Any function that is defined over the entire real line can be represented by a Fourier series if it is periodic. What if it is non-periodic?
For non-periodic functions, one usually represents them in some finite segment. This is sufficient for many purposes since the behaviour at infinity is typically investigated by other methods.
Yet, the answer is somewhat dependent on the precise definition of the class of the series that you want to use and whether you can extend your function into the complex plane. Indeed, the function sin(2 Pi a n z) has a real period for real a and integer n, but in general is not periodic on the real line for complex a.
What is the precise class of series that you want to use?
Regards,
Timur.
Thank you Prof.Timur.
Actually i would like to know if we want to integrate an extension coefficients along an interval from 0 to 1, and the result is a function which is non-periodic. Can we use Fourier series to find the desired function OR do we have to use Riemann sums instead to find the area (integral) of the extinction coefficients ?
Thanks again.
Dear Kulthoum Ismail,
I would need your precise definition of extension (or extinction?) coefficients that you are using to answer the additional question. As for the periodicity of the sum of a Fourier series understood as a converging linear combination of sines and cosines, it is always a periodic function on the real line. Extending the notion of the Fourier series into the complex plane yields interesting effects and the periodicity is then only in the direction of the real axes. A nice introduction into the topic can be found at http://www.math.lsa.umich.edu/~rauch/555/fouriercomplex.pdf
Regards,
Timur.
In this case Extinction coefficient is the fractional depletion of radiance per unit path length (also called attenuation especially in reference to radar frequencies).
What i would like to know is that if we can find tau (extinction coefficient) by using the Fourier transformation instead of Riemann sums or not ? The picture explains better what i mean.
**The aerosol optical depth or optical thickness (tau, τ) is defined as the integrated extinction coefficient over a vertical column of unit cross section. Extinction coefficient is the fractional depletion of radiance per unit path length (also called attenuation especially in reference to radar frequencies).
Hope this is clear. Thanks again Prof. Timur.
Unfortunately, I am not an expert in estimating optical thickness... The problem you describe appears to be totally real, so, unfortunately, I do not have anything to add to my previous answer. As far as I can see, Fourier series will only work on a finite segment in this case.
Regards,
Timur.
In general, the best one can do is the following: If function f is non-periodic, choose any interval [a,b] and adjust the Fourier series accordingly; Non-periodicity is not an issue now, as long as f itself is Riemann integrable over this interval. If you mean “represent” as the equality of the sum of the Fourier series of f and value of this function itself at all points of the interval, one needs standard assumptions on f, like being piecewise smooth. At the points of possible non-periodicity (a & b), we will have convergence to the average mean of values of the function at a and b.
In fact, in Fourier series, we do not investigate the nature of f(x) at the infinite branch because it follows from the periodicity of f(x). If it is non periodic and is defined in some finite interval, then we need to study the nature of f(x) at infinite branch, which is quite difficult even some times impossible. Therefore, we take the assumption of periodicity
of a function in entire real line and then find the Fourier series.
In the Fourier series can be expanded only function with a finite duration T. Then the coefficients of the Fourier series will be spaced at a distance of 1/T. Then a periodic function turns out the inverse Fourier transform. If T is infinite (the whole real axis) the function can only be represented by a Fourier integral. If this function is periodic, the Fourier integral will be a superposition of delta functions at frequencies that are multiples of 1/T. If the function is not periodic, then its spectrum will be blurry.
- Badges
- Science topic
- Similar topics
- Engineering
- Chemical Engineering