Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $\sigma$ be the "normalized Lebesgue" (Haar, really...) measure on the unit sphere $S=S^{n-1} \subset \mathbb R^n$. That is, $\sigma$ has support $S$, it is uniformly distributed, and $\int_S d\sigma = 1$.

I'm having a hard time figuring out how to understand and evaluate integrals $\int_S f\left(x\right) d\sigma\left(x\right)$ of non-trivial (let's say, even and non-constant) functions $f$. This is not really "the" $\mathbb{R}^n$ Lebesgue measure or a restriction thereof, so I can't easily convert it to a Riemann integral (unless there's something I haven't noticed).

As a simple example, how do I evaluate $\int_S (x_1)^2 d\sigma(x)$ when $n=2$? Please explain in a way that I can generalize.

Note: What I'm eventually really interested in is proving that the Fourier transform of $\sigma$, that is, $\hat\sigma(y) := \int_S \exp \left(2\pi i x\cdot y \right) d\sigma(x)$, which is always real (as $\sigma$ is symmetric), is negative on some sphere (in the Fourier-space) centered at zero. This is not supposed to be hard. I'm thinking it's going to look like a "sinc" function which is negative infinitely many times when you move away from the origin, but I don't know how to actually show that. Some pointers (but not a full proof please) would be welcome :)

share|cite|improve this question
The short answer is: polar coordinates. More generally, to integrate over a manifold, you parametrize it, and compute the volume form in coordinates (which corresponds to a Jacobian). – Nate Eldredge Dec 10 '12 at 20:06
That sounds like something I'd like to know how to do :) Do you have a reference where I can find a simple example of integrating over a manifold? – Yoni Rozenshein Dec 11 '12 at 8:32
up vote 1 down vote accepted

It sounds like you're on the wrong track to answer your question about the Fourier transform--these integrals can't be evaluated in any explicit sense (though one can derive formulas for them involving Bessel functions). The shortest solution to your problem that I can see would be to compute the integral $$\int_{{\mathbb R}^n} e^{-\beta \pi |y|^2} \hat \sigma(y) dy$$ exactly as a function of $\beta > 0$ using Fubini, then argue that the limiting behavior as $\beta \rightarrow 0^{+}$ of this integral contradicts the assumption that $\hat \sigma(y)$ is nonnegative.

share|cite|improve this answer
Interesting, I'll try that! Regarding the fact that the integral can't be evaluated in an explicit sense, well, okay, but what I was thinking of doing is using the fact that the Taylor series for $exp$ converges compactly uniformly, so if I know how to compute the integral of a polynomial, I may be able to work out a good enough approximation of the Fourier transform. Sort of the way you use the Dirichlet kernel to approach the delta function. – Yoni Rozenshein Dec 11 '12 at 8:30

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.