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Suppose in $\mathbb{R}^n$, $f(x)$ is homogeneous function of degree $a$, with $-n<a<0$, does the function always has Fourier transform? If not, what if $f$ is a continuous function

Pires Dankan
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1 Answers1

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If a function is continuous on $\mathbb{R}^n$ and homogeneous of negative degree, then it's identically zero. So the Fourier transform exists, and is identically zero.

If you mean "continuous on $\mathbb{R}^n\setminus \{0\}$", that is a more interesting case. Actually, just being bounded on the unit sphere is enough for $f$ to be locally integrable with decay at infinity. Hence, it is a tempered distribution and therefore has a well-defined Fourier transform. As mentioned here, the transform is also homogeneous, with the degree of homogeneity $-n-a$.

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