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

In summary: 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.The article http://en.wikipedia.org/wiki/Fourier_series#Fourier_series_on_a_general_interval_.5Ba.2C.C2.A0a_.2B_.CF.84.5D has the formulas you are looking for. Just make sure you do not give up too soon; you need to go almost half way through the article.In summary, there are general formulas for the coefficients of the Fourier series on a general interval [a, a + T
  • #1
glebovg
164
1
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
 
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  • #2
glebovg said:
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
 
  • #3
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.
 
  • #4
glebovg said:
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
 
  • #5
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.
 
  • #6
glebovg said:
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
 

1. What is a Fourier series on a general interval?

A Fourier series on a general interval is a mathematical representation of a periodic function as an infinite sum of sine and cosine functions. It allows us to decompose a complex function into simpler components, making it easier to analyze and manipulate.

2. How is a Fourier series on a general interval different from a Fourier series on a specific interval?

A Fourier series on a general interval [a, a + T] includes all possible values of the function within that interval, while a Fourier series on a specific interval only includes a finite number of values. This allows for a more precise representation of the function, but also requires more terms in the series.

3. What is the period (T) in a Fourier series on a general interval?

The period (T) in a Fourier series on a general interval is the length of the interval over which the function repeats itself. It is essential to determine the coefficients and frequencies of the sine and cosine terms in the series.

4. Can a Fourier series on a general interval represent any periodic function?

Yes, a Fourier series on a general interval can represent any periodic function, as long as the function satisfies certain conditions, such as being continuous and having a finite number of discontinuities within the interval [a, a + T].

5. How is a Fourier series on a general interval used in practical applications?

Fourier series on a general interval is used in various fields such as signal processing, image reconstruction, and differential equations. It allows us to approximate complex functions and analyze periodic patterns, making it a valuable tool in many scientific and engineering applications.

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