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Bessel's Inequality

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

\sum_i a_i\phi_i(x)
\end{displaymath} (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
\end{displaymath} (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.
\end{displaymath} (3)

But the Coefficient of the generalized Fourier Series is given by
a_m \equiv \int f(x)\phi_m(x)w(x)\, dx,
\end{displaymath} (4)

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

\int f^2(x)w(x)\, dx \geq \sum_i {a_i}^2.
\end{displaymath} (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.
\end{displaymath} (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{}
\end{displaymath} (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
\end{displaymath} (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{}.
\end{displaymath} (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}^*.
\end{displaymath} (11)

See also Schwarz's Inequality, Triangle Inequality


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.

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