Best Approximation in Function and Polynomial Spaces

Best Approximation in Function Spaces

Our goal is to perform approximation of a specific function within an appropriate subspace of function space using a certain norm with optimal results.

The set $C([a; b])$ of continuous functions $f : [a; b] \to \mathbb{R}$ forms an infinite-dimensional real vector space. Imagine this space like a massive library containing all possible continuous functions on the interval $[a; b]$.

Scalar Product and Euclidean Space

With the scalar product defined as

then $C([a; b])$ becomes a Euclidean vector space. The corresponding norm is the quadratic mean

This scalar product is like a way to measure similarity between two functions, similar to calculating how similar two songs are based on their harmonic resonance. While the norm provides a measure of the "magnitude" of a function in the quadratic sense, like measuring the average volume of a piece of music.

Definition of Best Approximation

Let $S \subset C([a; b])$ be a finite-dimensional vector subspace and $f \in C([a; b])$. Our task is to determine $g \in S$ such that

A function $g \in S$ with this property is called a Gauss approximation. Such $g \in S$ with this property is called the best approximation of $f$ in $S$ with respect to the norm $\| \cdot \|$.

Like searching for the photo that most closely resembles the original from a limited photo collection, the best approximation provides the function that most closely matches the original function within the available space.

Polynomial Spaces

Next, we will examine

which is the set of all polynomials of maximum degree $n$. Polynomial space is like a mathematical toolbox containing various simple curves to approximate more complex function shapes. The higher the degree $n$, the more "tools" are available to form more flexible curves.

Trigonometric Polynomials

Another possibility is the set of all trigonometric polynomials

for approximation of periodic functions $f : \mathbb{R} \to \mathbb{R}$ with period $\omega > 0$.

These trigonometric polynomials are like a mathematical orchestra where sine and cosine functions serve as musical instruments that harmonize to create the melody of repeating functions. Each term in this series adds a different harmonic frequency, just like musical instruments playing fundamental notes and their harmonics.