Short Essay 3: A study of the word “female” in Daodejing

May 19, 2022May 23, 2023Michael LiuLeave a comment

“雄兔脚扑朔，雌兔眼迷离。” A depiction of female (ci) and male (xiong) rabbits in the Ballad of Mulan.

The objective of this essay is to conduct an in-depth study of the term “female” as it appears in the Daodejing, with a particular focus on two specific characters – ci 雌 and pin 牝 – that are intrinsically associated with femininity in the text. Although typically used to describe female animals, plants, or feminine objects, an interpretation grounded in Daoist cosmology allows for the extension of the meaning of ci and pin to encompass the human female, in line with the Daoist view of a unitary whole, where humans and other species are contiguous. This usage is consistent with the fundamental Daoist principle that posits humans are not superior to other forms of life. The study will take a three-pronged approach: (1) a review of occurrences of ci and pin in Daodejing; (2) an interpretation of their meaning(s) based on the context of the chapters where they are found; and (3) an exploration of their broader significance within the text.

To initiate the study, I will examine the instances of ci and pin in Daodejing. Primarily, I will rely on D.C. Lau’s translation. ci appeared twice in chapters 10 and 28, respectively: “When the gates of heaven open and shut, are you capable of keeping to the role of the female (ci)?” and “Know the male (xiong 雄), but keep to the role of the female (ci), and be a ravine to the empire.” pin appeared five times in chapters 6, 55, and 61: “The spirit of the valley never dies. This is called the mysterious female (pin). The gateway of the mysterious female (pin) is called the root of heaven and earth”; “It does not know the union of male and female (pin), yet its male member will stir. This is because its virility is at its height”; “A large state is the lower reaches of a river: the place where all the streams of the world unite. In the union of the world, the female (pin) always gets the better of the male (xiong) by stillness. Being still, she takes the lower position”.

Drawing upon the selected passages, it emerges that Laozi’s portrayal of femininity can be categorized into the following dimensions:

1. The Female embodies qualities such as softness, passiveness, yieldingness, humility etc.

In Chapters 6 and 28, Laozi equates “the role of the female” with “the spirit of the valley” and “the ravine of the empire.” The metaphors of the valley and the ravine have a strong connection with the metaphor of water, a prominent motif in Daodejing. Valleys and ravines, being low-lying areas of land often traversed by streams, are embodiments of water, reflecting its inherent “qualities.” As per Laozi, water refrains from entering into contention^[1] (buzheng 不争) with other things and is compatible with all things (wanwu 万物) in nature, which signifies its receptiveness and passivity. In addition, Needham ascribes softness and yieldingness to water, as he points out that water is “yielding and assumes the shape of whatever vessel it is placed in^[2]“ (Needham, 1956). Laozi also asserts that humans, akin to water’s natural inclination to flow from higher to lower regions, should strive to maintain a “low” stance, implying the virtues of humility and discretion. This is exemplified in Chapter 61, where the act of assuming a lower position is associated with both water and pin. Consequently, one might argue that water, ravine, the spirit of the valley, and the female form all metaphorically epitomize attributes such as softness, yieldingness, receptiveness, passivity, humility, etc., which to a certain degree are traits commonly associated with the female gender.

2. The female overcomes the male

In chapter 28, Laozi urges one to “keep to the role of female”; in chapter 61, Laozi asserts that “the female always gets the better of the male.” It is clear that, in Laozi’s perception, the female overcomes the male. By corollary, this also implies that the feminine overcomes the masculine in such a way that: soft overcomes hard, weak overcomes strong^[3], and yieldingness to triumph over dominance. In chapter 61, Laozi takes the diplomacy between a large state and a small state as an example: “the large state, by taking the lower position, annexes the small state.” Laozi underscores that it is only when the larger state exhibits tolerance and submission, as opposed to aggression and militancy, can it garner the goodwill and trust of the smaller state. Hence, those who align their actions with these feminine principles place themselves in a more advantageous position.

3. Associating Female Sexual and Reproductive Functions with Dao

Laozi depicts Dao as the “mother of the world.” In chapter 6, Laozi attributes various characteristics of female, especially the female sexual and reproductive functions, to Dao: “The spirit of the valley never dies. This is called the mysterious female. The gateway of the mysterious female is called the root of heaven and earth. Dimly visible, it seems as if it were there, Yet use will never drain it”.The valley is connected to the mysterious female in two ways: firstly, valleys, brimming with rivers and fertile soils, foster and nourish diverse life forms, such as plants, animals, and trees. This is symbolically akin to the role of the female womb that provides nutrients to the embryo. Secondly, the overall structure of a valley – a moist, concave, and open ‘container’ – bears resemblance to the form of female reproductive organs. Consequently, when Laozi refers to the mysterious female, he may specifically be indicating the female womb. The “gateway” may symbolize the opening of the womb during childbirth, a process through which new life emerges. At a more profound level, Laozi suggests that Dao functions akin to a “womb.” The Dao, the principal concept of Daodejing, embodies these characteristics: (1) Dao is empty^[4] and inexhaustible^[5]; (2) Dao generates the world^[6] (dao-sheng-wan-wu 道生万物). Dao, with its generative emptiness and nourishing capabilities, is the source of all beings. For this reason, Dao is equated with the infinite maternal force accountable for the nourishment and sustenance of all entities. Therefore, it would be justified to perceive the female as an anthropomorphic representation of Dao.

Two additional points of note include the comparative reading of A. Charles Muller’s translation, which offers minor variances that do not significantly impact text interpretation, and the literal interpretation of the “union of male and female (pin)” in chapter 55 as sexual intercourse, which does not necessitate further explanation.

Works Cited

Lao Tzu, Trans. by D.C. Lau (1963). The Tao Te Ching.

Laozi, Trans. by A. Charles Muller (1991). Daode Jing.

Needham, J. (1956). The Water Symbol and the Feminine Symbol . In J. Needham, Science and Civilization in China (pp. 57-61). Cambridge: Cambridge University Press.

