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i'm studying Fourier & convolution, it's said convolution is using one function to smooth another function. i want to figure out How do math majors define smoothness? i see some answers and it seems it is related with Lebesgue integrals and L^p space. Although i have not studied Real analysis, i still want to know the accurate math-language of smoothness of a function. i guess it just a simple question for math-majored, hoping the question will not bother you. and I hope you can write it more completely, because I haven't studied real analysis, so if you omit some of the things you take for granted, it may bring me some ambiguity and misunderstanding. Thanks for your help

related links: https://math.stackexchange.com/a/398146/969078 (in this answer, authors said "Convolution combines the smoothness of two functions", what does it mean?) & Why convolution regularize functions?

Aerterliusi
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  • Probably you're already familiar with properties like continuous, differentiable, twice differentiable, etc. which are examples. – Funktorality Feb 09 '23 at 04:31
  • @Funktorality , you mean extent of smoothness of C^n+1 is always greater than the extent of smoothness of C^n? – Aerterliusi Feb 09 '23 at 04:36
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    The basic idea is that smoothness is related to how quickly the Fourier transform of a function decays; I think a function is $C^k$ iff its Fourier transform decays like $\frac{1}{\xi^k}$ or something like that. Convolution multiplies the Fourier transforms so it adds the exponents on these decays. – Qiaochu Yuan Feb 09 '23 at 04:54
  • @Qiaochu Yuan, Thanks, your idea inspires me, indeed high-frequency components often represent “blur” so seeing how quickly the Fourier transform of a function decays can be an angle to judge smoothness, but i want to confirm is it the standard way to judge smoothness in math? and another answer using C^n to judge smoothness, what do you think about the idea? correct? – Aerterliusi Feb 09 '23 at 08:15
  • Yes, the $k$ in $C^k$ is a standard way to judge smoothness. – Qiaochu Yuan Feb 09 '23 at 08:22
  • @Qiaochu Yuan, Why I feel like the relationship between C^k and the extent of smoothness is not so obvious? is it obvious for you? and is the way of C^k and the way of " how quickly the Fourier transform of a function decays" compatible? – Aerterliusi Feb 09 '23 at 09:38

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