Fourier series (2 same functions different inequality signs)

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Homework Help Overview

The discussion revolves around the properties of Fourier series, specifically whether two functions with different inequality signs can yield the same Fourier series. The original poster questions the similarity of the graphs of these functions and their corresponding Fourier series.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of changing function values at a finite number of points and how this affects the integral used to calculate Fourier coefficients. There is uncertainty about whether the coefficients would differ based on the inequality signs in the function definitions.

Discussion Status

Participants are actively engaging with the question, with some suggesting that the integral remains unchanged despite modifications at finite points, while others express uncertainty about this conclusion. The discussion reflects a mix of interpretations regarding the effects of inequality signs on the Fourier series.

Contextual Notes

There is mention of confusion stemming from lecture notes that only include examples with strict inequalities, leading to questions about the implications of non-strict inequalities in practice.

izen
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Homework Statement



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these two functions will give the same Fourier series? because when I write the graph they look the same?

Homework Equations





The Attempt at a Solution



in the picture

thank you
 
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The series coefficients are defined by an integral. If you change the value of a function at a finite number of points, can the integral give you a different result?
 
jbunniii said:
The series coefficients are defined by an integral. If you change the value of a function at a finite number of points, can the integral give you a different result?

I think integral give a different result so they are different ? not 100% sure

because in my lecture note all the example are < ,> there is no single example has <=, >=. so when I tried to do some exercise there are <=, >= on the functions so it kinda confused me. please explain more
 
Last edited:
izen said:
I think integral give a different result so they are different ? not 100% sure
No, you can change a function at a finite number of points, and the integral will still be the same. So your Fourier coefficients will be the same for both functions.
 
jbunniii said:
No, you can change a function at a finite number of points, and the integral will still be the same. So your Fourier coefficients will be the same for both functions.

thank you
 

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