Approximation of Functions Approximation problems divided into five
Approximation of Functions Approximation problems divided into five parts: ü Least square approximation of discrete functions or Regression ü Least square approximation of continuous function using various basis polynomials ü Orthogonal basis functions ü Approximation of periodic functions ü Interpolation
Theory of Approximation of Continuous Function f(x) Complicated Analytical Function, Analog Signal from a measuring device Missing Data, Derivative, Integration for tab (f): Interpolation Approximation of Discrete data or tab (f): Regression Discrete measurements of continuous experiments or phenomena Approximation of Discrete data or tab (f): Regression
Approximation of Functions •
Approximation of Functions When not to use polynomial basis? ü If the functional form or the model is known ü Sharp front ü Periodic function
Function Space vs. Vector Space •
Norm and Seminorm •
Inner Product •
Basis Functions: Linear Independence •
Orthogonal Functions •
Least Square Problem •
Schematic of Least Square Solution f(x)
Least Square Solution: Proof •
Least Square Solution: Proof •
Least Square Solution: Normal Equations •
Least Square Solution: Normal Equations •
Least Square Solution: Existence and Uniqueness •
Least Square Solution: Existence and Uniqueness •
Least Square Solution: Example (Continuous) •
Least Square Solution: Example (Continuous) •
- Slides: 34
