We have seen some types of approximations, such as Taylor and Fourier approximations. The type of convergence
used changes depending on the nature of the approximation. One of the most useful for Fourier approximations is
*L*^{2}-convergence:

Let *f* (*x*) be an integrable function on the interval
[- ,], such that

(*f* (*x*) - *f*_{N}(*x*))^{2}*dx* = (*f*^{2}(*x*) - 2*f* (*x*)*f*_{N}(*x*) + *f*^{2}_{N}(*x*))*dx*.

Easy calculations give
(*f* (*x*) - *f*_{N}(*x*))^{2}*dx* = *f*^{2}(*x*)*dx* - 2*a*^{2}_{0} + (*a*^{2}_{n} + *b*^{2}_{n}).

Since
(
2*a*^{2}_{0} + (*a*^{2}_{n} + *b*^{2}_{n}) *f*^{2}(*x*)*dx*,

for any
**Theorem.** **Bessel's Inequality** Let *f* (*x*) be a
function defined on
[- ,] such that *f*^{2}(*x*) has a finite
integral on
[- ,]. If *a*_{n} and *b*_{n} are the Fourier
coefficients of the function *f* (*x*), then we have

2*a*^{2}_{0} + (*a*^{2}_{n} + *b*^{2}_{n}) *f*^{2}(*x*)*dx*.

In particular, the series
(
**Remark.** The quantity
*A*_{n} = is called the **amplitude** of the *n*^{th} harmonic.
The square of the amplitude has a useful interpretation. Indeed, borrowing terminology from the study of periodic waves, we define the **energy** *E* of a 2-periodic function *f* (*x*) to be the number

2*a*^{2}_{0} + (*a*^{2}_{n} + *b*^{2}_{n}) *E*.

You may ask the following question: when does the inequality become an equality?

Note that for Fourier polynomials, the inequality does indeed become an equality. Using this, one can show that the answer to the question is in the affirmative if and only if

(*f* (*x*) - *f*_{N}(*x*))^{2}*dx* = 0.

In this case, we have
(2*a*^{2}_{0} + (*a*^{2}_{n} + *b*^{2}_{n})) = *f*^{2}(*x*)*dx*.

**Theorem.** **Parseval's Formula or the Energy Theorem.** Let
*f* (*x*) be a function defined on
[- ,] such that *f*^{2}(*x*)
has a finite integral on
[- ,]. If *a*_{n} and *b*_{n} are
the Fourier coefficients of *f* (*x*), then we have

2*a*^{2}_{0} + (*a*^{2}_{n} + *b*^{2}_{n}) = *f*^{2}(*x*)*dx* = *E*

if and only if
(*f* (*x*) - *f*_{N}(*x*))^{2}*dx* = 0.

**Remark.** One may wonder when does Parseval's Formula hold?
This is the case, for example, for piecewise smooth functions.
The reason behind is the uniform convergence of the Fourier
partial sums to *f* (*x*), ie.

| *f* (*x*) - *f*_{N}(*x*)| = 0.

The proof in this case is quite easy. Indeed, since
**Application: Least Square Error.**

One application of Parseval's Formula is the measure of the **least square error**
defined by

= (*f* (*x*) - *f*_{N}(*x*))^{2}*dx*.

If the function
= (*a*^{2}_{n} + *b*^{2}_{n}).

**Example.** Let
*f* (*x*) = | *x*| be defined on
[- ,]. Find
and its asymptotic behavior when *N* gets large.
**Answer.** Since *f* (*x*) is even, we have *b*_{n} = 0. On the other hand, easy calculations give

= = = .

Using the equality
= ,

we get
= *O*, as *N* .

Recall that the sequences {*u*_{n}} and {*v*_{n}} satisfy

**
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**Author**: M.A. Khamsi

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