# Fourier Transform of an even function

Let $$S^n$$ be an $$n$$-dimentional unit sphere.

Consider $$f: S^n \longrightarrow R_+$$, where $$f$$ is an even continuous function.

Denote $$F(f):=\int_0^{\infty}\int_{S^n}f(y)g\left(\frac{|xy|}{t}\right)dy\frac{dt}{t^{n+1}},$$ where $$x \in S^n, \, t>0$$, and function $$g$$ is such that $$\int_{0}^{\infty}s^jg(s)ds=0, \quad j=0,2,4,\ldots, 2\left[(n-1)/2\right]$$ $$\int_1^{\infty}s^{\alpha}|g(s)|ds< \infty, \quad \alpha>n-1.$$

Find the Fourier Transform of $$F$$.

• What does it mean for a function to be "even" on the unit sphere for $n>1$? – Michael Engelhardt Jun 13 '20 at 13:41
• I guess "even" just means $f(x)=f(-x)$ all $x$. – YCor Jun 13 '20 at 13:45
• Ah - parity-even as it's called in some quarters. Yes, that must be it. – Michael Engelhardt Jun 13 '20 at 14:45


These are the spaces of homogeneous harmonic polynomials or, equivalently, the eigenspaces of the Laplacian on the round $$n$$-dimensional sphere. As such they coincides with the restrictions to the sphere of homogeneous harmonic polynomilas.


C. Muller: Analysis of Spherical Symmetries in Euclidean Spaces, Springer Verlag, 1998.

Now observe that if $$f\in\bH_d$$ and $$x\in S^n$$ then $$F[f](x)=\int_0^\infty \eT_s[P] s^{-n-1} ds=\left(\int_0^\infty c_d(s) s^{-n-1} ds\right)P.$$

Thus, everything boils down to computing Fourier transforms of homogeneous functions of the form,

$$\frac{1}{|x|^{n+d}}P_d(x),$$

where $$P_d$$ is a homogenous harmonic polynomial of even degree $$d$$ in $$n+1$$ variables.