Daodejing, chapter 8. ↑
Science and Civilization in China, p.77. ↑
Daodejing, chapters 36 and 78. ↑
Ibid. chapter 4. ↑
Ibid. chapter 5. ↑
Ibid. chapter 42. ↑

Notes on Fourier analysis (part 2)

May 7, 2022May 7, 2022Michael LiuLeave a comment

1.3. Uniform convergence

Continuity is not necessarily preserved under pointwise limits. For example, Suppose that ${f_n:[0,1]\rightarrow \mathbb{R}}$ is defined by ${f_n(x)=x^n}$ . If ${0<x<1}$ , then ${x^n\rightarrow0}$ as ${n\rightarrow\infty}$ . If ${x=1}$ , then ${x^n\rightarrow1}$ as ${n\rightarrow\infty}$ . Thus, ${f_n}$ is continuous on ${[0,1]}$ but the pointwise limit ${f}$ is not (it is discontinuous at 1). On the other hand, uniform convergence of continuous functions does guarantee continuity.

Definition 3.1 Uniform convergence The series of continuous function ${\sum^{\infty}_{n=1} f_n}$ converges uniformly on the closed interval ${[a,b]}$ if the following two conditions are satisfied: First, the series converges to the sum ${S(x)}$ for each ${x}$ value in the interval; Second, given any number ${\epsilon >0}$ , there exists a ${N\in\mathbb{N}}$ such that for all ${m\geq N}$ the partial sums ${S_m=\sum^{m}_{n=1} f_n}$ satisfy ${\mid S(x)-S_m(x) \mid\leq\epsilon}$ for every ${x}$ value in the interval.

The following theorem gives an easily applied condition for determining uniform convergence. It is especially useful in its applicaion to Fourier series.

Theorem 3.2 Weierstrass’ ${M}$ -Test If for all points ${x}$ in ${[a,b]}$ we have ${\mid f_n(x)\mid \leq M_n }$ from a certain ${n=k}$ on, and the series of positive numbers ${\sum^{\infty}_{n=k}M_n}$ converges, then the series ${\sum^{\infty}_{n=1}f_n}$ converges uniformly on ${[a,b]}$ .

Proof For each value ${x}$ in the interval ${[a,b]}$ , since ${\mid f_n(x)\mid \leq M_n }$ and ${\sum^{\infty}_{n=k}M_n}$ converges, then by comparison test, the series ${\sum^{\infty}_{n=k}f_n}$ converges absolutely. Therefore, the series ${\sum^{\infty}_{n=k}f_n}$ converges absolutely for every ${x}$ on the interval ${[a,b]}$ .

Given some ${\epsilon>0}$ , then there exist an ${ \in \mathbb{N}}$ such that ${m>N}$ implies ${\sum^{\infty}_{n=m+1}M_n < \epsilon}$ .

By Triangle inequality, we have for each ${x}$ in ${[a,b]}$

$\displaystyle \lvert S(x)-S_m(x)\rvert \leq \lvert\sum^{\infty}_{n=m+1}f_n(x)\rvert \leq\sum^{\infty}_{n=m+1}\lvert f_n(x)\rvert \leq\sum^{\infty}_{n=m+1}M_n < \epsilon \ \ \ \ \ (62)$

Thus, for all ${m\geq\mathbb{N}}$ ,

$\displaystyle \mid S(x)-S_m(x) \mid < \epsilon \ \ \ \ \ (63)$

for every ${x}$ in ${[a,b]}$ .

${\square}$

In the case of Fourier series, we have the following estimates

$\displaystyle \hspace{0.5cm} \lvert a_n\mathrm{cos}n\theta\rvert\leq \lvert a_n \rvert \hspace{0.5cm} \lvert b_n\mathrm{sin}n\theta\rvert\leq \lvert b_n \rvert \hspace{0.5cm} \lvert c_ne^{in\theta}\rvert\leq \lvert c_n \rvert \ \ \ \ \ (64)$

Hence the Weierstrass’s ${M}$ -test will apply to a Fourier series in trigonometric form if ${\sum^{\infty}_0\lvert a_n \rvert<\infty}$ and ${\sum^{\infty}_1\lvert b_n \rvert<\infty}$ , and to Fourier series in exponential form if ${\sum^{\infty}_{-\infty}\lvert c_n \rvert<\infty}$ .

Another theorem that we will use for the proof of uniform convergence is the Cauchy-Schwarz Inequality.

Theorem 3.3 Cauchy-Schwarz Inequality Let ${ {a_n}^N_{n=1}}$ and ${ {b_n}^N_{n=1}}$ be two finite sets of real numbers. Then,

$\displaystyle (\sum^{N}_{n=1}a_nb_n)^2\leq (\sum^{N}_{n=1}a_n^2)(\sum^{N}_{n=1}b_n^2) \ \ \ \ \ (65)$

When expressed in vector form, the Cauchy-Schwarz inequality says that the dot product of two vectors is bounded by the product of their norms.

Proof Expanding out the brackets and collecting identical terms, we have

$\displaystyle \begin{aligned} \sum^{N}_{i=1}\sum^{N}_{j=1}(a_ib_j-a_jb_i)^2&=\sum^{N}_{i=1}a_i^2\sum^{N}_{j=1}b_j^2+\sum^{N}_{j=1}a_j^2\sum^{N}_{i=1}b_i^2-2\sum^{N}_{i=1}a_ib_i\sum^{N}_{j=1}a_jb_j\\ &= 2(\sum^{N}_{i=1}a_i^2)(\sum^{N}_{i=1}b_i^2)-2 (\sum^{N}_{i=1}a_ib_i)^2 \end{aligned} \ \ \ \ \ (66)$

The left-hand side of the equation is greater than or equal to zero since it is a sum of the squares of real numbers. Thus,

$\displaystyle (\sum^{N}_{i=1}a_i^2)(\sum^{N}_{i=1}b_i^2)\geq(\sum^{N}_{i=1}a_ib_i)^2 \ \ \ \ \ (67)$

${\square}$

The graph of periodic functions that are both continuous and piecewise smooth is a smooth curve except that is can have “corners” where the derivatives jump. The fundamental theorem of calculus,

$\displaystyle f(b)-f(a)=\int^b_a f'(\theta) \mathrm{d}\theta \ \ \ \ \ (68)$

applies to functions ${f}$ that are continuous and piecewise smooth, even though ${f'}$ is defined at the “corners”. To see this, let ${f}$ to be differentiable except at point ${c\in(a,b)}$ , we have

$\displaystyle \begin{aligned} \int^b_a f'(\theta) \mathrm{d}\theta &= \int^b_c f'(\theta) \mathrm{d}\theta+\int^c_a f'(\theta) \mathrm{d}\theta\\ &= (f(b)-f(c))+(f(c)-f(a))\\ &= f(b)-f(a) \end{aligned} \ \ \ \ \ (69)$

We introduce the following theorem as a preliminary step towards the main convergence theorem.

