Functions

Abstract

This chapter is about functions: how to construct them and how to approximate a continuous function by more regular functions such as polynomials. It is shown how to construct smooth (infinitely differentiable) functions with specific properties. Bernstein’s construction is used to give a constructive proof of the Weierstrass approximation theorem—uniform approximation of continuous functions by polynomials. The definition of a real analytic function is given and a number of foundational results are proved. Finally, and most importantly, the theory of Fourier series is developed and a proof given that the Fourier series of a periodic piecewise smooth continuous function converges to the function. An analysis is given of Gibb’s phenomenon that occurs when convergence of the Fourier series is not uniform (piecewise smooth functions with jumps). The chapter concludes with a second proof of the infinite product formula for sin x, this time using methods based on Fourier series.

Appendix: Second Weierstrass Approximation Theorem

In this appendix we prove Theorem 5.5.6: every continuous 2π-periodic function on $f: \mathbb{R} \rightarrow \mathbb{R}$ can be uniformly approximated by trigonometric polynomials (the second Weierstrass approximation theorem).

Since f is 2π-periodic, it is enough to show that we can uniformly approximate f by trigonometric polynomials on [−π, π]. We break the proof into a number of lemmas.

Lemma 5.7.1

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is even ( f(−x) = f(x), for all $x \in \mathbb{R}$ ), then we can uniformly approximate f by trigonometric polynomials.

Proof

Since f is even the values of f on [−π, π] are uniquely determined by the values of f on [0, π]. Therefore it suffices to uniformly approximate f on [0, π] by even trigonometric polynomials.

Define g(t) = f(cos⁻¹ t), t ∈ [−1, 1]. Since cos⁻¹: [−1, 1] → [0, π] is continuous, g is continuous on [−1, 1]. By the Weierstrass approximation theorem, we may uniformly approximate g on [−1, 1] by polynomials. That is, given ɛ > 0, there exists a $p \in P(\mathbb{R})$ such that

$$\displaystyle{ \sup _{t\in [-1,1]}\vert g(t) - p(t)\vert <\varepsilon. }$$

(5.8)

Set t = cosx, x ∈ [0, π]. We can rewrite (5.8) as sup _{x ∈ [0, π]} | g(cosx) − p(cosx) | < ɛ. Since g(cosx) = f(cos⁻¹(cosx)) = f(x), we have

$$\displaystyle{\sup _{x\in [0,\pi ]}\vert \,f(x) - p(\cos x)\vert <\varepsilon.}$$

Using standard trigonometric identities it is well-known (and easy) to show that every power of cosx can be written as linear combinations of cosjx, $j \in \mathbb{N}$. Hence p(cosx) can be written as a trigonometric polynomial with no sine terms:

$$\displaystyle{p(\cos x) = a_{0} +\sum _{ j=1}^{n}a_{ j}\cos jx.}$$

This function is even and so we have uniformly approximated f on [0, π] by an even trigonometric polynomial. □

Lemma 5.7.2

If f is even, then f(x)sin² ​x can be uniformly approximated by trigonometric polynomials.

Proof

Using Lemma 5.7.1, we first uniformly approximate f by trigonometric polynomials then we use standard trigonometric identities to obtain the required uniform approximations of f(x)sin²​x by trigonometric polynomials. □

Lemma 5.7.3

If f is odd (f(−x) = −f(x)) then f(x)sinx can be uniformly approximated by trigonometric polynomials.

Proof

Since f is odd, g(x) = f(x)sinx is even and so we may apply Lemma 5.7.1. □

Lemma 5.7.4

Every continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ may be written uniquely as a sum f _e + f _o of even and odd continuous functions. If f is 2π-periodic, so are f _e , f _o .

Proof

Define $f_{\mathrm{e}}(x) = \frac{\,f(x)+f(-x)} {2}$, $f_{\mathrm{o}}(x) = \frac{f(x)-f(-x)} {2}$. □

Lemma 5.7.5

If f is 2π-periodic, then we can uniformly approximate f(x)sin² ​x by trigonometric polynomials.

Proof

Using Lemmas 5.7.4 and 5.7.2, we reduce to the case when f is odd. Now apply Lemma 5.7.3 to f(x)sinx and finally multiply the approximating trigonometric polynomials by sinx and apply the trigonometric identities $\sin x\cos jx = \frac{1} {2}(\sin (\,j + 1)x -\sin (\,j - 1)x)$ to obtain the required uniform approximations to f(x)sin²​x. □

Lemma 5.7.6

If f is 2π-periodic then we can uniformly approximate $f( \frac{\pi }{2} - x)\sin ^{2}\!x$ by trigonometric polynomials.

Proof

Apply Lemma 5.7.4 to $\tilde{f}(x) = f( \frac{\pi }{2} - x)$. □

Proof of Theorem 5.5.6

Taking $y = \frac{\pi } {2} - x$ in Lemma 5.7.6, we see that f(x)cos² x​ can be uniformly approximated by trigonometric polynomials. Hence, by Lemma 5.7.2, f(x)sin²​x + f(x)cos²​x = f(x) can be uniformly approximated by trigonometric polynomials. □