Fourier Series of Functions: A Comprehensive Analysis of Convergence and Smoothness

0. From Functions to Series: The Fundamental Definition.

Our inquiry begins by defining a canonical map from a function to a trigonometric series. This definition is not arbitrary; it is derived by assuming a “perfect” series representation exists and then using the mathematical properties of the basis functions to uniquely determine the coefficients.

Theorem (Uniqueness of Coefficients for Uniformly Convergent Series).

Let $f:\mathbb{R}\to \mathbb{C}$ be a $2\pi$-periodic function. If $f(x)$ is the sum of a uniformly convergent trigonometric series

$$f(x)=\sum _{n=-\infty }^{\infty }{c}_n{e}^{inx}$$

then the coefficients $c_n$ are uniquely determined for each $n\in \mathbb{Z}$ by the formula

$$c_n=\frac{1}{2\pi }{\int }_0^{2\pi }f(x)e^{-inx}dx$$

Proof. The basis functions $\{e^{inx}{\}}_{n\in \mathbb{Z}}$ are orthogonal on the interval $[0,2\pi ]$. For any integers $n,k$, this property is expressed as:

$${\int }_0^{2\pi }{e}^{inx}{e}^{-ikx}dx={\int }_0^{2\pi }{e}^{i(n-k)x}dx=2\pi {\delta }_{nk}$$

where ${\delta }_{nk}$ is the Kronecker delta. To find the coefficient $c_k$ for a fixed integer $k$, multiply the series representation of $f(x)$ by $e^{-ikx}$ and integrate over $[0,2\pi ]$. The uniform convergence of the series permits the interchange of summation and integration.

$$\begin{array}{rl}{\int }_0^{2\pi }f(x)e^{-ikx}dx& ={\int }_0^{2\pi }\left(\sum _{n=-\infty }^{\infty }{c}_n{e}^{inx}\right)e^{-ikx}dx\\ & =\sum _{n=-\infty }^{\infty }{c}_n\left({\int }_0^{2\pi }{e}^{inx}{e}^{-ikx}dx\right)\\ & =\sum _{n=-\infty }^{\infty }{c}_n(2\pi {\delta }_{nk})\\ & =c_k\cdot 2\pi \end{array}$$

Solving for $c_k$ yields the stated formula. $\square$

Definition (Fourier Coefficients and Fourier Series).

Let $f:\mathbb{R}\to \mathbb{C}$ be a $2\pi$-periodic function that is integrable on $[0,2\pi ]$, i.e., $f\in L^1([0,2\pi ])$. The Fourier coefficients of $f$ are the sequence of complex numbers $\{c_n[f]{\}}_{n\in \mathbb{Z}}$ defined by

$$c_n[f]=\frac{1}{2\pi }{\int }_0^{2\pi }f(x)e^{-inx}dx$$

The Fourier series of $f$, denoted $S[f](x)$, is the trigonometric series constructed with these coefficients:

$$S[f](x)=\sum _{n=-\infty }^{\infty }{c}_n[f]e^{inx}$$

1. Convergence for $L^2$ Functions (Square-Integrable).

The space $L^2([0,2\pi ])$ consists of functions whose squared magnitude is integrable, representing functions of finite “energy”.

A. Convergence TO the Original Function.

Theorem ($L^2$ and Pointwise Convergence for $L^2$ Functions). Let $f\in L^2([0,2\pi ])$. Then:

1. The Fourier series $S[f]$ converges to $f$ in the $L^2$ norm.
2. The Fourier series $S[f](x)$ converges to $f(x)$ for almost every $x\in [0,2\pi ]$.

Proof.

1. $L^2$ Convergence: This is the Riesz-Fischer Theorem. It is a cornerstone of the Hilbert space theory of Fourier analysis, proving that the partial sums $S_N[f]$ form a Cauchy sequence in the complete space $L^2([0,2\pi ])$ which converges to $f$. Its proof is foundational to functional analysis and is stated here without derivation.

2. Pointwise Convergence: This proof perfectly links the function space to our theorems from Lecture 1. Analyze the Coefficients: A direct consequence of the $L^2$ theory is Parseval’s Identity: ${\sum }_{n=-\infty }^{\infty }|c_n[f]{|}^2=\frac{1}{2\pi }{\int }_0^{2\pi }|f(x){|}^{2}dx$. Since $f\in L^2$, the integral on the right is finite by definition. Therefore, the sum of the squared magnitudes of the coefficients is finite. This proves that the sequence of Fourier coefficients $\{c_n[f]\}$ is a member of the space ${\ell }^2(\mathbb{Z})$.

   • Apply Lecture 1 Theorem: Our “series-first” lecture presented Carleson’s Theorem, a deep result stating that if a sequence $\{c_n\}$ is in ${\ell }^2$, its trigonometric series $\sum c_n{e}^{inx}$ converges pointwise for almost every $x$. * Conclusion: Since the coefficients of our $L^2$ function belong to ${\ell }^2$, Carleson’s theorem directly applies and guarantees that $S[f](x)$ converges pointwise almost everywhere to $f(x)$. $\square$

B. Smoothness of the Resulting Series.

The resulting function defined by the sum of the series, $g(x)=S[f](x)$, is precisely $f(x)$ (almost everywhere). Therefore, the sum function $g$ belongs to the same space as the original function, $L^2$. There is no inherent gain in regularity.

2. Convergence for C¹ Functions (Continuously Differentiable).

Requiring one degree of smoothness—a continuous derivative—radically improves convergence.

