Bessel's Inequality for the Sum of Coefficients of a Fourier Series

Recall from The Fourier Series of Functions Relative to an Orthonormal System page that if $\mathcal S = \{ \varphi_0(x), \varphi_1(x), ... \}$ is an orthonormal system of functions on $I$ and $f \in L^2(I)$ then the Fourier series of $f$ relative to $\mathcal S$ is $\displaystyle{f(x) \sim \sum_{n=0}^{\infty} c_n \varphi_n(x)}$ where $\displaystyle{c_n = (f(x), \varphi_n(x)) = \int_I f(x) \overline{\varphi_n(x)} \: dx}$ for all $n \in \{0, 1, 2, ... \}$.

We will now look at a very important inequality which relates the sum of the squared absolute values of the Fourier coefficients of the Fourier series of a function $f$ relative to an orthonormal system AND the square of the norm of that function $f$ known as Bessel's inequality.

Theorem 1 (Bessel's Inequality): Let $\{ \varphi_0(x), \varphi_1(x), ... \}$ be an orthonormal system of functions on $I$ and let $f \in L^2(I)$. If $\displaystyle{f(x) \sim \sum_{n=0}^{\infty} c_n \varphi_n(x)}$ then we have that $\displaystyle{\sum_{n=0}^{\infty} \mid c_n \mid^2 \leq \| f(x) \|^2}$.

• Proof: Recall from The Best Approximation of a Function from an Orthonormal System page that for all $n \in \mathbb{N}$ we have that:

(1)

\begin{align} \quad \| f(x) - t_n(x) \|^2 = \| f(x) \|^2 + \sum_{k=0}^{n} \mid b_k - c_k \mid^2 - \sum_{k=0}^{n} \mid c_k \mid^2 \end{align}

• Where $\displaystyle{t_n = \sum_{k=0}^{n} b_k \varphi_k(x)}$ for any arbitrary $b_0, b_1, ..., b_n \in \mathbb{C}$. So let $b_k = c_k$ for all $k \in \{0, 1, ..., n \}$. Then for $\displaystyle{s_n(x) = \sum_{k=0}^{n} c_k \varphi_k(x)}$ we have that:

(2)

\begin{align} \quad \| f(x) - s_n(x) \|^2 &= \| f(x) \|^2 + \sum_{k=0}^{n} \mid c_k - c_k \mid^2 - \sum_{k=0}^{n} \mid c_k \mid^2 \\ \quad \| f(x) - s_n(x) \|^2 &= \| f(x) \|^2 - \sum_{k=0}^{n} \mid c_k \mid^2 \\ \quad \sum_{k=0}^{n} \mid c_k \mid^2 + \| f(x) - s_n(x) \|^2 &= \| f(x) \|^2 \end{align}

• Notice that $\| f(x) - s_n(x) \|^2 \geq 0$ and so we have that:

(3)

\begin{align} \quad \sum_{k=0}^{n} \mid c_k \mid^2 \leq \| f(x) \|^2 \end{align}

• But this inequality holds for all $n \in \mathbb{N}$ and thus:

(4)

\begin{align} \quad \sum_{n=0}^{\infty} \mid c_n \mid^2 \leq \| f(x) \|^2 \quad \blacksquare \end{align}

Corollary 1: Let $\{ \varphi_0(x), \varphi_1(x), ... \}$ be an orthonormal system of functions on $I$ and let $f \in L^2(I)$. If $\displaystyle{f(x) \sim \sum_{n=0}^{\infty} c_n \varphi_n(x)}$ then $\displaystyle{\lim_{n \to \infty} c_n = 0}$.

• Proof: Note that $f \in L^2(I)$ and so $\int_I [f(x)]^2 \: dx$ exists (is finite). However we see that:

(5)

\begin{align} \quad \| f(x) \|^2 = \int_I [f(x)]^2 \: dx \end{align}

• So by Bessel's inequality we have that the series $\displaystyle{\sum_{n=0}^{\infty} \mid c_n \mid^2}$ converges. But from the Sequence of Terms Divergence Criterion for Infinite Series we know that then $\lim_{n \to \infty} \mid c_n \mid^2 = 0$, which happens only when:

(6)

\begin{align} \quad \lim_{n \to \infty} c_n = 0 \quad \blacksquare \end{align}

We will also state the following corollary (which is the contrapositive to Bessel's inequality) which gives us a criterion for determining if an infinite linear combination of a system of orthonormal functions is not a Fourier series for any function $f \in L^2(I)$.

Corollary 2: Let $\{ \varphi_0(x), \varphi_1(x), ... \}$ be an orthonormal system of functions on $I$ and let $c_0, c_1, ... \in \mathbb{C}$. If $\displaystyle{\sum_{n=0}^{\infty} \mid c_n \mid^2}$ diverges then $\displaystyle{\sum_{n=0}^{\infty} c_n \varphi_n(x)}$ is not a Fourier series relative to this system for any function $f \in L^2(I)$.