An orthonormal set of functions is a collection of functions where every distinct pair of functions is orthogonal to each other, and each individual function within the set is normalized to have a magnitude (or "length") of one.
Defining Orthonormal Sets
A set of functions, often denoted as ${\phi_k(t)}$, where $k$ represents an index (like an integer), forms an orthonormal set if it satisfies two critical conditions over a specified interval:
- Orthogonality: Any two distinct functions from the set, $\phi_k(t)$ and $\phi_m(t)$ (where $k \ne m$), are orthogonal. This means their inner product (typically an integral over the given interval) is zero.
- Normalization: Every function within the set, $\phi_k(t)$, is normalized. This means its inner product with itself (or its squared norm) is equal to one.
Together, these properties ensure that the functions are "independent" in a mathematical sense and have a consistent "unit length," which simplifies many mathematical analyses.
Understanding Orthogonality
Two functions, $\phi_k(t)$ and $\phi_m(t)$, are considered orthogonal over an interval $[a, b]$ if their inner product over that interval is zero. For real-valued functions, the inner product is typically defined by an integral:
$$ \int_a^b \phi_k(t) \phi_m(t) \, dt = 0 \quad \text{for } k \ne m $$
For complex-valued functions, the inner product includes a complex conjugate of one of the functions:
$$ \int_a^b \phi_k(t) \overline{\phi_m(t)} \, dt = 0 \quad \text{for } k \ne m $$
Orthogonality is a fundamental concept in functional analysis and is analogous to perpendicular vectors in Euclidean geometry.
Understanding Normalization
A function $\phi_k(t)$ is normalized if its norm, or "length," is equal to one. For functions, the squared norm is typically calculated as the inner product of the function with itself. Over an interval $[a, b]$, this means:
$$ \int_a^b |\phi_k(t)|^2 \, dt = 1 $$
where $|\phi_k(t)|^2 = \phi_k(t) \overline{\phi_k(t)}$ for complex functions, and simply $\phi_k(t)^2$ for real functions. Normalizing a function means scaling it so that its "energy" or "power" over the interval is unity.
Why Orthonormal Sets Matter
Orthonormal sets of functions are incredibly important across various scientific and engineering disciplines due to their unique properties:
- Basis for Function Spaces: They often form a convenient basis for infinite-dimensional function spaces, allowing any function in that space to be expressed as a linear combination of the orthonormal basis functions. This is similar to how unit vectors ($\hat{i}, \hat{j}, \hat{k}$) form a basis for 3D space.
- Simplified Calculations: The orthogonality property greatly simplifies calculations involving inner products, as most terms vanish.
- Signal Processing: Used in Fourier analysis, wavelet transforms, and other techniques for decomposing signals into their constituent components.
- Quantum Mechanics: Eigenfunctions of Hermitian operators are often orthonormal, forming the basis for describing quantum states.
- Numerical Methods: Essential in methods like the Gram-Schmidt process for constructing orthonormal bases.
Examples of Orthonormal Function Sets
Many well-known sets of functions, when properly normalized, form orthonormal sets over specific intervals.
| Orthonormal Set | Functions | Interval | Application Examples |
|---|---|---|---|
| Complex Fourier Series Basis | $\left{ \frac{1}{\sqrt{2\pi}} e^{ikt} \right}_{k=-\infty}^{\infty}$ | $[-\pi, \pi]$ | Signal analysis, solving PDEs, image compression |
| Real Fourier Series Basis | $\left{ \frac{1}{\sqrt{2\pi}}, \frac{1}{\sqrt{\pi}}\cos(kt), \frac{1}{\sqrt{\pi}}\sin(kt) \right}_{k=1}^{\infty}$ | $[-\pi, \pi]$ | Electrical engineering, vibrations |
| Normalized Legendre Polynomials | $\left{ \sqrt{\frac{2k+1}{2}} Pk(x) \right}{k=0}^{\infty}$ | $[-1, 1]$ | Electromagnetism, potential theory, numerical integration |
| Haar Wavelets | A specific construction of orthonormal wavelets | Varies with level | Image processing, data compression |
For example, for the complex Fourier basis over $[-\pi, \pi]$:
- Orthogonality: $\int{-\pi}^{\pi} \left(\frac{1}{\sqrt{2\pi}} e^{ikt}\right) \overline{\left(\frac{1}{\sqrt{2\pi}} e^{imt}\right)} dt = \frac{1}{2\pi} \int{-\pi}^{\pi} e^{i(k-m)t} dt = 0$ for $k \ne m$.
- Normalization: $\int{-\pi}^{\pi} \left|\frac{1}{\sqrt{2\pi}} e^{ikt}\right|^2 dt = \int{-\pi}^{\pi} \frac{1}{2\pi} dt = \frac{1}{2\pi} [t]_{-\pi}^{\pi} = \frac{1}{2\pi} (\pi - (-\pi)) = \frac{2\pi}{2\pi} = 1$.
How to Verify Orthonormality
To check if a given set of functions ${\phi_k(t)}$ forms an orthonormal set over a specified interval $[a, b]$:
-
Check for Orthogonality:
- Pick any two distinct functions from the set, say $\phi_k(t)$ and $\phi_m(t)$ where $k \ne m$.
- Calculate their inner product over the interval $[a, b]$.
- If $\int_a^b \phi_k(t) \overline{\phi_m(t)} \, dt = 0$ for all distinct pairs, the set is orthogonal.
-
Check for Normalization:
- For each function $\phi_k(t)$ in the set, calculate its squared norm over the interval $[a, b]$.
- If $\int_a^b |\phi_k(t)|^2 \, dt = 1$ for all functions in the set, the set is normalized.
If both conditions are met, the set is an orthonormal set of functions.
