using Fourier series to ensure models are as accurate as possible.

Mathematical Functions

Mathematical functions represent the relationships between sets of values and are indispensable tools in this context. The exercise focuses on two key functions: \( f(t)\), a piecewise function, and \( S(t;n)\), a Fourier series approximation.

The original function \( f(t)\) is defined for specific intervals to be constant at 1, 0, or -1. It represents a type of waveform that is constant over each interval but has abrupt changes at the boundaries. Such function characteristics often appear in square waves or pulse sequences. \( S(t;n)\) is designed to approximate \( f(t) \) using a sum of sines which inherently are smooth and continuous.

• Piecewise functions like \( f(t)\) can be challenging to approximate precisely at their discontinuities.
• However, smoothing them out through Fourier series enables analysis using continuous waveforms.

The study of these functions and their approximations helps in understanding diverse phenomena, from signal processing to solving differential equations.