Minimax approximation

1. Sablon:Matematika minimax közelítés

Minimax approximation is a technique used in approximation theory, particularly in the context of function approximation. The goal of minimax approximation is to find a function (typically a polynomial or another type of simpler function) that approximates a given target function in such a way that the maximum error (the worst-case error) between the two functions is minimized.

Key Concepts

1. Approximation Problem: Given a target function $f(x)$, we want to find an approximating function $p(x)$ such that the maximum absolute error is minimized: $$E=\underset{x\in [a,b]}{\max }|f(x)-p(x)|$$ where $[a,b]$ is the interval of interest. 2. Minimax Criterion: The minimax approximation aims to minimize the maximum error $E$ over the interval, leading to the formulation: $$\underset{p}{\min }\underset{x\in [a,b]}{\max }|f(x)-p(x)|$$ 3. Chebyshev Nodes: In many cases, the approximation is evaluated at specific points known as Chebyshev nodes, which are designed to minimize the error in polynomial interpolation. These nodes are given by: $$x_k=\cos \left(\frac{2k+1}{2n}\pi \right),k=0,1,\dots ,n-1$$ 4. Chebyshev Polynomials: The use of Chebyshev polynomials in minimax approximation is particularly beneficial because they have properties that help achieve optimal approximations. They can minimize the maximum deviation from the target function over the interval. 5. Applications: Minimax approximation is used in various fields such as numerical analysis, control theory, signal processing, and optimization problems, where it's crucial to have a robust approximation that performs well in the worst-case scenario.

Example

For instance, if you want to approximate the function $f(x)=e^x$ on the interval $[0,1]$ with a polynomial $p(x)$, you would: 1. Determine the Chebyshev nodes in the interval. 2. Formulate the minimax problem by finding $p(x)$ that minimizes the maximum absolute error. 3. Use techniques like the Remez algorithm to compute the coefficients of $p(x)$ that achieve this minimax property.

Summary

Minimax approximation provides a powerful framework for creating robust function approximations by minimizing the worst-case error, which is particularly useful in applications requiring high reliability. Sablon:Engl