Problem 17 - Advanced Engineering Mathematics

Question

Let \(\left\\{\phi_{n}(x)\right\\}\) be an orthogonal set of functions on \([a, b]\). Show that \(\left\|\phi_{m}(x)+\phi_{n}(x)\right\|^{2}=\left\|\phi_{m}(x)\right\|^{2}+\left\|\phi_{n}(x)\right\|^{2}, m \neq n\).

Step-by-Step Solution

Answer

Use orthogonality: \(\|\phi_m + \phi_n\|^2 = \|\phi_m\|^2 + \|\phi_n\|^2\).

Step 1: Understand Orthogonality

Orthogonality of two functions, \(\phi_{m}(x)\) and \(\phi_{n}(x)\), on the interval \([a, b]\) means that their inner product is zero: \(\int_{a}^{b} \phi_{m}(x) \phi_{n}(x) \, dx = 0\) for \(m eq n\).

Key Concepts

Orthogonal Functions

Orthogonal functions are a central concept in mathematics and engineering, especially in the analysis of signals and vibrations. Two functions, say \( \phi_{m}(x) \) and \( \phi_{n}(x) \), are defined as orthogonal over an interval \([a, b]\) if their inner product equals zero. This is expressed mathematically by:\[\int_{a}^{b} \phi_{m}(x) \phi_{n}(x) \, dx = 0\]for \( m eq n \).
This property of orthogonality is quite similar to the concept of perpendicular vectors in Euclidean space. Just as perpendicular vectors have a dot product of zero, orthogonal functions have an integral or inner product of zero over specified intervals.
Orthogonal functions can simplify problems significantly. They allow us to decompose complicated functions into simpler, non-overlapping components, much like using a basis in vector spaces. When you work with orthogonal functions, you can manage data, functions, and signals more effectively without interference from other parts of the system.

Inner Product

The concept of an inner product extends the idea of the dot product of vectors to functions. It provides a way to quantify the 'overlap' or similarity between two functions. For two functions \( \phi_{m}(x) \) and \( \phi_{n}(x) \), their inner product over an interval \([a, b]\) is defined as:

• \(\int_{a}^{b} \phi_{m}(x) \phi_{n}(x) \, dx\)

This expression calculates the area under the curve of the product of the two functions. In the context of orthogonality, if the inner product is zero, it indicates that the functions do not overlap significantly; they are orthogonal.
Understanding and calculating the inner product is crucial in many areas such as quantum mechanics, signal processing, and statistics. It helps in determining relationships between functions and can be used to extract independent components from data, similar to how different notes in music can be identified in a complex sound wave.

Norm of a Function

The norm of a function is a measure of its size or length, akin to the magnitude of a vector. Mathematically, the norm \( \|\phi(x)\| \) of a function \( \phi(x) \) over an interval \([a, b]\) is given by:

• \(\sqrt{\int_{a}^{b} \phi(x)^2 \, dx}\)

This formula offers a scalar value that represents the total 'energy' of the function over the given interval.
The norm is crucial when discussing orthogonal functions, especially in problems where you sum functions together. For example, if \( \phi_{m}(x) \) and \( \phi_{n}(x) \) are orthogonal, the norm of their sum follows the Pythagorean theorem, reflecting the additive nature of energy: \[\left\|\phi_{m}(x) + \phi_{n}(x)\right\|^{2} = \left\|\phi_{m}(x)\right\|^{2} + \left\|\phi_{n}(x)\right\|^{2}\]This relationship highlights how orthogonal components combine without confounding one another, making it a critical aspect of functional analysis, signal processing, and other areas where maintaining clarity and separation between distinct components is vital.