Convergence of General Trigonometric Series

0. Preliminaries.

Definition. A general trigonometric series is a formal mathematical expression written as

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

where $\{c_n{\}}_{n\in \mathbb{Z}}$ is a sequence of complex numbers, called the coefficients, and $x$ is a real variable. The symbol ~ is used to denote that this is a formal series, and we make no initial assumptions about its convergence or its relationship to any function $f$.

Definition. The N-th symmetric partial sum of the series is the function $S_N:\mathbb{R}\to \mathbb{C}$ defined by the finite sum

$$S_N(x)=\sum _{k=-N}^N{c}_k{e}^{ikx}.$$

The series is said to converge pointwise at a specific value $x$ if the sequence of complex numbers $\{S_N(x){\}}_{N\ge 0}$ has a well-defined limit as $N\to \infty$.

1. Necessary Conditions and Divergence.

Before establishing conditions that guarantee convergence, we identify the most basic prerequisite. A series cannot converge if its individual terms do not shrink to nothing.

Theorem. If the series $\sum c_n{e}^{inx}$ converges at a point $x$, then the terms corresponding to indices $N$ and $-N$ must jointly vanish as $N\to \infty$. Specifically,

$$\underset{N\to \infty }{\lim }\left(c_N{e}^{iNx}+c_{-N}{e}^{-iNx}\right)=0.$$

Proof. Let $\{S_N(x)\}$ be the sequence of partial sums. The hypothesis that the series converges at $x$ means that ${\lim }_{N\to \infty }{S}_N(x)=L$ for some finite complex number $L$.

1. A convergent sequence is necessarily a Cauchy sequence. This implies that the difference between successive terms of the sequence must approach zero.

   $$\underset{N\to \infty }{\lim }\left(S_N(x)-S_{N-1}(x)\right)=L-L=0.$$

2. We compute this difference explicitly from the definition of the partial sum:

   $$S_N(x)-S_{N-1}(x)=\left(\sum _{k=-N}^N{c}_k{e}^{ikx}\right)-\left(\sum _{k=-(N-1)}^{N-1}{c}_k{e}^{ikx}\right).$$

   All terms except those with index $N$ and $-N$ cancel, leaving

   $$S_N(x)-S_{N-1}(x)=c_N{e}^{iNx}+c_{-N}{e}^{-iNx}.$$

3. Combining these two points yields the stated result. $\square$

Theorem 2 (Necessity of Decaying Coefficients). The condition ${\lim }_{|n|\to \infty }{c}_n=0$ is necessary for the series $\sum c_n{e}^{inx}$ to converge on a set of positive measure.

Proof. Let the series converge on a set $E$ with Lebesgue measure $\mu (E)>0$. We first show the coefficient sequence $\{c_n\}$ is bounded, then show it converges to zero.

Step 1: The sequence $\{c_n\}$ is bounded. Assume for contradiction that $\{c_n\}$ is unbounded. Then there exists a subsequence of indices $\{n_k\}$ such that ${\gamma }_k^2=|c_{n_k}{|}^2+|c_{-n_k}{|}^2\to \infty$. By Theorem 1, $T_k(x)=c_{n_k}{e}^{i{n}_{k}x}+c_{-n_k}{e}^{-i{n}_{k}x}\to 0$ for $x\in E$. By Egorov’s theorem, convergence is uniform on a subset $F\subseteq E$ with $\mu (F)>0$. The normalized functions $g_k(x)=T_k(x)/{\gamma }_k$ also converge to 0 uniformly on $F$. The coefficients of $g_k(x)$, namely $a_k=c_{n_k}/{\gamma }_k$ and $b_k=c_{-n_k}/{\gamma }_k$, satisfy $|a_k{|}^2+|b_k{|}^2=1$. Uniform convergence implies ${\lim }_{k\to \infty }{\int }_F|g_k(x){|}^{2}dx=0$. We expand the integral:

$${\int }_F|g_k(x){|}^{2}dx=(|a_k{|}^2+|b_k{|}^2)\mu (F)+a_k\overline{b_k}{\int }_F{e}^{2i{n}_{k}x}dx+\overline{a_k}{b}_k{\int }_F{e}^{-2i{n}_{k}x}dx$$ $$=\mu (F)+a_k\overline{b_k}{\int }_F{e}^{2i{n}_{k}x}dx+\overline{a_k}{b}_k{\int }_F{e}^{-2i{n}_{k}x}dx$$

The sequences $\{a_k\}$ and $\{b_k\}$ are bounded. By the Riemann-Lebesgue lemma, ${\int }_F{e}^{\pm 2i{n}_{k}x}dx\to 0$. The last two terms thus vanish as $k\to \infty$. Taking the limit yields $0=\mu (F)$, a contradiction. The sequence $\{c_n\}$ must be bounded.

