Fourier Series

Abstract

Our approach to Fourier series is based on some rudimentary facts about linear spaces equipped with an inner product. Our approach to pointwise convergence is based on Dini’s criterion. We discuss uniform convergence and Cesàro summability of Fourier series. We also show the Fourier series of a Riemann integrable function convergences in the mean. We establish Weyl’s criterion for uniform distribution of sequences. As an application, we establish the uniform distribution in the unit interval of the fractional parts of the integer multiples of an irrational number.

Appendices

Problems

1.1 Problems for Sect. 12.1

• 1. Let $f(x):=x$ on the interval $[0,1].$ Calculate $\widehat {f}(k).$

• 2. Same as the previous problem, but for

  $$ f(x):=\begin{cases} 1 & \text{when }0<x<1/2\\ 0 & \text{when }1/2<x<1 \end{cases}. $$

1.2 Problems for Sect. 12.2

• 1. State the triangle inequality and the Cauchy–Schwarz inequality in the case of $\mathbb {C}^{d}.$

• 2. If $x_{k}>0$ for all $k$ and $\sum _{k=1}^{\infty }x_{k}$ is convergent, then $\sum _{k=1}^{\infty }x_{k}^{2}$ is convergent.

1.3 Problems for Sect. 12.3

• 1. Show that $S_{N}(af+bg)=aS_{N}f+bS_{N}g$ for any integrable functions $f,g$ and any constants $a,b.$

• 2. Calculate $D_{N}\left (\frac {1}{2}\right )$ and $D_{N}\left (\frac {1}{4}\right ).$

1.4 Problems for Sect. 12.4

• 1. Suppose $f$ is integrable and $g$ is determined by (12.23). If $f$ has a jump discontinuity at $x_{0},$ then $g$ is not integrable.

• 2. (Riemann Localization) If $f$ is integrable on $[0,1]$ and $f(t)=0$ for all $t$ in $(a,b),$ then $S_{N}f(t)\to 0$ for all $t\in (a,b).$

• 3. If two continuous functions are equal except possibly at one point in an interval, then they are equal everywhere on that interval.

• 4. Let $f(x):=\begin {cases} -1 & \text {for }-\frac {1}{2}\leq x<0\\ 1 & \text {for }0\leq x<\frac {1}{2} \end {cases}.$

  • a. Find $S_{N}f(x).$

  • b. Why does $S_{N}f$ not converge uniformly to $f.$

  • c. For which $-\frac {1}{2}\leq x<\frac {1}{2}$ does $S_{N}f(x)$ converge to $f(x)?$

1.5 Problems for Sect. 12.5

• 1. If $a_{k}\to a$ as $n\to \infty ,$ then $\frac {1}{N}\sum _{k=1}^{N}a_{k}\to a$ as $N\to \infty .$

• 2. If $a_{k}=(-1)^{k},$ then $(a_{k})$ is divergent and the sequence $\left (\frac {1}{N}\sum _{k=1}^{N}a_{k}\right )$ is convergent.

• 3. Find $a_{N,k}$ such that $\sigma _{N}f\left (x\right )=\sum _{k=-N}^{N}a_{N,k}e^{i2\pi kx}.$

• 4. Show, if $f$ is continuous and periodic, then $f$ is uniformly continuous on $\mathbb {R}.$

• 5. Show, if $f$ is continuous and periodic, then $f$ is bounded.

1.6 Problems for Sect. 12.6

• 1. Prove $(\{\sin (k)\})$ is dense in $[0,1],$ but not uniformly distributed in $[0,1].$

• 2. $\left (\left \{ \sqrt {2}^{k}\right \} \right )$ is not uniformly distributed in $[0,1].$ [Hint: Since $\left \{ \sqrt {2}^{k}\right \} =0$ when $k$ is even, this follows from the definition of uniform distribution.]

• 3. If $\alpha =\frac {1+\sqrt {5}}{2},$ then $\left (\left \{ \alpha ^{k}\right \} \right )$ is not uniformly distributed. [Hint: If $\beta :=\frac {1-\sqrt {5}}{2}$ and $f_{k}:=\alpha ^{k}+\beta ^{k},$ then $f_{0}=2,$ $f_{1}=1,$ and $f_{k+2}=f_{k+1}+f_{k}.$ In particular, $f_{k}$ is an integer $\geq 2$ for all $k\geq 2.$ Hence, $\alpha ^{2k+1}=f_{2k+1}+|\beta |^{2k+1}$ implies $\left \{ \alpha ^{2k+1}\right \} =|\beta |^{2k+1}\to 0$ as $k\to \infty .$ And $\alpha ^{2k}=f_{2k}-|\beta |^{2k}$ implies $\left \{ \alpha ^{2k}\right \} =1-|\beta |^{2k}\to 1$ as $k\to \infty .$ Consequently, $\lim _{N\to \infty }\frac {\#\{k\in \mathbb {N}\mid k\leq N\text { and }\frac {1}{3}<\left \{ \alpha ^{k}\right \} <\frac {2}{3}\}}{N}=0.$]

• 4. Replacing the closed intervals in the definition of uniformly distributed by open intervals gives an equivalent concept.

• 5. $\left (\left \{ \sqrt {k}\right \} \right )$ is uniformly distributed in $[0,1].$

• 6. Let $\gamma _{k}:=\left \{ \sqrt {k}\right \} .$ By uniform distribution there is a subsequence $\left (\gamma _{n_{k}}\right )$ of the sequence $\left (\gamma _{k}\right )$ such that $\gamma _{n_{k}}\to 1.$ Construct such a subsequence.