Theorem 3.4 Suppose ${f}$ is ${2\pi}$ -periodic, continuous, and piecewise smooth. Let ${a_n}$ , ${b_n}$ , and ${c_n}$ be the Fourier coefficients of ${f}$ defined in (2.5) and (2.6), and let ${a_n'}$ , ${b_n'}$ , and ${c_n'}$ be the corresponding Fourier coefficients of ${f'}$ . Then

$\displaystyle \hspace{0.5cm} a_n'=nb_n, \hspace{0.5cm} b_n'=-na_n, \hspace{0.5cm} c'_n=inc_n \ \ \ \ \ (70)$

Proof This is a simple matter of integration by parts.

${\square}$

From theorem 3.4 we obtain the following results on differentiation and integration of Fourier series.

Theorem 3.5 Suppose ${f}$ is ${2\pi}$ -periodic, continuous, and piecewise smooth, and suppose also that ${f'}$ is piecewise smooth. If

$\displaystyle \frac{1}{2}a_0+\sum_{n=1}^{\infty}(a_n\mathrm{cos}(n\theta)+b_n\mathrm{sin}(n\theta))=\sum_{-\infty}^{\infty}c_ne^{in\theta} \ \ \ \ \ (71)$

is the Fourier series of ${f(\theta)}$ , then ${f'(\theta)}$ is the sum of the derived series

$\displaystyle \frac{1}{2}a_0+\sum_{n=1}^{\infty}(nb_n\mathrm{cos}(n\theta)-na_n\mathrm{sin}(n\theta))=\sum_{-\infty}^{\infty}inc_ne^{in\theta} \ \ \ \ \ (72)$

for all ${\theta}$ at which ${f'(\theta)}$ exists. At the exceptional points ${f'}$ has jumps, the series converges to ${\frac{1}{2}[f'(\theta_-)+f'(\theta_+)]}$ .

Proof Since ${f'}$ is piecewise smooth, by Theorem 2.6, it is the sum of it Fourier series at every point. By theorem 3.4, the coefficients of ${e^{in\theta}}$ , ${\mathrm{sin}(n\theta)}$ , ${\mathrm{cos}(n\theta)}$ in this series are ${inc_n}$ , ${-na_n}$ , and ${nb_n}$ respectively. Thus theorem 3.5 follows.

${\square}$

Theorem 3.6 Suppose ${f}$ is ${2\pi-}$ periodic and piecewise continuous, with Fourier coefficients ${a_n}$ , ${b_n}$ , ${c_n}$ , and let ${F(\theta)=\int^\theta_{0}f(\phi)\mathrm{d}\phi}$ . If ${c_0(=\frac{1}{2}a_0)=0}$ , then for all ${\theta}$ we have

$\displaystyle F(\theta)=\frac{1}{2}A_0+\sum_{n=1}^{\infty}(\frac{a_n}{n}\mathrm{sin}(n\theta)-\frac{b_n}{n}\mathrm{cos}(n\theta))=C_0+\sum_{n \neq 0}\frac{c_n}{in}e^{in\theta} \ \ \ \ \ (73)$

where the constant term is the mean value of ${F}$ on ${[-\pi,\pi]}$ :

$\displaystyle C_0=\frac{1}{2}A_0=\frac{1}{2\pi}\int^{\pi}_{-\pi}F(\theta)\mathrm{d}\theta \ \ \ \ \ (74)$

The series on the right of (72) is the series obtained by formally integrating the Fourier series of ${f}$ term by term, whether the latter series actually converges or not. If ${c_0 \neq 0}$ , the sum of the series on the right of (72) is ${F(\theta)-c_0\theta}$ .

Proof ${F}$ is continuous and piecewise smooth, being the integral of a piecewise continuous function. Moreover, if ${c_0=0}$ , ${F}$ is ${2\pi-}$ periodic, for

$\displaystyle F(\theta+2\pi)-F(\theta)=\int^{\theta+2\pi}_{\theta} f(\phi)\mathrm{d}\phi=\int^{\pi}_{-\pi} f(\phi)\mathrm{d}\phi=2\pi c_0=0 \ \ \ \ \ (75)$

Hence, by theorem 2.6, ${F(\theta)}$ is the sum of its Fourier series at every ${\theta}$ . But by theorem 3.4 applied to F, the Fourier coefficients ${A_n, B_n}$ , and ${C_n}$ of ${F}$ are related to those of ${f}$ by

$\displaystyle \hspace{0.5cm} A_n'=-\frac{b_n}{n}, \hspace{0.5cm} B_n'=\frac{a_n}{n}, \hspace{0.5cm}C_n=\frac{c_n}{in} \hspace{0.5cm} (n\neq0). \ \ \ \ \ (76)$

The formula (74) for the constant ${C_0}$ or ${\frac{1}{2}A_0}$ is just the usual formula for the zeroth Fourier coefficient of ${F}$ . If ${c\neq0}$ , these arguments can be applied to the function ${f(\theta)-c_0}$ rather than ${f(\theta)}$ , yielding the final assertion.

${\square}$

We will now prove a theorem on uniform convergence of Fourier series.

Theorem 3.7 If ${f}$ is ${2\pi-}$ periodic, continuous, and piecewise smooth, then the Fourier series of ${f}$ converges to ${f}$ absolutely and uniformly on ${\mathbb{R}}$ .