Step 2: The coefficients converge to zero. Assume for contradiction that ${\lim }_{|n|\to \infty }{c}_n\ne 0$. Then ${\mathrm{limsup}}_{|n|\to \infty }(|c_n{|}^2+|c_{-n}{|}^2)>\delta$ for some $\delta >0$. There exists a subsequence $\{n_k\}$ with $|c_{n_k}{|}^2+|c_{-n_k}{|}^2>\delta$. As in Step 1, $T_k(x)\to 0$ uniformly on a set $F$ with $\mu (F)>0$, which implies ${\lim }_{k\to \infty }{\int }_F|T_k(x){|}^{2}dx=0$. The integral is:

$${\int }_F|T_k(x){|}^{2}dx=(|c_{n_k}{|}^2+|c_{-n_k}{|}^2)\mu (F)+c_{n_k}\overline{c_{-n_k}}{\int }_F{e}^{2i{n}_{k}x}dx+\overline{c_{n_k}}{c}_{-n_k}{\int }_F{e}^{-2i{n}_{k}x}dx$$

From Step 1, $\{c_n\}$ is bounded, so $\{c_{n_k}\overline{c_{-n_k}}\}$ is a bounded sequence. The Riemann-Lebesgue lemma ensures the integrals vanish. The last two terms thus tend to zero. Taking the limit gives ${\lim }_{k\to \infty }(|c_{n_k}{|}^2+|c_{-n_k}{|}^2)\mu (F)=0$. Since $\mu (F)>0$, this implies ${\lim }_{k\to \infty }(|c_{n_k}{|}^2+|c_{-n_k}{|}^2)=0$, contradicting that $|c_{n_k}{|}^2+|c_{-n_k}{|}^2>\delta$. Thus, ${\lim }_{|n|\to \infty }{c}_n=0$. $\square$

Theorem (Kolmogorov). There exists a sequence $\{c_n\}$ with $c_n\to 0$ for which the series $\sum c_n{e}^{inx}$ diverges for every $x\in \mathbb{R}$.

Proof. The proof is a landmark result in harmonic analysis that involves the intricate construction of an $L^1$ function whose Fourier series diverges everywhere. This construction is highly advanced and is omitted. The significance of this theorem is profound: it demonstrates that there is no simple condition on the magnitude of the coefficients alone that can guarantee pointwise convergence for a general series. $\square$

2. Absolute and Uniform Convergence.

The strongest and most straightforward type of convergence occurs when the coefficients themselves are absolutely summable. This guarantees excellent behavior for the series.

Theorem (Weierstrass M-Test for Trigonometric Series). If the sequence of coefficients is absolutely summable, meaning ${\sum }_{n=-\infty }^{\infty }|c_n|<\infty$, then the series converges absolutely and uniformly on $\mathbb{R}$ to a function $f(x)$ which is continuous.

Proof. The goal is to show that the sequence of functions $\{S_N(x)\}$ converges uniformly. We work within the space of bounded, continuous functions on $\mathbb{R}$, denoted $C_b(\mathbb{R})$, which is a complete metric space (a Banach space) under the supremum norm, $||g|{|}_{\infty }={\sup }_{x\in \mathbb{R}}|g(x)|$.

1. Cauchy Sequence Argument: We show that $\{S_N\}$ is a Cauchy sequence in this space. Let $M>N\ge 0$. Consider the norm of the difference:

   $$||S_M-S_N|{|}_{\infty }=\underset{x\in \mathbb{R}}{\sup }\left|\sum _{|k|=N+1}^M{c}_k{e}^{ikx}\right|.$$

2. Applying the Triangle Inequality: By the triangle inequality and the fact that $|e^{ikx}|=1$:

   $$\underset{x\in \mathbb{R}}{\sup }\left|\sum _{|k|=N+1}^M{c}_k{e}^{ikx}\right|\le \sum _{|k|=N+1}^M|c_k||e^{ikx}|=\sum _{|k|=N+1}^M|c_k|.$$

3. Using the Hypothesis: The hypothesis is that the series of real numbers $\sum |c_n|$ converges. A convergent series must have a Cauchy sequence of partial sums. This means that for any $ϵ>0$, there exists an integer $N_0$ such that for all $M>N>N_0$, the tail sum is small: ${\sum }_{|k|=N+1}^M|c_k|<ϵ$.

