Fourier Transforms: Jump Discontinuities & Continuous Functions in G(R)

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Homework Help Overview

The discussion revolves around the properties of Fourier transforms in relation to functions belonging to the space G(R), particularly focusing on continuity and jump discontinuities.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand the implications of continuity in G(R) on the Fourier transform's membership in G(R) and seeks a counterexample to their assumption.

Discussion Status

Some participants are clarifying the definition of G(R) and its implications, while others are exploring the original poster's claims regarding continuity and Fourier transforms. There is an ongoing search for ideas and examples.

Contextual Notes

Participants are discussing the specific characteristics of functions in G(R), including the nature of discontinuities and integrability, which may influence the Fourier transform's properties.

maria clara
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If a function belongs to G(R) but has points that are jump discontinuities, it's Fourier transform will not belong to G(R).
But would it be correct to claim that if a function in G(R) is continuous than its Fourier transform also belongs to G(R)? I guess it's not true, but can someone suggest a counterexample?
Thanks.
 
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Frst, what do YOU mean by "G(R)"?
 
Sorry, I wasn't sure whether this sign is well known.
G(R) is the space of functions that might have points of discontinuity only of first kind, and which are absolutely integrable.
 
ideas? anyone?...:frown:
 

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