Proof Let ${c_n'}$ denote the Fourier coefficients of ${f'}$ . By theorem 3.4 we know that ${c_n=(in)^-1c_n'}$ for ${n \neq 0}$ , and by (29) (Bessel’s inequality) applied to ${f'}$ ,

$\displaystyle \sum^{\infty}_{-\infty}|c_n'|^2\leq \frac{1}{2\pi}\int^{\pi}_{-\pi}|f'(\theta)|^2\mathrm{d}\theta<\infty \ \ \ \ \ (77)$

Hence, by the Cauchy-Schwartz inequality,

$\displaystyle \sum^{\infty}_{-\infty}|c_n|=\lvert c_0\rvert+\sum_{n\neq0}\lvert \frac{c_n'}{in} \rvert\leq \lvert c_0\rvert+(\sum_{n\neq 0}\frac{1}{n^2})^{\frac{1}{2}}(\sum_{n\neq 0}\lvert c_n'\rvert^2)^{\frac{1}{2}} <\infty \ \ \ \ \ (78)$

since ${\sum_{n\neq 0}(\frac{1}{n^2})=2\sum_{n=1}^{\infty}(\frac{1}{n^2})<\infty}$ . This completes the proof.

${\square}$

Notes on Fourier analysis

May 3, 2022May 25, 2022Michael LiuLeave a comment

1.1. Definition

When solving heat equations with linear and homogeneous boundary conditions, we obtained product solutions in terms of infinite series of sines and cosines. Joseph Fourier developed his ideas on the convergence of these trigonometric series while studying heat flow.

We now study Fourier series of a periodic function. Suppose ${f(\theta)}$ is a function defined on the real line such that ${f(\theta+2\pi)=f(\theta)}$ for all ${\theta}$ , then we say that ${f(\theta)}$ is ${2\pi}$ -periodic. We shall assume that ${f}$ is Riemann integrable on every bounded interval; this will be the case if ${f}$ is bounded and is continuous except at finitely many points in each bounded interval. We allow ${f}$ is be complex-valued (we will see why later). We wish to obtain the following series expansion:

$\displaystyle f(\theta)=\frac{1}{2}a_0+\sum_{n=1}^{\infty}(a_n\mathrm{cos}(n\theta)+b_n\mathrm{cos}(n\theta)) \ \ \ \ \ (1)$

The series expansion of ${f(\theta)}$ can also be written in terms of complex exponential functions ${e^{i\theta}}$ . We use the following properties:

$\displaystyle \mathrm{cos}(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2} \ \ \ \ \ (2)$

$\displaystyle \mathrm{sin}(\theta)=\frac{e^{i\theta}-e^{-i\theta}}{2i} \ \ \ \ \ (3)$

$\displaystyle e^{i\theta}= \mathrm{cos}(\theta)+i\mathrm{sin}(\theta) \ \ \ \ \ (4)$

Equation (1) can be rewritten as

$\displaystyle f(\theta)=\sum_{-\infty}^{\infty}c_ne^{in\theta} \ \ \ \ \ (5)$

where

$\displaystyle c_0=\frac{1}{2}a_0 {\hspace{0.5cm}} c_n=\frac{1}{2}(a_n-ib_n) {\hspace{0.5cm}} c_{-n}=\frac{1}{2}(a_n+ib_n) {\hspace{0.5cm}} n=1,2,3... \ \ \ \ \ (6)$

We can consider (5) as the sum of two infinity series, one going from ${n=0}$ to ${+\infty}$ and one going from ${n=-1}$ to ${-\infty}$ .

Alternatively, we can express the coefficients in the following way:

$\displaystyle a_0=2c_0 {\hspace{0.5cm}} a_n=c_n+c_{-n} {\hspace{0.5cm}} b_n=i(c_n-c_{-n}){\hspace{0.5cm}} {\hspace{0.5cm}} n=1,2,3... \ \ \ \ \ (7)$

How can the coefficient ${c_n}$ be expressed in terms of ${f}$ ? Let’s first assume that the series is term by term integrable, we multiply equation (5) by ${e^{-ik\theta}}$ , ${k}$ is an integer, and integrate both sides from ${-\pi}$ to ${\pi}$ . We obtain

$\displaystyle \int^{\pi}_{-\pi}f(\theta)e^{-ik\theta}\mathrm{d}\theta = \sum_{-\infty}^{\infty}c_n \int^{\pi}_{-\pi}e^{i(n-k)\theta}\mathrm{d}\theta \ \ \ \ \ (8)$

Note that

$\displaystyle \int^{\pi}_{-\pi}e^{i(n-k)\theta}\mathrm{d}\theta=\frac{1}{{i(n-k)}}e^{i(n-k)\theta}\Big|^\pi_{-\pi}=0 {\hspace{0.5cm}} n\neq k \ \ \ \ \ (9)$

$\displaystyle \int^{\pi}_{-\pi}{d}\theta = 2\pi {\hspace{0.5cm}} n= k \ \ \ \ \ (10)$

Hence, the only non-zero term after the integration is the term ${n=k}$ , and thus we have

$\displaystyle \int^{\pi}_{-\pi}f(\theta)e^{-ik\theta}\mathrm{d}\theta=c_k2\pi \ \ \ \ \ (11)$

Relabeling the term ${c_k}$ as ${c_n}$ we get

$\displaystyle c_n=\frac{1}{2\pi}\int^{\pi}_{-\pi}f(\theta)e^{-in\theta}\mathrm{d}\theta \ \ \ \ \ (12)$

This means that we can also find ${a_0}$ , ${a_n}$ and ${b_n}$ easily

$\displaystyle a_0=2c_0=\frac{1}{\pi}\int^{\pi}_{-\pi}f(\theta)\mathrm{d}\theta \ \ \ \ \ (13)$

$\displaystyle a_n=c_n+c_{-n}= \frac{1}{2\pi}\int^{\pi}_{-\pi}f(\theta)(e^{-in\theta}+e^{in\theta})\mathrm{d}\theta = \frac{1}{\pi}\int^{\pi}_{-\pi}f(\theta)\mathrm{cos}(n\theta)\mathrm{d}\theta \ \ \ \ \ (14)$

