A generalized function whose kth derivative is 0

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Homework Help Overview

The problem involves a distribution on R whose kth derivative is zero, and the task is to prove that this distribution must be a polynomial. The subject area relates to distributions and their properties in mathematical analysis.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of treating the distribution like a regular function and question the validity of integrating distributions. There is also a focus on understanding the definition of the derivative of a distribution.

Discussion Status

The discussion is exploring foundational concepts related to distributions and their derivatives. Some participants are questioning assumptions about integration and the nature of derivatives in this context, while others are clarifying definitions.

Contextual Notes

There is an indication that the original poster feels uncertain about how to approach the problem, and there may be constraints related to the properties of distributions that are under consideration.

e(ho0n3
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Homework Statement


Let f be a distribution on R and suppose that its kth derivative is 0. Prove that f is a polynomial.

2. The attempt at a solution
I honestly haven't a clue how to start. If I could treat f like a "regular" function, this would so easy.
 
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Maybe worth to try integrating k times?=)
 
I believe integration does not exist for distributions so that will not work.
 
Well the first question to ask it "what does the derivative of a distribution mean?", i.e. what's the definition of the derivative of a distribution?
 
The derivative f' is the distribution defined by (f', g) = -(f, g'), where g is any test function.
 

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