Let $u\in L^1(\mathbb{S}^{d-1})$. I want to show that
\begin{align\*}
\lim\_{|\xi|\to \infty}
\int\_{\mathbb{S}^{d-1}}(1-\cos(\xi\cdot w))u(w)d \sigma\_{d-1}(w)
= \int\_{\mathbb{S}^{d-1}}u(w)d \sigma\_{d-1}(w). 
\end{align\*}
Basically, the question can be reduced into showing that
\begin{align\*}
\lim\_{|\xi|\to \infty}
\int\_{\mathbb{S}^{d-1}}\cos(\xi\cdot w)u(w)d \sigma\_{d-1}(w)
= 0. 
\end{align\*}
This looks like a Riemann-Lebesgue lemma. But I don't know how to tackle it.


I intuitively guessed this from the classical Riemann-Lebesgue Lemma which infers that
\begin{align\*}
\lim\_{|\xi|\to \infty}
\int\_{B\_1(0)}(1-\cos(|\xi| z\cdot x))u(x)d x
= \int\_{B\_1(0)}u(x)dx\quad \text{for fixed $z\in \Bbb R^d$}. 
\end{align\*}


More generally, if $f$ is $T^d$-periodic, then $f\_\lambda(x)= f(\lambda x)$ weakly converge in $L^p$ to its mean value as $\lambda\to\infty$ that is
$$f\_\lambda \rightharpoonup \bar f,\quad \quad \bar f=\frac{1}{T^d}\int\_{[0,T]^d}f(x) dx.$$


Is there any good reference for this type of limit?
Any help is welcome



Computing a limit on the unit sphere: Riemann Lebesgue?
The key fact here is the (surprising, initially, but well known) (power) decay of $\widehat{\sigma}(\xi)$. If $u\in C^{\infty}(S)$, we can extend to a function $u\_0\in C^{\infty}\_0(\mathbb R^d)$, and then
$$
\widehat{u\,d\sigma}=\widehat{u\_0\,d\sigma}=\widehat{u\_0}\*\widehat{\sigma}
$$
still decays. See [here](https://en.wikipedia.org/wiki/Convolution_theorem#Convolution_theorem_for_tempered_distributions) for the general version of the convolution theorem needed here.


We can then extend this to arbitrary $u\in L^1$ by the argument from the traditional Riemann-Lebesgue lemma: given $\epsilon>0$, pick a $v\in C^{\infty}(S)$ with $\|u-v\|\_1<\epsilon$. Since $|\widehat{u\, d\sigma}(\xi) -\widehat{v\, d\sigma}(\xi)|<\epsilon$ and $\widehat{v\, d\sigma}\to 0$, we also have $|\widehat{u\, d\sigma}(\xi)|<2\epsilon$ for all large $\xi$.


