Uniqueness of coefficients in infinite sin series $a(x) = \sum\limits_{n=1}^{\infty} a_{i} \sin(n x) =0 $

Question

if we can say

$$a(x) = \sum\limits_{n=1}^{\infty} a_{n} \sin(n x) =0, \forall x$$

does this generally imply that all constant coefficients $a_i$ should be zero, or can I construct any number of solutions that may provide the right cancellations to make it zero?

My intuition for superposition of different modes of $\sin(nx)$ is that if $a_i$ is always same-signed (I might be able to show this in my application), then this should be unique, but if each coefficient can have a different sign then it could be anything.

I was looking for someone to confirm or deny what I was thinking, and maybe point me towards more mathematical ideas about why that is so.

Answer 1

Note that $$\int_0^{2\pi}\sin(m\theta)\sin(n\theta)\,d\theta=\frac12\int_0^{2\pi}\cos((n-m)\theta) - \cos((n+m)\theta)\,d\theta = 0$$if $m\ne \pm n$. And when $m = n$, the integral is $\pi$.

Now, what is $\int_0^{2\pi}a(x)\sin(mx)\,dx$?

Answer 2

Let me expand the comments in an answer: there are two concepts here - trigonometric series and Fourier series:

Let $S$ be $\frac{1}{2}a_0+\sum_{n \geq 1}{(a_n\cos(nx) + b_n\sin(nx))}$, where $a_n, b_n$ are complex numbers; then S is called a trigonometric series and the Riemann theory of such series tries to associate to $S$ some meaning and deal with issues of unicity of the meaning etc.

IF $a_n, b_n$ come from an absolutely integrable (Lebesgue) periodic function $f$ on $[0, 2\pi]$ by the usual formulas $a_n = \frac{1}{\pi}\int_{0}^{2\pi} f(x)\cos(nx)dx$ and $b_n = \frac{1}{\pi}\int_{0}^{2\pi} f(x)\sin(nx)dx$, S is called a Fourier series and it satisfies some properties like $a_n, b_n$ converge to zero (Riemann-Lebesgue lemma), $S$ is Cesaro (also called Feijer in this context) and Abel (also called radial) summable to $f$ though it may not be convergent anywhere to $f$ (Kolmogorov 1920's or so), while if $f$ is continuos, a famous theorem of Carleson (1960's) shows that $S$ converges a.e to $f$ (while a classic theorem of Feijer (1910's) shows that $S$ is uniformly Cesaro (and Abel) summable to $f$.

The distinction between trigonometric and Fourier series is non-trivial even if we are given that $a_n, b_n$ converge to zero in a positive decreasing manner as the series $\sum_{n \geq 2}\frac{\sin(nx)}{\log n}$ is convergent everywhere, uniformly convergent on any compact set that doesn't contain an integral multiple of $2\pi$ including zero, so its sum is a continuous function except at $0$ and integral multiples of $2\pi$, but it is NOT a Fourier series; on the other hand its conjugate cosine sum $\sum_{n \geq 2}\frac{\cos(nx)}{\log n}$ is uniformly convergent on compact sets outside integral multiples of $2\pi$, $\infty$ at such and its sum is integrable so the cosine series IS a Fourier series.

In general, if $S$ can be associated to a function $f$ in a strong enough manner (absolute convergence, uniform convergence, $L^p$ convergence, $p \geq 1$, other more esoteric forms of convergence) then $S$ is the Fourier series of $f$, so people generally study only Fourier series for that reason but as noted pointwise convergence or a.e uniform convergence are not good enough.

As unicity goes: for Fourier series a.e convergence is enough to give it - so if we know apriori that a trigonometric series comes from an integrable function by the formulas above and the series converges a.e to $0$, all the coefficients are $0$. However the proof requires work (follows from Feijer theorem) as simple integration of $S$ against $\cos(nx), \sin(nx)$ and the use of the ortogonality relations for those to conclude is not justified unless we have strong assumptions on $S$ like uniformly bounded partial sums a.e etc

However for a general trigonometric series while convergence on a set of non-zero measure implies $a_n, b_n$ going to zero (Cantor-Lebesgue lemma), the rest of the issues are considerably more complicated and they are treated by what is called the Riemann theory of trigonometric series which was at its apogee in the 1920's when the Russian school led by N Bary obtained almost definitive results regarding unicity and the like; in essence convergence everywhere or even on a set with countable complement implies unicity (so indeed the answer to the original question is true) but the proof is complicated (though known from Riemann), while for more general sets of zero measure, convergence a.e outside them may or may not imply unicity and while a full classification is not quite known as the property of such to be "unicity sets" seem to depend on arithmetic not topological properties, lots of results are known. However simple summability (Abel, Cesaro) etc is not very useful here as for example the Poisson kernel is Abel summable to $0$ everywhere except at $1$ but its coefficients definitely do not go to zero at infinity.