Problem 17 - Differential Equations and Linear Algebra

Question

Show that the given functions are orthonormal on [-1,1]. $$\begin{array}{l} f_{1}(x)=\cos \pi x, f_{2}(x)=\cos 2 \pi x \\ f_{3}(x)=\cos 3 \pi x \end{array}$$

Step-by-Step Solution

Answer

To show that the given functions \(f_1(x)=\cos(\pi x)\), \(f_2(x)=\cos(2\pi x)\), and \(f_3(x)=\cos(3\pi x)\) are orthonormal on the interval \([-1,1\), we first compute the norm of each function by evaluating the integral \(\int_{-1}^1 f_i(x)^2 dx\) for \(i = 1, 2, 3\). We verify that each norm is equal to 1, indicating the functions are normalized. Next, we compute the inner products of all pairs of functions using the formula \(\langle f, g\rangle = \int_{-1}^1 f(x)g(x) dx\). We check that the inner products are equal to 0, confirming the orthogonality of the functions. Since all functions are normalized and orthogonal to each other, we conclude that the given functions are orthonormal on the interval \([-1,1\).

Step 1: Compute the norm of each function

Compute the following integrals for every function: $$\int_{-1}^1 f_i(x)^2 dx$$ where \(i = 1, 2, 3\).

Step 2: Show that the functions are normalized

For each function, check if the norm is equal to 1. If not, normalize the function by dividing it by its square root.

Step 3: Check the orthogonality by computing the inner product

Compute the inner products for all pairs of functions: $$\langle f_1, f_2 \rangle,\, \langle f_1, f_3 \rangle,\, \langle f_2, f_3 \rangle$$

Step 4: Verify that inner products equal 0

Make sure all inner products calculated in Step 3 are equal to 0. This confirms that functions are orthogonal to each other.

Step 5: Conclude Orthonormality

After showing that all functions are normalized and orthogonal to each other, we can conclude that the given functions are orthonormal on the interval \([-1,1\]).

Key Concepts

Inner Product

The 'inner product' is a fundamental concept in linear algebra and functional analysis, providing a way to quantify the 'angle' and 'length' within a vector space. Specifically, for functions, the inner product is an integral that combines two functions and provides a scalar output.

For real-valued functions such as those given in the exercise, the inner product is defined as:
\[\langle f, g \rangle = \int_{-1}^{1} f(x)g(x)\, dx\]
where \(f(x)\) and \(g(x)\) are functions defined on the interval \([-1, 1]\).

The inner product has several important properties, including commutativity \( \langle f, g \rangle = \langle g, f \rangle \), linearity in both arguments, and positivity which implies that \( \langle f, f \rangle \) is always non-negative.

When evaluating the orthonormality of functions, the inner product is used as a tool to measure orthogonality—functions are orthogonal if their inner product equals zero. In the context of the provided exercise, this means that if the inner product of any two distinct functions \(f_i\) and \(f_j\) (where \(i eq j\)) equals zero, these functions are orthogonal over the interval.

Normalization of Functions

Normalization in the context of functions means adjusting a function so that its 'norm' (or 'magnitude') equals one. The norm of a function, denoted as \(||f||\), is essentially the function's 'length' and is obtained by taking the square root of its inner product with itself:
\[||f|| = \sqrt{\langle f, f \rangle}\]
For a function to be normalized, its norm must equal one, \( ||f|| = 1 \).

The solution to the exercise involves normalizing functions by ensuring that when you integrate the square of the function over the interval, the result should be one. If the norm is not one, you can normalize the function by dividing the function by its norm.

This process can be visualized as scaling the function in such a way that it fits perfectly into a unit length on its domain, preserving the function's shape. Normalization is crucial when dealing with orthonormal functions, as it ensures that each function not only is orthogonal to others but also has a consistent, standard size.

Orthogonal Functions

Orthogonal functions are functions that, when paired and evaluated using an inner product, result in zero, meaning they 'point' in directions that are completely independent of each other within the function space.
In mathematical terms, two functions \(f\) and \(g\) are orthogonal over an interval if their inner product over that interval is zero:
\[\langle f, g \rangle = 0\]
If this condition is satisfied, no amount of scaling of one function can make it align with the other—the functions are inherently different in terms of the 'directions' they occupy in the function space.

Orthogonality is a crucial concept when discussing function spaces because it allows for the creation of a basis for a space, where new functions can be expressed as linear combinations of these orthogonal functions with no redundancy or overlap. The textbook exercise provided asked you to verify orthogonality by computing the inner products and checking if they equated to zero, thereby confirming that the functions are, in fact, orthogonal.