$\displaystyle b_n=i(c_n-c_{-n})=\frac{i}{2\pi}\int^{\pi}_{-\pi}f(\theta)(e^{-in\theta}-e^{in\theta})\mathrm{d}\theta = \frac{1}{\pi}\int^{\pi}_{-\pi}f(\theta)\mathrm{sin}(n\theta)\mathrm{d}\theta \ \ \ \ \ (15)$

for ${n=1,2,3...}$

To recapitulate: if ${f}$ has a series expansion of the form in (1), and if the series converges decently such that term by term integration is permissible, then the coefficients ${a_n, b_n}$ and ${c_n}$ are given by (14), (15), (12) respectively. But now if ${f}$ is any Riemann-integrable periodic functions, the integrals in (14), (15), and (12) make perfectly good sense, and we use them to define the coefficients ${a_n, b_n}$ and ${c_n}$ . We are able to give the formal definition as follows:

Definition 1.1 Suppose ${f}$ is periodic with period ${2\pi}$ and integrable over ${[-\pi,\pi]}$ . The numbers ${a_n}$ and ${b_n}$ defined in (14) and (15), or the number ${c_n}$ defined in (12), are called the Fourier coefficients of ${f}$ , and the corresponding series

$\displaystyle \frac{1}{2}a_0+\sum_{n=1}^{\infty}(a_n\mathrm{cos}(n\theta)+b_n\mathrm{sin}(n\theta)) {\hspace{0.5cm}} \mathrm{or} {\hspace{0.5cm}} \sum_{-\infty}^{\infty}c_ne^{in\theta} \ \ \ \ \ (16)$

are called the Fourier series of ${f}$ .

We also present the following Lemmas:

Lemma 1.2 With reference to the formulas (14) and (15),

if f is even,

$\displaystyle a_n=\frac{2}{\pi}\int^{\pi}_{0}f(\theta)\mathrm{cos}(n\theta)\mathrm{d}\theta {\hspace{0.5cm}}and {\hspace{0.5cm}} b_n=0 \ \ \ \ \ (17)$

if f is odd,

$\displaystyle b_n=\frac{2}{\pi}\int^{\pi}_{0}f(\theta)\mathrm{sin}(n\theta)\mathrm{d}\theta {\hspace{0.5cm}}and {\hspace{0.5cm}} a_n=0 \ \ \ \ \ (18)$

For Fourier series of a ${2\pi}$ -periodic function, the constant term of series is

$\displaystyle c_0=\frac{1}{2}a_0=\frac{1}{2\pi}\int^{\pi}_{-\pi}f(\theta)\mathrm{d}\theta \ \ \ \ \ (19)$

which is equivalent to the mean value of ${f}$ in the interval ${[-\pi,\pi]}$ . Thus,

Lemma 1.3 The constant term in the Fourier series of a ${2\pi}$ -periodic function ${f}$ is the mean value of ${f}$ on an interval of length ${2\pi}$ .

We now turn to the discussion of periodic functions with an arbitrary period ${2L}$ . Similarly, we obtain the following definition of the Fourier series.

Definition 1.4 For a function ${f}$ with period ${2L}$ the Fourier coefficients of ${f}$ are

$\displaystyle a_n=\frac{1}{L}\int^{L}_{-L}f(x)\mathrm{cos}(\frac{n\pi x}{L})\mathrm{d}x \ \ \ \ \ (20)$

$\displaystyle b_n=\frac{1}{L}\int^{L}_{-L}f(x)\mathrm{sin}(\frac{n\pi x}{L})\mathrm{d}x \ \ \ \ \ (21)$

$\displaystyle c_0=\frac{1}{2L}\int^{L}_{-L}f(x)e^{-i(\frac{n\pi x}{L})}\mathrm{d}x \ \ \ \ \ (22)$

and the corresponding series

$\displaystyle \frac{1}{2}a_0+\sum_{n=1}^{\infty}(a_n\mathrm{cos}(n\frac{\pi x}{L})+b_n\mathrm{sin}(\frac{n\pi x}{L})) {\hspace{0.5cm}} \mathrm{or} {\hspace{0.5cm}} \sum_{-\infty}^{\infty}c_ne^{i\frac{n\pi x}{L}} \ \ \ \ \ (23)$

are called the Fourier series of function ${f}$ .

Figure 1. Graph of partial sums (n=10, 30, 80) of the Fourier series of a square wave.

Definition 1.5 Let ${f}$ be a function defined on the interval ${[-L,L]}$ . The periodic extension of ${f}$ is that function, denoted by ${f_p}$ , which satisfies for ${p=2L}$

$\displaystyle f_p(x+kp)=f(x) \ \ \ \ \ (24)$

for each integer k.

For an even function ${f}$ over ${[-L,L]}$ , the Fourier series for ${f}$ has the form

$\displaystyle f(x)=\frac{1}{2}a_0+\sum_{n=1}^{\infty}a_n\mathrm{cos}(n\frac{\pi x}{L}) \hspace{0.5cm} a_n=\frac{2}{L}\int^{L}_{0}f(x)\mathrm{cos}(\frac{n\pi x}{L})\mathrm{d}x \ \ \ \ \ (25)$

Formula (25) is called the cosine series of ${f}$ over ${[0,L]}$ .

For an odd function ${f}$ over ${[-L,L]}$ , the Fourier series for ${f}$ has the form

$\displaystyle f(x)=\frac{1}{2}a_0+\sum_{n=1}^{\infty}b_n\mathrm{sin}(n\frac{\pi x}{L}) \hspace{0.5cm} a_n=\frac{2}{L}\int^{L}_{0}f(x)\mathrm{sin}(\frac{n\pi x}{L})\mathrm{d}x \ \ \ \ \ (26)$

Formula (26) is called the sine series of ${f}$ over ${[0,L]}$ .

