Fourier analysis Basis (ii)--derivation of continuous Fourier transformation from series

Derivation reference here (copy) A first Course in wavelets with Fourier analysis Second Edition, Albert boggess& Francis j.narcowich

It takes only one step to generalize from Fourier series to Fourier transform--to find a limit.

When approaching the positive infinity, the entire Fourier series inverse transformation (or called reduction) becomes an integral, at this time the positive parameter sequence of the formula is a natural integral, but at this time with the approaching positive infinity, from the sequence to the function, we call it the spectrum, generally written.

First consider the Fourier series defined on the top:

which

Substituting:

Note that we might as well plug in a bunch of the following indices:

And then just a little bit to find the limit:

Defined at this time,

This time notice a place, that is, when it occurs, approaching 0, the density of the sequence is gradually becoming larger, and finally become a continuous function, and, regardless of how the change, multiplied by the number is always equal to the whole real axis measure.

Well, then let me take it in:

Further calculations are given:

This time, has changed from a arithmetic progression to R, using a continuous variable to call it more appropriate, applauded, commonly known as frequency. For the mathematical symmetry of beauty, change the formula to this:

The part inside the brackets:, called:

This is the means transformation.

and change back (or call it all, that is, to change a variable), called the Fourier inverse transformation:

Positive and inverse transformations have a minus sign on an exponent, they must be conjugated, and there's no limit, you can do it. When the inverse transformation is reversed, the inverse transformation is the same, the effect is the same, and the frequency domain function will be slightly different. and called the Fourier nucleus. Why is Fourier the nucleus? Mainly because of its good orthogonal characteristics, if there are other functions to meet such characteristics, can also do the kernel.

In the actual numerical calculation, we do not require the function to have the uniform expression, at this time can use the Gram-schmidt method constructs the orthogonal function group to decompose.

Fourier analysis Basis (ii)--derivation of continuous Fourier transformation from series