Bessel's Inequality

If $f(x)$ is piecewise Continuous and has a general Fourier Series

$$\sum_i a_i\phi_i(x)$$ (1)

with Weighting Function $w(x)$, it must be true that

$$\int \left[{f(x)-\sum_i a_i\phi_i(x)}\right]^2 w(x)\, dx \geq 0$$ (2)

$$\int f^2(x)w(x)\, dx-2\sum_i a_i\int f(x)\phi_i(x)w(x)\, dx+\sum_i {a_i}^2\int {\phi_i}^2(x) w(x)\,dx \geq 0.$$ (3)

But the Coefficient of the generalized Fourier Series is given by

$$a_m \equiv \int f(x)\phi_m(x)w(x)\, dx,$$ (4)

so

$$\int f^2(x) w(x)\, dx-2\sum_i {a_i}^2+\sum_i {a_i}^2 \geq 0$$ (5)

$$\int f^2(x)w(x)\, dx \geq \sum_i {a_i}^2.$$ (6)

Equation (6) is an inequality if the functions $\phi_i$ are not Complete. If they are Complete, then the inequality (2) becomes an equality, so (6) becomes an equality and is known as Parseval's Theorem. If $f(x)$ has a simple Fourier Series expansion with Coefficients $a_0$, $a_1$, $\ldots$ , $a_n$ and $b_1$, ..., $b_n$, then

$${\textstyle{1\over 2}}{a_0}^2 + \sum\limits_{k=1}^\infty ({a_k}^2+{b_k}^2) \leq {1\over\pi}\int^\pi_{-\pi} [f(x)]^2\,dx.$$ (7)

The inequality can also be derived from Schwarz's Inequality

$$\vert\left\langle{f\vert g}\right\rangle{}\vert^2\leq \left\... ...{f\vert f}\right\rangle{}\left\langle{g\vert g}\right\rangle{}$$ (8)

by expanding $g$ in a superposition of Eigenfunctions of $f$, $g=\sum_i a_if_i$. Then

$$\left\langle{f\vert g}\right\rangle{} = \sum_i a_i\left\langle{f\vert f_i}\right\rangle{}\leq \sum_i a_i$$ (9)

$$\vert\left\langle{f\vert g}\right\rangle{}\vert^2\leq \left\... ...f\vert f}\right\rangle{}\left\langle{g\vert g}\right\rangle{}.$$ (10)

If $g$ is normalized, then $\left\langle{g\vert g}\right\rangle{}=1$ and

$$\left\langle{f\vert f}\right\rangle{}\geq \sum_i {a_i}{a_i}^*.$$ (11)

See also Schwarz's Inequality, Triangle Inequality

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 526-527, 1985.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1102, 1980.