Definition 1.6 Let the function ${f}$ be defined on ${[0,L]}$ . The even extension ${f_e}$ and the odd extension ${f_o}$ of ${f}$ are the following functions

$\displaystyle f_e(x) \left\{ \begin{array}{lr} f(x) \\ f(-x) \\ \end{array} \right. \ \ \ \ \ (27)$

$\displaystyle f_o(x) \left\{ \begin{array}{lr} f(x) \\ -f(-x) \\ \end{array} \right. \ \ \ \ \ (28)$

${\square}$

1.2. A convergence theorem

Now that we have formally introduced the Fourier series. However, one important question remains unresolved: How do we know whether Fourier series converges for all function ${f(x)}$ ? Proving the convergence of the Fourier series is not a simple matter. Firstly, let’s derive an estimate of the Fourier coefficients that will be needed to establish convergence.

Theorem 2.1 (Bessel’s Inequality) If ${f}$ is ${2\pi}$ -periodic and Riemann integrable on ${[-\pi,\pi]}$ , and the Fourier coefficients are defined by (16), then

$\displaystyle \sum^{\infty}_{-\infty}|c_n|^2\leq \frac{1}{2\pi}\int^{\pi}_{-\pi}|f(\theta)|^2\mathrm{d}\theta \ \ \ \ \ (29)$

Proof We use the following property of complex numbers,

$\displaystyle \mid z \mid^2=z\overline{z} \ \ \ \ \ (30)$

and let

$\displaystyle \begin{aligned} \mid f(\theta)-\sum^{N}_{-N}c_ne^{in\theta}. \mid^2 & =( f(\theta)-\sum^{N}_{-N}c_ne^{in\theta})( \overline{f(\theta)}-\sum^{N}_{-N}\overline{c_n}e^{-in\theta}) \\ & = \mid f(\theta) \mid^2 -\sum^{N}_{-N} [f(\theta)\overline{c_n}e^{-in\theta}+\overline{f(\theta)}c_ne^{in\theta}]+\sum^{N}_{-N}c_n\overline{c_n}\\ \end{aligned} \ \ \ \ \ (31)$

Divide both sides of (31) by ${2\pi}$ and integrate from ${-\pi}$ to ${\pi}$ , we obtain the following using formula (12),

$\displaystyle \begin{aligned} \frac{1}{2\pi}\int^{\pi}_{-\pi}\mid f(\theta)-\sum^{N}_{-N}c_ne^{in\theta}. \mid^2 \mathrm{d}\theta & = \frac{1}{2\pi}\int^{\pi}_{-\pi} \mid f(\theta) \mid^2\mathrm{d}\theta -\sum^{N}_{-N} \overline{c_n}c_n+c_n\overline{c_n}+\sum^{N}_{-N}c_n\overline{c_n}\\ & = \frac{1}{2\pi}\int^{\pi}_{-\pi} \mid f(\theta) \mid^2\mathrm{d}\theta -\sum^{N}_{-N} \overline{c_n}c_n\\ & = \frac{1}{2\pi}\int^{\pi}_{-\pi} \mid f(\theta) \mid^2 \mathrm{d}\theta-\sum^{N}_{-N}\mid c_n\mid^2\\ \end{aligned} \ \ \ \ \ (32)$

It is obvious that the left-hand-side of (32) is nonnegative, thus

$\displaystyle \frac{1}{2\pi}\int^{\pi}_{-\pi} \mid f(\theta) \mid^2 \mathrm{d}\theta-\sum^{N}_{-N} \mid c_n\mid^2 \geq 0 \ \ \ \ \ (33)$

Letting ${n\rightarrow \infty}$ , we have

$\displaystyle \frac{1}{2\pi}\int^{\pi}_{-\pi} \mid f(\theta) \mid^2 \mathrm{d}\theta \geq \sum^{\infty}_{-\infty}\mid c_n\mid^2 \ \ \ \ \ (34)$

${\square}$

Bessel’s inequality can also be stated in terms of ${a_n}$ and ${b_n}$ defined by (16). By equation (7), for ${n\geq1}$ we have

$\displaystyle \begin{aligned} \mid a_n \mid^2 + \mid b_n \mid^2 & = a_n\overline{a_n}+b_n\overline{b_n}\\ & = (c_n+c_{-n})(\overline{c_n}+\overline{c_{-n}})+i(c_n-c_{-n})(-i)(\overline{c_n}-\overline{c_{-n}})\\ & = 2c_n\overline{c_n}+2c_{-n}\overline{c_{-n}} \end{aligned} \ \ \ \ \ (35)$

so that

$\displaystyle \mid a_0 \mid^2=4\mid c_0 \mid^2, {\hspace{0.5cm}} \mid a_n \mid^2+\mid b_n \mid^2 = 2(\mid c_n \mid^2 + \mid c_{-n} \mid^2) {\hspace{0.3cm}} \mathrm{for} {\hspace{0.3cm}} n\geq1 \ \ \ \ \ (36)$

Therefore,

$\displaystyle \frac{1}{4} \mid a_0 \mid^2+\frac{1}{2}\sum^{\infty}_{n=1} (\mid a_n \mid^2+\mid b_n \mid^2)= \sum^{\infty}_{-\infty} \overline\mid c_n\mid^2 \leq \frac{1}{2\pi}\int^{\pi}_{-\pi} \mid f(\theta) \mid^2 \mathrm{d}\theta \ \ \ \ \ (37)$

The significance of (37) is that the series ${\sum \mid a_0 \mid^2}$ , ${\sum \mid b_0 \mid^2}$ and ${\sum \mid c_0 \mid^2}$ are all convergent. As a consequence, we obtain the following result.

Corollary 2.2 The Fourier coefficients ${a_n}$ , ${b_n}$ , and ${c_n}$ all tend to ${0}$ as ${n\rightarrow\infty}$ (and as ${n\rightarrow-\infty}$ in the case of ${c_n}$ ).

Proof ${\mid a_n \mid^2}$ , ${\mid b_n \mid^2}$ and ${\mid c_n \mid^2}$ are the ${n}$ th term of convergent series, so they all tend to ${0}$ as ${n\rightarrow\infty}$ ; hence so do ${a_n}$ , ${b_n}$ , and ${c_n}$ .

