Fourier series on a general interval [a, a + T]

[glebovg]

Are there general formulas for Fourier coefficients on an integral [a, a + T], where T is the period. There is a general formula for the coefficients of exponential Fourier series. Are there general formulas for the coefficients of the trigonometric Fourier series that would work on any interval [a, a + T]?

http://en.wikipedia.org/wiki/Fourie...general_interval_.5Ba.2C.C2.A0a_.2B_.CF.84.5D

 

[Ray Vickson]

Are there general formulas for Fourier coefficients on an integral [a, a + T], where T is the period. There is a general formula for the coefficients of exponential Fourier series. Are there general formulas for the coefficients of the trigonometric Fourier series that would work on any interval [a, a + T]?

http://en.wikipedia.org/wiki/Fourie...general_interval_.5Ba.2C.C2.A0a_.2B_.CF.84.5D

Of course there are; the article http://en.wikipedia.org/wiki/Fourier_series contains all the formulas you need. Just make sure you do not give up too soon; you need to go almost half way through the article.

RGV

 

[glebovg]

I cannot find it. I could not find it anywhere. I am looking for the most general formulas. Like the one for the exponential Fourier series (formulas which can be applied to any interval of the form [a, a + T]). Usually, there are different formulas for different intervals. I am looking the the most general one.

 

[Ray Vickson]

I cannot find it. I could not find it anywhere. I am looking for the most general formulas. Like the one for the exponential Fourier series (formulas which can be applied to any interval of the form [a, a + T]). Usually, there are different formulas for different intervals. I am looking the the most general one.

Look again. The link I gave you has a section entitled "Fourier Series on an Interval [a,a+T]". I just now looked at it again! Anyway, if you cannot find it on the web you can find it in many books.

RGV

 

[glebovg]

Yes, I know. I actually have the link to that section in my first post. I am looking for similar formulas for the coefficients of the trigonometric Fourier series.

 

[Ray Vickson]

Yes, I know. I actually have the link to that section in my first post. I am looking for similar formulas for the coefficients of the trigonometric Fourier series.

It is simple.
(1) Either express the exponentials in terms of sin and cos and use the formulas given in the link (exactly the same way as one does for the interval [-π,π]); or (2) change variables to x = α + βt, so that a <= t <= a+T becomes -π <= x <= π, then just change variables in the integrations, etc.

RGV