Question

Let \(M\) be the set of all trigonometric polynomials, regarded as a subspace of \(C_{2 \pi}\). Show that \(M\) is dense in \(C_{2 \pi}\). [Hint: use the transformation \(t=\cos \theta\) between \([0, \pi]\) and \([-1,1]\) together with the Weierstrass result on the density of the algebraic polynomials in \(C[-1,1] .]\)

Short answer

Question: Show that the set of trigonometric polynomials is dense in the set of continuous functions C_{2π}. Answer: We showed that for any continuous function f(t) defined on [-1, 1], the corresponding even extension of the continuous function g(θ) = f(cos θ) can be approximated by a trigonometric polynomial in the set M to within ε/2 for all θ in [0, 2π]. Thus, the set M of trigonometric polynomials is dense in the space of continuous functions C_{2π}.

Step-by-step solution

State the Weierstrass Approximation Theorem

The Weierstrass Approximation Theorem states that the algebraic polynomials are dense in the space of continuous functions C[a, b] for any closed interval [a, b]. In other words, for any continuous function f(x) defined on [a, b] and any ε > 0, there exists a polynomial P(x) such that |f(x) - P(x)| < ε for all x in [a, b].

Define M and C_{2π}

Let M be the set of all trigonometric polynomials, which are finite linear combinations of sine and cosine functions with integer frequencies. C_{2π} is the space of continuous functions defined on [0, 2π] with f(0) = f(2π).

Define the transformation

Define t = cos θ, where t ∈ [-1, 1] and θ ∈ [0, π]. This transformation maps the interval [0, π] to the interval [-1, 1].

Using Weierstrass Approximation Theorem

For any continuous function f(t) defined on [-1, 1] and any ε > 0, there exists a polynomial P(t) such that |f(t) - P(t)| < ε for all t in [-1, 1] due to the Weierstrass Approximation Theorem. The corresponding function g(θ) = f(cos θ) is continuous on [0, π].

Even extension

Extend the function g(θ) to an even function G(x) defined on the interval [-π, π]. The extended function G(x) is continuous on [-π, π], G(0) = G(2π), and G(x) = G(2π - x) for all x in [0, 2π].

Show that M approximates G

Since G(x) is an even function with periodicity 2π, it can be represented as a Fourier cosine series. For a sufficiently large N, the partial sum of the Fourier cosine series will approximate G(x) to within ε/2. Thus, the trigonometric polynomial formed by sine and cosine functions with integer frequencies approximates G(x) to within ε/2 for all x in [0, 2π].

Conclude that M is dense in C_{2π}

It has been shown that for any continuous function f(t) defined on [-1, 1], the corresponding even extension of the continuous function g(θ) = f(cos θ) can be approximated by a trigonometric polynomial in M to within ε/2 for all θ in [0, 2π]. Thus, M is dense in the space of continuous functions C_{2π}, as required.

Key concepts

Understanding Trigonometric Polynomials

Trigonometric polynomials are foundational components in the study of periodic functions, crucial for subjects like Fourier analysis and signal processing. In essence, a trigonometric polynomial is a finite sum of sine and cosine functions, each multiplied by a coefficient. These polynomials are incredibly useful for approximating more complex periodic functions.

Formally, such a polynomial can be expressed as
\[ T_n(x) = a_0 + \sum_{k=1}^n (a_k \cos(kx) + b_k \sin(kx)) \]
where \( n \) is a non-negative integer and the coefficients \( a_0, a_k, b_k \) are real numbers. The degree of this trigonometric polynomial is the highest value of \( k \) for which either \( a_k \) or \( b_k \) is non-zero.

This concept becomes particularly interesting because of a property known as density. Polynomials are dense in the space of continuous functions, meaning that for any given continuous function, it's possible to find a trigonometric polynomial that approximates it as closely as one desires over a closed interval. This fact is a cornerstone of the Weierstrass Approximation Theorem and will be further explored in relation to continuous functions.

Density in Continuous Functions

The notion of density in relation to continuous functions is pivotal in understanding function approximation. Density, in this context, does not refer to physical mass per volume, but to the concept that a set of functions (like trigonometric polynomials) is 'dense' in the space of continuous functions. This means that for any continuous function defined on a closed interval (and satisfying certain conditions), there exists a sequence of functions from our set that converges uniformly to it.

To break it down, any continuous function \( f(x) \) defined on a specific interval can be approximated as closely as you'd like by a trigonometric polynomial. Whether for analysis or practical applications, like synthesizing sounds or compressing images, this concept assures us that we can work with simpler, more manageable functions without significant loss of precision.

It's this very property that the Weierstrass Approximation Theorem exploits. The theorem transforms the complex into the accessible, consolidating the idea that algebraic and trigonometric polynomials are capable tools for approximating continuous functions across various domains of mathematics and engineering.

Exploring Fourier Series

Fourier series play an integral role in understanding periodic functions and serve as a tool for the Weierstrass Approximation Theorem. A Fourier series is a way to represent a periodic function using a sum of sine and cosine terms. Essentially, it decomposes any periodic function into a sum of simple oscillating functions, greatly simplifying the study of such functions.

Mathematically, the Fourier series of a function \( f(x) \) is given by:
\[ f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty [a_n \cos(nx) + b_n \sin(nx)] \]
where \( a_n \) and \( b_n \) are Fourier coefficients, calculated based on the integral of \( f(x) \) with the appropriate sine and cosine functions over one period of the function.

A critical aspect of Fourier series in the context of the Weierstrass Approximation Theorem is that they allow us to approximate any reasonable function as closely as desired by truncating the series after a finite number of terms. Each truncated series is, in fact, a trigonometric polynomial. This is directly connected to the theorem's assertion of the density of trigonometric polynomials: the Fourier series confirms that by taking sufficiently many terms, a trigonometric polynomial can provide an approximation to a continuous function with arbitrary precision.