5.4 Orthonormal bases and Fourier series

Orthonormal bases are the building blocks of Hilbert spaces. They're like a special set of tools that let us break down complex objects into simpler parts. These bases have two key features: the vectors are perpendicular to each other and have a length of 1.

Using orthonormal bases, we can represent any vector in the space as a unique combination of these basic elements. This idea extends to Fourier series, where we can express functions as sums of sines and cosines. It's a powerful way to analyze and solve problems in many areas of math and physics.

Orthonormal Bases

Orthonormal bases in Hilbert spaces

• Set of vectors in a Hilbert space satisfying two conditions:
  • Orthogonality: Inner product of any two distinct vectors equals zero (perpendicular)
  • Normality: Each vector has a norm (length) equal to 1 (unit vectors)
• Key properties of orthonormal bases:
  • Unique representation: Every vector in the Hilbert space can be expressed as a unique linear combination of the basis vectors (no ambiguity)
  • Parseval's identity: Sum of the squares of the coefficients in the basis expansion equals the square of the norm of the vector (energy conservation)
  • Completeness: Linear span of the orthonormal basis is dense in the Hilbert space (can approximate any vector arbitrarily well)

Linear combinations of orthonormal elements

• Given an orthonormal basis $\{e_n\}_{n=1}^{\infty}$ in a Hilbert space $H$ and a vector $x \in H$, express $x$ as:
  • $x = \sum_{n=1}^{\infty} \langle x, e_n \rangle e_n$ (infinite sum of basis vectors scaled by Fourier coefficients)
  • Fourier coefficients $\langle x, e_n \rangle$ represent the contribution of each basis vector to the vector $x$
• Calculate Fourier coefficients using the inner product:
  • $\langle x, e_n \rangle = \int_a^b x(t) \overline{e_n(t)} dt$ for functions in $L^2([a,b])$ (square-integrable functions on the interval $[a,b]$)

Fourier Series

Fourier series with orthonormal bases

• Expand a periodic function $f(x)$ as an infinite sum of sines and cosines:
  • $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(n \omega x) + b_n \sin(n \omega x))$ (Fourier series)
  • Fundamental frequency $\omega = \frac{2\pi}{T}$, where $T$ is the period of $f(x)$ (number of cycles per unit length)
• Calculate coefficients $a_n$ and $b_n$ using the inner product with the orthonormal basis functions:
  • $a_n = \frac{2}{T} \int_0^T f(x) \cos(n \omega x) dx$ (cosine coefficients)
  • $b_n = \frac{2}{T} \int_0^T f(x) \sin(n \omega x) dx$ (sine coefficients)

Parseval's identity for Fourier series

• Parseval's identity for a function $f(x)$ with Fourier coefficients $a_n$ and $b_n$:
  • $\int_0^T |f(x)|^2 dx = \frac{T}{2} \left(\frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)\right)$ (energy conservation)
• Interpretation in the context of Fourier series:
  • Left-hand side: Total energy of the function $f(x)$ over one period
  • Right-hand side: Sum of the energies of the individual Fourier components
  • Parseval's identity demonstrates that the energy of a function is conserved when decomposed into its Fourier components (no energy lost or gained)

Fourier series in boundary value problems

• Use Fourier series to solve boundary value problems (BVPs) in partial differential equations (PDEs)
• Steps to solve a BVP using Fourier series:
  1. Assume a solution in the form of a Fourier series with unknown coefficients
  2. Substitute the assumed solution into the PDE and boundary conditions
  3. Use the orthogonality of the basis functions to determine the Fourier coefficients
  4. Construct the final solution using the Fourier series with the calculated coefficients
• Example: Solving the heat equation $\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}$ with boundary conditions $u(0,t) = u(L,t) = 0$ (insulated ends) and initial condition $u(x,0) = f(x)$ (initial temperature distribution)