Fourier Series Interval Points

In summary, the Fourier sine series of any function satisfying Dirichlet's theorem is not defined on discontinuous points because the odd extension of the function is not continuous at those points. On the other hand, we can define the Fourier cosine series for these discontinuous points because the even extension of the function is continuous at those points. This results in the Fourier series converging to different values at these points.
  • #1
zorro
1,384
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Why is Fourier sine series of any function satisfying Dirichlet's theorem, not defined on the discontinuous points whereas we define it for Fourier cosine series?

ex - sine series of f(x) = cosx, 0<=x<=∏ is defined on 0<x<∏

whereas cosine series of f(x) = sinx, 0<=x<=∏ is defined on 0<=x<=∏
 
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  • #2
Abdul Quadeer said:
Why is Fourier sine series of any function satisfying Dirichlet's theorem, not defined on the discontinuous points whereas we define it for Fourier cosine series?

ex - sine series of f(x) = cosx, 0<=x<=∏ is defined on 0<x<∏

whereas cosine series of f(x) = sinx, 0<=x<=∏ is defined on 0<=x<=∏

Not sure what you are getting at. The half range Fourier series you mention both converge for all x. In the first case the FS converges to the average of the right and left hand limits at x = 0 of the odd extension of cos(x). In the second case the FS converges to sin(0) = 0 at x = 0. That is because the even extension of sin(x) is continuous at x=0 while the odd extension of cos(x) is not continuous at x = 0.
 

What is a Fourier Series?

A Fourier series is a mathematical representation of a periodic function as a sum of sines and cosines. It is used to analyze and approximate periodic functions in various fields of science and engineering.

What are interval points in a Fourier Series?

Interval points refer to the points at which the function being represented by the Fourier series is sampled. These points are evenly spaced along the interval of the periodic function and are used to calculate the coefficients for the sine and cosine terms in the series.

How are interval points chosen in a Fourier Series?

The interval points are typically chosen to be evenly spaced along the interval of the periodic function. The more interval points used, the more accurate the Fourier series will be in representing the function.

What is the significance of the interval points in a Fourier Series?

The interval points are crucial in determining the accuracy of the Fourier series in representing the periodic function. They are used to calculate the coefficients for the sine and cosine terms, which in turn affect the shape and amplitude of the function in the series.

Can the interval points be adjusted in a Fourier Series?

Yes, the interval points can be adjusted to improve the accuracy of the Fourier series. Adding more interval points or adjusting their spacing can result in a more precise representation of the periodic function. However, too many interval points may also lead to computational difficulties.

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