Jump discontinuity with fourier series

In summary, a jump discontinuity in a Fourier series is a sudden change in the function being represented, resulting in a discontinuity in the graph. This can cause the series to converge slowly or not converge at all, as it relies on smooth, continuous functions. While a jump discontinuity can be represented in a Fourier series, it may require an infinite number of terms and may not be practical for real-world applications. Alternative methods, such as piecewise functions or Taylor series, may be more accurate and efficient for representing jump discontinuities.
  • #1
yoq_bise
3
0
Hi all,
I couldn't find any proof of the following statement: The Fourier series expansion of f(x), which has a discontinuity at y, takes on the mean of the left and right limits
i.e. f(y)= (1/2)(f(y+)+f(y-))

is there anyone who can help me?
Thanks
 
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  • #2
Have you tried looking in a book?
 
  • #3
Yes I looked in Hassani, Arfken and Diprima but I couldn't find
Can anyone suggest other books or web sites?
 
Last edited:

1. What is a jump discontinuity in a Fourier series?

A jump discontinuity in a Fourier series occurs when there is a sudden change in the function being represented. This results in a discontinuity in the graph of the function, where the function "jumps" from one value to another.

2. How does a jump discontinuity affect the Fourier series representation?

A jump discontinuity in a Fourier series can cause the series to converge slowly or not converge at all. This is because the Fourier series relies on smooth, continuous functions and a jump discontinuity violates this condition.

3. Can a jump discontinuity be represented in a Fourier series?

Yes, a jump discontinuity can be represented in a Fourier series. However, it may require an infinite number of terms to accurately represent the jump discontinuity, making the series impractical for practical applications.

4. How can a jump discontinuity be approximated in a Fourier series?

A jump discontinuity can be approximated in a Fourier series by using a technique called Gibbs phenomenon. This involves taking a partial sum of the series and adding a correction term to account for the jump discontinuity. However, the approximation may still not be very accurate.

5. Are there any alternative methods for representing a jump discontinuity?

Yes, there are alternative methods for representing a jump discontinuity, such as using piecewise functions or other types of series, such as Taylor series. These methods may be more accurate and efficient for representing jump discontinuities compared to Fourier series.

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