In fact, corollary 2.2 is a special case of the Riemann-Lebesgue lemma. Next, we define the class of functions with which we shall be working.

Definition 2.3 Suppose ${-\infty<a<b<\infty}$ , we say that a function ${f}$ on the closed interval ${[a,b]}$ is piecewise continuous provided that (i) f is continous on ${[a,b]}$ except perhaps at finitely many points ${x_1,...,x_k}$ ;(ii) at each of the points ${x_1,...,x_k}$ the left-hand side and right-hand side limits of ${f}$ ,

$\displaystyle f(x_j-)=\lim_{h\rightarrow0, h>0}(x_j-h) \ \ \ \ \ (38)$

and

$\displaystyle f(x_j+)=\lim_{h\rightarrow0, h>0}(x_j+h) \ \ \ \ \ (39)$

exists. (If ${a}$ (or ${b}$ ) are one of the exception points ${x_j}$ , then we require only the left-hand (or the right-hand) limit to exist). Thus, we say that ${f}$ is piecewise continuous on the ${[a,b]}$ if f is continuous there except for finitely many finite jump continuities.

Figure 2. A piecewise continuous function.

Definition 2.4 A function ${f}$ on the closed interval ${[a,b]}$ is piecewise smooth if ${f}$ and its first derivative ${f'}$ are both piecewise continuous.

Pictorially, ${f}$ is piecewise smooth if its graph is a smooth curve except for finitely many jumps (where ${f}$ is discontinuous) and corners (where ${f'}$ is discontinuous). We do not allow infinite discontinuities or sharp cusps.

Figure 3. A piecewise smooth function (left) and a function that is not piecewise smooth (right).

A very useful tool in Fourier analysis is the Dirichlet’s kernel for the ${n}$ th partial sum

$\displaystyle S_N^f(\theta)=\frac{1}{2}a_0+\sum_{n=1}^{N}(a_n\mathrm{cos}(n\theta)+b_n\mathrm{cos}(n\theta))=\sum_{-N}^{N}c_ne^{in\theta} \ \ \ \ \ (40)$

of the Fourier series for ${f}$ . Substitute ${c_n}$ as defined in (16) into (31), we get

$\displaystyle S_N^f(\theta)=\sum_{-N}^{N}\frac{1}{2\pi}\int^{\pi}_{-\pi}f(\psi)\mathrm e^{in(\theta-\psi)} {d}\psi \ \ \ \ \ (41)$

Note that we changed the variable of integration from ${\theta}$ to ${\psi}$ for later convenience. We obtain the following

$\displaystyle S_N^f(\theta)= \frac{1}{2\pi}\sum_{-N}^{N}\int^{\pi}_{-\pi}f(\psi) e^{in(\psi-\theta)} \mathrm{d}\psi \ \ \ \ \ (42)$

by replacing ${n}$ with ${-n}$ . The sum is not affected since it ranges from ${-N}$ to ${N}$ . Now, letting ${\phi=\psi-\theta}$ , we have

$\displaystyle S_N^f(\theta)= \frac{1}{2\pi}\sum_{-N}^{N}\int^{\pi}_{-\pi}f(\phi+\theta) e^{in\phi} \mathrm{d}\phi\\ \ \ \ \ \ (43)$

We can write (34) as

$\displaystyle S_N^f(\theta)=\int^{\pi}_{-\pi}f(\phi+\theta)D_N(\phi)\mathrm{d}\phi \hspace{0.5cm} \mathrm{where} \hspace{0.5cm} D_N(\phi)= \frac{1}{2\pi}\sum_{-N}^{N}e^{in\phi} \ \ \ \ \ (44)$

The function ${D_N(\phi)}$ is called the ${N}$ th Dirichlet kernel. We introduce the following properties of the Dirichlet kernal:

${D_N(\phi)=\frac{1}{2\pi}+\frac{1}{\pi}\sum^{N}_{n=1}\mathrm{cos}(n\theta)}$
${D_N(\phi)=\frac{1}{2\pi}\frac{\mathrm{sin}(N+\frac{1}{2})\phi}{\mathrm{sin}\frac{1}{2}(\phi)}}$

Proof For (1), we use the property ${e^{ix}+e^{-ix}=\mathrm{cos}(x)}$ to obtain

$\displaystyle \begin{aligned} D_N(\phi)&=\frac{1}{2\pi}\sum^{N}_{-N}e^{in\theta}\\ &= \frac{1}{2\pi}(\sum^{N}_{1}[e^{in\theta}+e^{-in\theta}]+e^{i(0)\theta})\\ &= \frac{1}{\pi}\sum^{N}_{1}\mathrm{cos}(n\theta)+\frac{1}{2\pi} \end{aligned} \ \ \ \ \ (45)$

For (2), we recognize that ${D_N(\phi)}$ is the sum of a finite geometric series:

$\displaystyle D_N(\phi)= \frac{1}{2\pi}e^{-iN\theta}(1+e^{i\theta}+...+e^{i(2N+1)\theta}) \ \ \ \ \ (46)$

Since ${\sum^n_1r^{k}=\frac{r^{n+1}-1}{r-1}}$ , we have

$\displaystyle D_N(\phi)= \frac{1}{2\pi}e^{-iN\phi}\frac{e^{i(2N+1)\phi}-1}{e^{i\phi}-1}=\frac{1}{2\pi}\frac{e^{i(N+1)\phi}-e^{-iN\phi}}{e^{i\phi}-1} \ \ \ \ \ (47)$

Multiplying top and bottom by ${e^{-\frac{i\theta}{2}}}$ ,

$\displaystyle D_N(\phi)=\frac{1}{2\pi}\frac{e^{i(N+\frac{1}{2})\phi}-e^{-i(N-\frac{1}{2})\phi}}{e^{\frac{i\phi}{2}}-e^{-\frac{i\theta}{2}}} \ \ \ \ \ (48)$

using the property ${e^{ix}-e^{-ix}=\mathrm{sin}(x)}$ , we get

$\displaystyle D_N(\phi)=\frac{1}{2\pi}\frac{\mathrm{sin}(N+\frac{1}{2})\phi}{\mathrm{sin}\frac{1}{2}(\phi)} \ \ \ \ \ (49)$

${\square}$

Figure 4. Graph of the Dirichlet kernel (solid) with N=25 on the interval from -π to π .

