The Fourier series of a function is given by $$ \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos n \theta + \sum_{n=1}^\infty b_n \sin n \theta . $$ Here what does the statement " $\sum_{n=1}^\infty b_n \sin n \theta $ is complete" mean? And would you tell me how can I prove this statement?
[Math] About completeness of the Fourier series.
fourier analysisfourier series
Related Solutions
$\newcommand{\Vector}[1]{\mathbf{#1}}\newcommand{\vece}{\Vector{e}}$The linked questions provide good answers, but may be at the technical end of "intuitive". Here's a fast-and-loose conceptual motivation:
If $\bigl(V, \langle\ ,\ \rangle\bigr)$ is an $N$-dimensional real inner product space, and if $\{\vece_{n}\}_{n=1}^{N}$ is an (ordered) orthonormal basis, then an arbitrary vector $v$ in $V$ may be written as a linear combination $$ v = \sum_{n=1}^{N} \langle v, \vece_{n}\rangle \vece_{n}. \tag{1} $$ Indeed, $\{\vece_{n}\}$ is a basis of $V$, so there exist real coefficients $a_{k}$ such that $$ v = \sum_{k=1}^{N} a_{k} \vece_{k}. \tag{2} $$ Taking the inner product of each side with $\vece_{n}$ gives $\langle v, \vece_{n}\rangle = a_{n}$ because the basis $\{\vece_{n}\}$ is orthonormal.
Loosely, one might expect a similar conclusion to hold if $V$ is infinite-dimensional. Getting the definitions and hypotheses right, and proving a version of (1) in this new setting, is why any "honest" answer is bound to be technical. Phrases in quotes below are not mathematically correct, and therefore require careful inspection and/or justification.
Intuitively, let $L > 0$ be real, let $V$ be "the space of real-valued functions" on $[-L, L]$, and define an "inner product" by $$ \langle f, g\rangle = \frac{1}{L} \int_{-L}^{L} f(t) g(t)\, dt. $$ The functions $$ C_{n}(t) = \begin{cases} 1/\sqrt{2}, & n = 0, \\ \cos(n\pi t/L), & n > 0; \end{cases}\qquad S_{n}(t) = \sin(n\pi t/L),\quad n > 0; $$ turn out (by elementary calculus and trigonometry) to form an "orthonormal basis" of $V$.
Loosely, we expect that if $f$ is a function, we can express $f$ as an infinite sum of these basis functions, and the coefficients are the inner products of $f$ with the basis elements, i.e. (for $n > 0$), \begin{align*} a_{0} &= \langle f, 1\rangle = \frac{1}{L} \int_{-L}^{L} f(t)\, dt, \\ a_{n} &= \langle f, C_{n}\rangle = \frac{1}{L} \int_{-L}^{L} f(t) \cos(n\pi t/L)\, dt, \\ b_{n} &= \langle f, S_{n}\rangle = \frac{1}{L} \int_{-L}^{L} f(t) \sin(n\pi t/L)\, dt, \\ f(t) &= \frac{a_{0}}{2} + \sum_{n=1}^{\infty} (a_{n} C_{n}(t) + b_{n} S_{n}(t), \\ &= \frac{a_{0}}{2} + \sum_{n=1}^{\infty} a_{n} \cos(n\pi t/L) + b_{n} \sin(n\pi t/L). \end{align*} (The "special" factor of $1/2$ on the constant term arises because $C_{0} = 1/\sqrt{2} \neq 1$.)
Try replacing $L$ with $\pi$ in the definition you have for the interval $[-L,L]$. Notice that you get back the original definition for $[-\pi,\pi]$. The generalization comes from wanting the cosine and sine parts to still be periodic over the interval $[-L,L]$, just as they were over $[-\pi,\pi]$.
For the series on the interval $[0,2\pi]$, a useful trick is to look at a new function, $g:[-\pi,\pi]\rightarrow\mathbb{R}$ defined by $g(x)=f(x+\pi)$. Then compute the Fourier series for $g(x)$ instead of $f(x)$, using the formula you already know.
Best Answer
I don't think it makes sense to call a Fourier series complete.
But what you can call complete is the basis of your space.
For example, if the functions you're computing Fourier series of are in $L^2(G)$, where $G$ is a compact Abelian topological group, one can say that the characters $\chi$ of $G$ are complete and it means two things:
(i) that for every function $f$ in $L^2$ there exists a sequence of continuous characters $\chi_n \in \mathrm{Hom}(G, S^1)$ such that $\|f - \sum_{k=0}^n a_k \chi_k \|_{L^2} \xrightarrow{n \to \infty} 0$ and
(ii) that $\chi_n$ are orthonormal, that is, $\langle \chi_n , \chi_m \rangle = \delta_{nm}$ where $\langle \cdot, \cdot \rangle$ is the inner product of $L^2$, namely, $\langle f, g \rangle = \int_G f \;\overline{g}\; d \mu$ (where $\overline{g}$ denotes complex conjugation)