bibkey: tao2021stripfourier authors: Terence Tao year: 2021 title: 246B, Notes 2 — Some connections with the Fourier transform doi: null url: https://terrytao.wordpress.com/2021/01/23/246b-notes-2-some-connections-with-the-fourier-transform/ claim: Proposition 3(i)–(ii) transports horizontal holomorphic shifts to Fourier weights and gives exponential Fourier decay under uniform integrable polynomial decay on smaller closed strips. strata_touched: [] license: citation-only triage: anchor

Holomorphic strips and Fourier transport

Proposition 3 assumes a holomorphic function on $|\mathrm{Im}z|<a$, $a>0$. On each smaller closed strip $|y|\le b<a$ it requires

$$|f(x+iy)|\le \frac{C_b}{1+|x{|}^{\eta }},\eta >1.$$

With the source convention $\hat{f}(\xi )=\int f(x)e^{-2\pi ix\xi }dx$, part (i) gives $\widehat{f(\cdot +w)}(\xi )=e^{2\pi iw\xi }\hat{f}(\xi )$. Part (ii) gives $|\hat{f}(\xi )|\le C_{b,\eta }{e}^{-2\pi b|\xi |}$. The integrable strip decay is a hypothesis; holomorphy alone is not the stated theorem. Parts (iii)–(v) concern partial/full inversion and Poisson summation under the same hypotheses.

For the unitary angular-frequency convention used in the original-theta root supplier, part (i) reads $\widehat{f(\cdot +iy)}(\xi )=e^{-y\xi }\hat{f}(\xi )$. Plancherel then expresses the two horizontal-line squared norms as $\int 2\cosh (2b\xi )|\hat{f}(\xi ){|}^{2}d\xi$ when those norms are finite. This is a normalization and norm application of the published method, not a new Fourier theorem or a Lean certification.

The source does not establish nonvanishing of the original theta kernel, its square-root branch or the quantitative line envelopes. Those are separate model-specific inputs in the linked supplier. Exact sharp Fourier projections preserve the weighted $L^2$ norm structure, while they can lose rapid spatial decay; the supplier checks the integrable decay again after multiplication by $s$.

The public HTML containing Proposition 3 was inspected at the source URL. No implementation code or full source document is copied here.