Lemma 2.5 For any ${N}$ ,

$\displaystyle \int^{0}_{-\pi}D_N(\theta)d\theta = \int^{\pi}_{0}D_N(\theta)d\theta=\frac{1}{2} \ \ \ \ \ (50)$

Proof We integrate both sides of property (45) from ${0}$ to ${\pi}$ ,

$\displaystyle \begin{aligned} \int^{\pi}_{0}D_N(\theta)d\theta&= \int^{\pi}_{0} \frac{1}{\pi}\sum^{N}_{1}\mathrm{cos}(n\theta)+\frac{1}{2\pi}d\theta\\ &= \bigg[\frac{1}{\pi}\sum^{N}_{1}\frac{1}{n}\mathrm{sin}(n\theta)+\frac{\theta}{2\pi}\bigg]^\pi_0\\ &= \frac{1}{2} \end{aligned} \ \ \ \ \ (51)$

${\square}$

We are now ready to present the main convergence theorem. It says that the Fourier series of a function ${f}$ that is piecewise smooth on every bounded interval ${[a,b]}$ converges pointwise to ${f}$ , provided that we redefine ${f}$ at its points of discontinuities to be the average of its left and right hand limits.

Theorem 2.6 If ${f}$ is ${2\pi}$ periodic and piecewise smooth on ${\mathbb{R}}$ , and ${S^{f}_N}$ is defined by (40), then

$\displaystyle \lim_{n\rightarrow\infty} S^{f}_N(\theta)=\frac{1}{2}[f(\theta-)+f(\theta+)] \ \ \ \ \ (52)$

for every ${theta}$ . In particular, ${\lim_{N \rightarrow \infty} S^{f}_N(\theta) = f(\theta)}$ for every ${\theta}$ at which ${f}$ is continuous.

Proof By formula (50), we have

$\displaystyle \frac{1}{2}(f(\theta-))= \int^{0}_{-\pi}f(\theta-)D_N(\phi)d\phi \ \ \ \ \ (53)$

$\displaystyle \frac{1}{2}(f(\theta+))= \int^{\pi}_{0}f(\theta+)D_N(\phi)d\phi \ \ \ \ \ (54)$

and hence by equation (44)

$\displaystyle S^{f}_N(\theta)-\frac{1}{2}[f(\theta-)+f(\theta+)] \ \ \ \ \ (55)$

$\displaystyle \begin{aligned} &=\int^{\pi}_{0}f(\phi+\theta)D_N(\phi)d\phi-\int^{\pi}_{0}f(\theta+)D_N(\phi)d\phi\\ &+\int^{0}_{-\pi}f(\phi+\theta)D_N(\phi)d\phi-\int^{0}_{-\pi}f(\theta-)D_N(\phi)d\phi\\ &=\int^{\pi}_{0}(f(\phi+\theta)-f(\theta+))D_N(\phi)d\phi+\int^{0}_{-\pi}(f(\phi+\theta)-f(\theta-))D_N(\phi)d\phi \end{aligned} \ \ \ \ \ (56)$

We wish to show that for each fixed ${\theta}$ , equation (56) tends to 0 as ${N\rightarrow\infty}$ . By (47) we can write it as

$\displaystyle\frac{1}{2\pi} \int^{\pi}_{-\pi}g(\phi)(e^{i(N+1)\phi}-e^{-iN\phi})d\phi \ \ \ \ \ (57)$

where

$\displaystyle g(\phi)= \frac{f(\phi+\theta)-f(\theta+)}{e^{i\phi}-1}, -\pi<\phi<0 \ \ \ \ \ (58)$

$\displaystyle g(\phi)= \frac{f(\phi+\theta)-f(\theta-)}{e^{i\phi}-1}, 0<\phi<\pi \ \ \ \ \ (59)$

We see that ${g(\phi)}$ is a well-behaved function except when near ${\phi=0}$ . However, since ${\lim_{\phi\rightarrow0} f(\phi+\theta)-f(\theta-)=0}$ and ${\lim_{\phi\rightarrow0} e^{i\phi}-1=0}$ , by L’Hôpital’s rule,

$\displaystyle \lim_{\phi\rightarrow0} g(\phi) = \lim_{\phi\rightarrow0} \frac{f(\phi+\theta)-f(\theta-)}{e^{i\phi}-1} = \lim_{\phi\rightarrow0}\frac{f'(\phi+\theta)}{ie^{i\phi}}=\frac{f'(\theta)}{i} \ \ \ \ \ (60)$

Likewise, ${g(\phi)}$ approaches ${\frac{f'(\theta)}{i}}$ as ${\phi}$ approaches ${0}$ from the right. Thus, ${g(\phi)}$ is piecewise continuous on ${[-\pi,\pi]}$ . By Bessel’s inequality, the Fourier coefficient

$\displaystyle c_n=\frac{1}{2\pi}\int^{\pi}_{-\pi}g(\phi)e^{-in\theta}\mathrm{d}\phi \ \ \ \ \ (61)$

tends to 0 as ${n\rightarrow\pm\infty}$ . However, we know that formula (57) is just ${C_{-(N+1)}-C_N}$ . Thus, equation (55) tends to ${0}$ as ${n\rightarrow\pm\infty}$ . This is what we needed to show.

${\square}$

Theorem 2.6 says that the Fourier series of a ${2\pi}$ -periodic piecewise smooth function ${f}$ converges to ${f}$ everywhere, provided that ${f}$ is (re)defined at each of its points of discontinuity to be the average of its left- and right-hand limits there. With this understanding, we have the following uniqueness theorem.

Corollary 2.7 If ${f}$ and ${g}$ are 2 ${\pi}$ -periodic and piecewise smooth, and ${f}$ and ${g}$ have the same Fourier coefficients, then ${f=g}$ .

Proof ${f}$ and ${g}$ are both the sums of the same Fourier series.

${\square}$