4. Conclusion: Combining these steps, for $M>N>N_0$, we have $||S_M-S_N|{|}_{\infty }<ϵ$. Thus, $\{S_N\}$ is a Cauchy sequence in the Banach space $C_b(\mathbb{R})$. Because the space is complete, the sequence must converge to a limit function $f\in C_b(\mathbb{R})$. The uniform limit of a sequence of continuous functions (each $S_N$ is a finite sum of continuous functions, hence continuous) is itself continuous. $\square$

Theorem (Moment Conditions for Smoothness). Let $k\ge 1$ be an integer. If the “k-th moment” of the coefficients is finite, i.e., ${\sum }_{n=-\infty }^{\infty }|n{|}^k|c_n|<\infty$, then the series converges uniformly to a function $f\in C^k(\mathbb{R})$ (meaning $f$ has $k$ continuous derivatives). Furthermore, these derivatives can be computed by term-by-term differentiation of the series.

Proof. We proceed by induction on $k$.

1. Base Case (k=1): We are given $\sum |n||c_n|<\infty$. Since $|c_n|\le |n||c_n|$ for $|n|\ge 1$, the condition $\sum |c_n|<\infty$ also holds. By the previous theorem, the series for $f(x)$ converges uniformly to a continuous function $f$. Now consider the formal series of derivatives:

   $$g(x)\sim \sum _{n=-\infty }^{\infty }(in)c_n{e}^{inx}.$$

   Let $g_N(x)=S_N^{\prime }(x)$ be its partial sums. The sum of the absolute values of the coefficients of this new series is $\sum |in{c}_n|=\sum |n||c_n|$. By hypothesis, this sum is finite. Applying the Weierstrass M-Test to this new series, we conclude that the functions $\{g_N(x)\}$ converge uniformly to a continuous function $g(x)$. A fundamental theorem of analysis states that if $\{S_N\}$ converges to $f$ (even just pointwise) and the sequence of derivatives $\{S_N^{\prime }\}$ converges uniformly to $g$, then $f$ must be differentiable and $f^{\prime }=g$. Since $g$ is continuous, $f\in C^1(\mathbb{R})$.

2. Inductive Step: Assume the theorem holds for $k-1$. Suppose $\sum |n{|}^k|c_n|<\infty$. Let $g(x)=\sum (in)c_n{e}^{inx}$. The coefficients of this series are $d_n=in{c}_n$. We check the $(k-1)$-th moment condition for this new series:

   $$\sum |n{|}^{k-1}|d_n|=\sum |n{|}^{k-1}|in{c}_n|=\sum |n{|}^k|c_n|.$$

   This is finite by our hypothesis. By the inductive hypothesis, the series for $g(x)$ converges to a function $g\in C^{k-1}(\mathbb{R})$. Since $f^{\prime }=g$, it follows that $f\in C^k(\mathbb{R})$. The proof is complete by induction. $\square$

3. Conditional and Pointwise Convergence.

When absolute convergence fails, the series can still converge if the coefficients decay in a sufficiently structured manner. The key tool here is summation by parts.

Definition. A sequence of complex numbers $\{c_n{\}}_{n\in \mathbb{Z}}$ is of bounded variation if the sum of the magnitudes of its successive differences is finite: ${\sum }_{n=-\infty }^{\infty }|c_n-c_{n+1}|<\infty$.

Theorem (Jordan-Dirichlet Test). If the sequence of coefficients $\{c_n\}$ is of bounded variation and ${\lim }_{|n|\to \infty }{c}_n=0$, then the series converges pointwise for every $x$ which is not an integer multiple of $2\pi$.

Proof. It suffices to prove the convergence of the sum over positive indices, ${\sum }_{n=1}^{\infty }{c}_n{e}^{inx}$, as the argument for negative indices is identical.

1. Summation by Parts (Abel’s Transformation): Let $D_k(x)={\sum }_{j=1}^k{e}^{ijx}$. The formula for summation by parts states:

   $$\sum _{n=1}^N{c}_n{e}^{inx}=c_N{D}_N(x)-\sum _{n=1}^{N-1}(c_{n+1}-c_n)D_n(x).$$

2. Boundedness of the Dirichlet Sums: The sum $D_k(x)$ is a finite geometric series. For $x\notin 2\pi \mathbb{Z}$, $e^{ix}\ne 1$.

   $$|D_k(x)|=\left|e^{ix}\frac{1-e^{ikx}}{1-e^{ix}}\right|=\frac{|1-e^{ikx}|}{|1-e^{ix}|}\le \frac{2}{|1-e^{ix}|}=\frac{1}{|\sin (x/2)|}.$$

   Let $M_x=1/|\sin (x/2)|$. This bound $M_x$ is finite for our chosen $x$ and is independent of $k$.

