Key Concepts of Basis Functions to Know for Representation Theory

Basis functions are essential tools in Representation Theory, helping to express complex functions in simpler forms. They include Fourier series, spherical harmonics, and various orthogonal polynomials, each playing a key role in fields like physics, engineering, and signal processing.

1. Fourier basis functions

   • Represent periodic functions as sums of sine and cosine functions.
   • Fundamental in signal processing, allowing for frequency analysis.
   • Convergence properties ensure that functions can be approximated accurately.
   • Basis for Fourier series and Fourier transforms, linking time and frequency domains.

2. Spherical harmonics

   • Functions defined on the surface of a sphere, used in solving problems in physics and engineering.
   • Serve as the angular part of solutions to Laplace's equation in spherical coordinates.
   • Orthogonal functions that can represent complex shapes and patterns on a sphere.
   • Important in fields like quantum mechanics and geophysics for modeling spherical data.

3. Wavelet basis functions

   • Provide a multi-resolution analysis of functions, capturing both frequency and location information.
   • Useful for signal compression and noise reduction in various applications.
   • Allow for localized analysis, making them suitable for non-stationary signals.
   • Basis for wavelet transforms, which decompose signals into different frequency components.

4. Legendre polynomials

   • Orthogonal polynomials defined on the interval [-1, 1], used in solving problems in physics and engineering.
   • Arise in the context of solving Laplace's equation in spherical coordinates.
   • Useful in numerical methods, particularly in polynomial approximation and quadrature.
   • Form a complete basis for functions in the space of square-integrable functions.

5. Hermite polynomials

   • Orthogonal polynomials associated with the Gaussian weight function, important in probability and statistics.
   • Arise in the context of quantum mechanics, particularly in the solution of the quantum harmonic oscillator.
   • Useful in approximation theory and numerical analysis.
   • Form a complete basis for functions in the space of square-integrable functions with respect to the Gaussian measure.

6. Laguerre polynomials

   • Orthogonal polynomials associated with the exponential weight function, used in quantum mechanics and optics.
   • Arise in the context of solving the radial part of the Schrödinger equation for a hydrogen atom.
   • Useful in numerical integration and approximation of functions.
   • Form a complete basis for functions in the space of square-integrable functions with respect to the exponential measure.

7. Chebyshev polynomials

   • Orthogonal polynomials defined on the interval [-1, 1] with respect to the weight function (1-x²)⁻¹/².
   • Minimize the error in polynomial interpolation, making them ideal for numerical methods.
   • Useful in approximation theory, particularly in Chebyshev approximation.
   • Form a complete basis for functions in the space of square-integrable functions.

8. Zernike polynomials

   • Orthogonal polynomials defined on the unit disk, used in optics and image analysis.
   • Useful for representing wavefronts and analyzing optical aberrations.
   • Provide a complete basis for functions defined on the unit circle.
   • Important in applications such as surface fitting and pattern recognition.

9. Bessel functions

   • Solutions to Bessel's differential equation, important in problems with cylindrical symmetry.
   • Arise in various fields, including acoustics, electromagnetism, and heat conduction.
   • Have orthogonality properties that make them useful in series expansions.
   • Serve as basis functions for representing solutions to boundary value problems.

10. Eigenfunctions of linear operators

    • Functions that remain scaled (not changed in form) when acted upon by a linear operator.
    • Fundamental in quantum mechanics, where they represent measurable states of a system.
    • Provide a framework for solving differential equations and understanding stability.
    • Form a complete basis for the function space, allowing for expansion of arbitrary functions.