A. Convergence TO the Original Function.

Theorem (Absolute and Uniform Convergence for C¹ Functions). Let $f\in C^1([0,2\pi ])$. Its Fourier series $S[f]$ converges absolutely and uniformly to $f(x)$.

Proof. The strategy is to show that the coefficients $\{c_n[f]\}$ are absolutely summable ($\in {\ell }^1$). We then invoke the strongest convergence theorem from Lecture 1.

1. Relate Coefficients of f and f’: Using integration by parts for $n\ne 0$:

   $$c_n[f^{\prime }]=\frac{1}{2\pi }{\int }_0^{2\pi }{f}^{\prime }(x)e^{-inx}dx=\frac{1}{2\pi }\left([f(x)e^{-inx}{]}_0^{2\pi }+in{\int }_0^{2\pi }f(x)e^{-inx}dx\right).$$

   Periodicity of both $f(x)$ and $e^{-inx}$ makes the boundary term zero. The remaining integral is $2\pi c_n[f]$. This gives the crucial decay-rate relationship: $c_n[f]=\frac{c_n[f^{\prime }]}{in}$.

2. Establish Absolute Summability: We must prove $\sum |c_n[f]|<\infty$.

   • Since $f\in C^1$, its derivative $f^{\prime }$ is continuous on $[0,2\pi ]$. Any continuous function on a compact interval is in $L^2$. Thus, $f^{\prime }\in L^2$.
   • This implies the coefficients of $f^{\prime }$ are square-summable: $\{c_n[f^{\prime }]\}\in {\ell }^2$.
   • To bound the sum $\sum |c_n[f]|={\sum }_{n\ne 0}\frac{1}{|n|}|c_n[f^{\prime }]|$, we use the Cauchy-Schwarz inequality: $(\sum a_k{b}_k{)}^2\le (\sum a_k^2)(\sum b_k^2)$. $${\left(\sum _{n\ne 0}\frac{1}{|n|}|c_n[f^{\prime }]|\right)}^2\le \left(\sum _{n\ne 0}\frac{1}{n^2}\right)\left(\sum _{n\ne 0}|c_n[f^{\prime }]{|}^2\right).$$
   • The first bracket is a convergent p-series ($=2\cdot {\pi }^2/6$). The second bracket converges because $\{c_n[f^{\prime }]\}\in {\ell }^2$. The product is finite, proving that $\sum |c_n[f]|$ is finite. Thus, $\{c_n[f]\}\in {\ell }^1$.

3. Apply Lecture 1 Theorem: From Lecture 1, the Weierstrass M-Test proves that if a coefficient sequence $\{c_n\}$ is in ${\ell }^1$, the series $\sum c_n{e}^{inx}$ converges absolutely and uniformly.

4. Conclusion: The Fourier series $S[f]$ converges absolutely and uniformly. Since $f$ is continuous and the convergence is uniform, the limit of the series must be $f(x)$. $\square$

B. Smoothness of the Resulting Series.

Since we proved that $S[f]$ converges uniformly to $f$, and $f$ is a $C^1$ function, the resulting sum function is itself in $C^1$.

3. The Hierarchy of Smoothness: $C^k$ and $C^{\infty }$ Functions.

Theorem (Rapid Convergence for $C^k$ Functions). Let $f\in C^k([0,2\pi ])$ for $k\ge 1$. Its Fourier series $S[f]$ converges absolutely and uniformly to $f$, and the coefficients exhibit rapid polynomial decay: $|c_n[f]|=O(|n{|}^{-k})$.

Proof.

1. Analyze the Coefficients: Iterating the differentiation rule $c_n[g^{\prime }]=in\cdot c_n[g]$ for $k$ times yields:

   $$c_n[f]=\frac{c_n[f^{(k)}]}{(in{)}^k},n\ne 0.$$

2. Establish Decay Rate: The derivative $f^{(k)}$ is continuous by hypothesis, hence its Fourier coefficients $\{c_n[f^{(k)}]\}$ are bounded by some constant $M$. This gives a precise bound on the decay of the original coefficients:

   $$|c_n[f]|=\frac{|c_n[f^{(k)}]}{|n{|}^k}\le \frac{M}{|n{|}^k}.$$

3. Show Absolute Summability: The sum $\sum |c_n[f]|\le \sum M/|n{|}^k$. For any $k\ge 2$, this p-series converges rapidly. (The case $k=1$ was handled above). Therefore, for any $f\in C^k$ with $k\ge 1$, its coefficients are in ${\ell }^1$.

4. Apply Lecture 1 Theorem: The Weierstrass M-Test again implies absolute and uniform convergence of $S[f]$ to $f$. $\square$

B. Smoothness of the Resulting Series. The sum function is $f$, which belongs to $C^k$.

4. Comprehensive Summary of Convergence

This table details both the convergence of the series to the original function and the intrinsic smoothness class of the function defined by the series sum itself.

Function Space of $f$	$L^2$ Conv. to $f$	Pointwise Conv. to $f$	Uniform Conv. to $f$	Absolute Conv.	Smoothness Class of the Sum Function $S[f]$
$L^2$ (Square-Integrable)	Yes	Almost Everywhere	Not Guaranteed	Not Guaranteed	$L^2$
$C^0$ (Continuous)	Yes	Not Guaranteed	Not Guaranteed	Not Guaranteed	Not Guaranteed
$C^1$ (Continuously Diff.)	Yes	Yes	Yes	Yes	$C^1$
$C^k$ ($k\ge 2$ times Diff.)	Yes	Yes	Yes	Yes	$C^k$
$C^{\infty }$ (Smooth)	Yes	Yes	Yes	Yes	$C^{\infty }$