3. Analyzing the Limit: We take the limit of the summation-by-parts formula as $N\to \infty$.

   • First Term: $|c_N{D}_N(x)|\le |c_N|M_x$. Since $c_N\to 0$ by hypothesis, this term vanishes: ${\lim }_{N\to \infty }{c}_N{D}_N(x)=0$.
   • Second Term: We examine the infinite series ${\sum }_{n=1}^{\infty }(c_{n+1}-c_n)D_n(x)$. We show it converges absolutely: $$\sum _{n=1}^{\infty }|(c_{n+1}-c_n)D_n(x)|\le \sum _{n=1}^{\infty }|c_{n+1}-c_n||D_n(x)|\le M_x\sum _{n=1}^{\infty }|c_{n+1}-c_n|.$$ This final sum is finite by the hypothesis that $\{c_n\}$ is of bounded variation. Since the series of absolute values converges, the series itself converges.

4. Conclusion: Both terms on the right-hand side of the summation-by-parts formula have a well-defined limit. Therefore, the left-hand side must also converge. $\square$

4. L² and Almost Everywhere Convergence.

A paradigm shift in modern analysis was to consider convergence not at every point, but in an average sense, which requires tools from Hilbert space theory.

Theorem (Riesz-Fischer). Let the coefficients be square-summable, i.e., $\{c_n\}\in {\ell }^2(\mathbb{Z})$, meaning ${\sum }_{n=-\infty }^{\infty }|c_n{|}^2<\infty$. Then there exists a unique function $f\in L^2([0,2\pi ])$ such that the partial sums $S_N$ converge to $f$ in the $L^2$ norm.

Proof. The space $L^2([0,2\pi ])$ of square-integrable functions is a complete Hilbert space. The proof shows that $\{S_N\}$ is a Cauchy sequence in this space.

1. Orthogonality: The basis functions $\{e^{inx}{\}}_{n\in \mathbb{Z}}$ are orthogonal over $[0,2\pi ]$, satisfying ${\int }_0^{2\pi }{e}^{ijx}\overline{e^{ikx}}dx=2\pi {\delta }_{jk}$.

2. Computing the Norm of the Difference: For $M>N$, we compute the squared $L^2$-norm of the difference of partial sums:

   $$\begin{array}{rl}||S_M-S_N|{|}_{L^2}^2& ={\int }_0^{2\pi }{\left|\sum _{|k|=N+1}^M{c}_k{e}^{ikx}\right|}^{2}dx\\ & ={\int }_0^{2\pi }\left(\sum _{|j|=N+1}^M{c}_j{e}^{ijx}\right)\left(\sum _{|k|=N+1}^M\overline{c_k}{e}^{-ikx}\right)dx\\ & =\sum _{|j|=N+1}^M\sum _{|k|=N+1}^M{c}_j\overline{c_k}{\int }_0^{2\pi }{e}^{i(j-k)x}dx.\end{array}$$

   Due to orthogonality, the integral is zero unless $j=k$, in which case it is $2\pi$.

3. Result: The expression simplifies to:

   $$||S_M-S_N|{|}_{L^2}^2=2\pi \sum _{|k|=N+1}^M|c_k{|}^2.$$

4. Conclusion: The hypothesis $\sum |c_n{|}^2<\infty$ means that the tail of this real series must go to zero. Thus, for any $ϵ>0$, we can find $N_0$ so that for $M>N>N_0$, $||S_M-S_N|{|}_{L^2}^2<ϵ$. This shows $\{S_N\}$ is a Cauchy sequence in $L^2$. By the completeness of the space, a limit function $f\in L^2$ must exist. $\square$

Theorem (Carleson-Hunt). If $\{c_n\}\in {\ell }^2(\mathbb{Z})$, then the partial sums $S_N(x)$ converge pointwise to the limit function $f(x)$ for almost every $x\in [0,2\pi ]$.

Proof. The proof of this theorem, which settled a 50-year-old conjecture by Lusin, is one of the most complex and profound achievements in 20th-century mathematical analysis. It is far beyond the scope of this summary and is omitted. Its statement, however, is a beautiful bridge between the worlds of average ($L^2$) convergence and pointwise convergence. $\square$