How to obtain the decay of Fourier transform of the Bochner-Riesz multipliers? For $\lambda>0$ define: $\hat{m_{\lambda}}(x)=\int_{\mathbb{R}^d} (1-|\xi|^2)_{+}^{\lambda}e^{2\pi i x\cdot \xi}d\xi$, then $|\hat{m_{\lambda}}(x)|\sim |x|^{-\frac{d+1}{2}-\lambda},|x|\rightarrow \infty$.
The classic way to do this is to compute them exactly using Bessel functions, while Tao mentions in his notes (1999) that there is another more robust way called "fuzzier", but I don't get his point. Can someone explain this in details or provide other methods without using special functions?
PS: The "fuzzier" method he mentions: we can assume $x=\lambda e_d$, then:
First: $\int_{|\xi|\leq 1}(1-|\xi|^2)_{+}^{\lambda}e^{2\pi i x\cdot \xi}d\xi=\int_{|\xi_{d}-1|\ll1}+\int_{|\xi_{d}+1|\ll1}+\int_{other}$, the "other" term is easy.
Second: $(1-|\xi|^2)_{+}^{\lambda}=(fd\sigma)*\mu+error$, where f is an appropriate function on $\mathbb{S}^{d-1}$ and $\mu(\xi)=\delta_0(\xi^{'})\eta(\xi_d)(-\xi_d)_{+}^{\lambda}$ ($\xi=(\xi^{'},\xi_d)$) such that the "error" term vanishes to order $\lambda+1$.
Third: we can repeat these procedures above and get error term with order enough to get the decay, and the front term could also be tackled.
I don't understand what happens in the second and the third. (How to select the $f$? What is the "vanishing order" which can lead to decay?)