• 7. Is $\gamma _{k}:=\left \{ \sqrt [3]{k}\right \}$ uniformly distributed?

• 8. For which $x\in \mathbb {R}$ is $\gamma _{k}:=\left \{ \sqrt [x]{k}\right \}$ uniformly distributed?

1.7 Problems for Sect. 12.7

• 1. Let $f(x):=x$ on $[0,1].$ Calculate both sides of Parseval’s Identity.

• 2. Same as the previous problem, but with $f$ replaced by

  $$ g(x):=\begin{cases} 0 & \text{when }-\frac{1}{2}<x<0\\ 1 & \text{when }0<x<\frac{1}{2} \end{cases}. $$

• 3. Calculate both sides of Plancherel’s Formula, if $f(x):=x$ on $[0,1]$ and

  $$ g(x):=\begin{cases} 0 & \text{when }-\frac{1}{2}<x<0\\ 1 & \text{when }0<x<\frac{1}{2} \end{cases}. $$

• 4. Let $a$ be a complex number and let $f,g$ be bounded functions on $\left [0,1\right ].$ Prove that $\left \Vert af\right \Vert _{\infty }=\left |a\right |\left \Vert f\right \Vert _{\infty }$ and $\left \Vert f+g\right \Vert _{\infty }\leq \left \Vert f\right \Vert _{\infty }+\left \Vert g\right \Vert _{\infty }.$

• 5. Let $f_{k},$ $k\in \mathbb {N},$ and $f$ be bounded functions. Prove $f_{k}$ converges uniformly to $f$ on $\left [0,1\right ]$ iff $\left \Vert f-f_{k}\right \Vert _{\infty }\to 0$ as $k\to \infty .$

• 6. Let $f$ be integrable.

  • a. Why is $\widehat {f}(k)=1$ for $k\geq 1$ not possible?

  • b. Why is $\widehat {f}(k)=\frac {1}{\sqrt {k}}$ for $k\geq 1$ not possible?

Solutions and Hints for the Exercises

Exercise 12.2.2. $\int _{0}^{1}|f|=0$ implies $\int _{0}^{1}\left |f\overline {g}\right |=0,$ since $g$ is bounded. But $\int _{0}^{1}\left |f\right |>0$ implies $\int _{0}^{1}\left |f\right |^{2}>0.$ One way to see this is to use that, if $\sum _{k=1}^{n}m_{k}\chi _{\left ]x_{k-1},x_{k}\right [}$ is a lower step function for $|f|,$ then $\sum _{k=1}^{n}m_{k}^{2}\chi _{\left ]x_{k-1},x_{k}\right [}$ is a lower step function for $|f|^{2}.$

Exercise 12.3.3 Since $f$ has period one we may assume $0\leq a<1.$ Since $f$ has period one, it follows by substitution that $\int _{1}^{a+1}f=\int _{0}^{a}f.$

Exercise 12.3.5 Evaluate the geometric series used to define $D_{N}.$

Exercise 12.4.2 Let $\varepsilon>0$ be given. Let $M$ be an upper bound for $|g|.$ Pick $0<\delta <1/2$ such that $4M\delta <\varepsilon /2.$ Then $g$ is integrable on the interval $I_{+}:=[\delta ,1/2]$ because it is a product of two integrable functions this interval. Hence we can find upper and lower step functions $s_{+},S_{+}$ for $g$ restricted to $I_{+}$ such that $\int _{\delta }^{1/2}S_{+}-\int _{\delta }^{1/2}s_{+}<\varepsilon /4.$ Similarly, we can find upper and lower step functions $s_{-},S_{-}$ for $g$ restricted to $I_{-}:=[-1/2,-\delta ]$ such that $\int _{\delta }^{1/2}S_{-}-\int _{\delta }^{1/2}s_{-}<\varepsilon /4.$ Let $S_{0}(t)=M$ and $s_{0}(t)=-M$ for $t$ in the interval $I_{0}:=[-\delta ,\delta ].$ Combining $s_{-},$ $s_{0},$ and $s_{+}$ we get an upper step function $s$ for $g$ on $[-1/2,1/2].$ Similarly, combining $s_{-},$ $s_{0},$ and $s_{+}$ we get a lower step function $s$ for $g$ on $[-1/2,1/2].$ By construction $\int _{-1/2}^{1/2}S-\int _{-1/2}^{1/2}s<\varepsilon .$

Exercise 12.5.2 Since $\int _{-1/2}^{1/2}D_{n}=1$ for all $n\geq 1,$ this follows from the definition of $K_{N}.$

Exercise 12.5.4 Since $t\to \sin (\pi t)$ is increasing on the interval $[0,1],$ we have

$$ \begin{aligned} K_{N}(t)\leq\frac{1}{N+1}\left(\frac{1}{\sin(\pi t)}\right)^{2}\leq\frac{1}{N+1}\left(\frac{1}{\sin(\pi\delta)}\right)^{2} \end{aligned} $$

for $\delta \leq |t|\leq 1/2.$

Exercise 12.5.6 Imitating the proof of the Approximate Identity Lemma.

Exercise 12.6.3 (12.26) implies (12.25) because any characteristic function is a step function.

Since any step function is a linear combination of characteristic functions the converse follows from theorems about linear combinations of limits.

Exercise 12.7.5 A consequence of the Pythagorean Theorem and Theorem 12.7.3.

Exercise 12.7.8 Expand the right-hand side of the Polarization Identity using $\|f\|^{2}=\langle f\mid f\rangle$ and equation (12.10).

Exercise 12.7.9 Apply the Polarization Identity and Parseval’s Identity.