Fourier Series at Discontinuities

However, the Fourier series does not necessarily converge at the midpoint between y1 and y2. It may converge at any point within that range, depending on the function. In summary, the Fourier Series may converge at any point within the range of the discontinuity of a periodic function, and not necessarily at the midpoint between the two limiting values.
  • #1
Hermes10
5
0
Dear all,

I am wondering why the Fourier Series converges at a finite discontinuity of a periodic function at 1/2*(y1+y2) at the point f(x1), where x1 is the point at which the discontinuity occurs and y1 is the limiting value for the function when we approach x=x1 from one side and y2 is the limiting value when we approach x=x1 from the other side?

Say, in a particular case y2 is 5 and y1 is 2, shouldn't the Fourier series converge to 1/2*(5-2)? I would have though that the Fourier series just converges at the midpoint between y1 and y2 on the graph that is if you draw the function I would have draw the value for x1 at which the discontinuity occurs to be in the middle of the two limiting values. Is that correct?

All the
Hermes10
 
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  • #2
Your idea is correct, except for an error (typo?) y2=5 and y1=2 gives 1/2(5+2) as the midpoint.
 

FAQ: Fourier Series at Discontinuities

1. What is a Fourier series at a discontinuity?

A Fourier series at a discontinuity is a mathematical representation of a function that has a jump or abrupt change in value at a specific point. The series is used to approximate the function in the region around the discontinuity by using a combination of sine and cosine functions.

2. How is a Fourier series at a discontinuity different from a regular Fourier series?

A regular Fourier series is used to represent a function that is continuous and periodic, while a Fourier series at a discontinuity is used to represent a function that has a discontinuity or jump at a specific point. This means that the coefficients and the formulas used in the two series are different.

3. Why are Fourier series at discontinuities useful?

Fourier series at discontinuities are useful because they allow us to approximate functions that have discontinuities, which are common in real-world applications. They also provide a way to analyze and understand the behavior of these functions in the regions around the discontinuities.

4. How are Fourier series at discontinuities calculated?

To calculate a Fourier series at a discontinuity, we first identify the location of the discontinuity and determine the coefficients for the sine and cosine terms using the jump conditions. We then use the standard formulas for Fourier series to find the coefficients for the rest of the terms.

5. What are some practical applications of Fourier series at discontinuities?

Fourier series at discontinuities are used in various fields such as signal processing, image and audio compression, and data analysis. They are also used in physics and engineering to model and analyze systems with discontinuities such as electrical circuits and mechanical systems.

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