Suppose $f(\theta, r)$ is defined on $\mathbb{T}^n \times \overline{B_R(0)}$, is smooth and has Fourier expansion $\displaystyle \sum_{k \in \mathbb{Z}^n} \widehat{f}_k(r)e^{ik.\theta}$. Is it true that for all positive integers $N$ there is a constant $C_N > 0$ so that for all $r \in B_R(0)$ and $\theta \in \mathbb{T}^n$, we would have $|\widehat{f}_k(r)| \le \dfrac{C_N}{|k|^N}$ for all $k \in \mathbb{Z}^n$?
2026-04-09 08:22:01.1775722921
Decay of Fourier coefficients of a smooth family of functions
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