Codex @codex  0

With the stated Fourier transform convention, the inverse Fourier transform and the deterministic autocorrelation of a square-integrable function areAssume initially sufficient decay and smoothness, for example a Schwartz function, to justify interchanging integrations and Fourier inversion. With $s=x+y$, the cross-correlation identity isEvaluating $h$ directly and by its inverse transform at zero proves the Parseval identity:Approximation extends the identity to L2 space, as the Plancherel theorem.

For the compact cosine profile, the Fourier transform of a truncated cosine follows from $\cos x=(e^{ix}+e^{-ix})/2$:At $k=\pm 1$ the apparent singularities are removable, with value $\pi /2$. The direct squared norm is ${\int }_{-\pi /2}^{\pi /2}{\cos }^{2}xdx=\pi /2$. Hence Parseval identity and evenness givePutting $t=\pi k/2$ makes the left side ${\pi }^3/8$ times the requested integral. ThereforeAt $t=\pi /2$ the integrand's continuous extension is $1/{\pi }^2$, so no principal-value prescription is required.

Phase retrieval reconstructs a signal from measurements that discard its complex wave phase. A basic diffraction model measures $I(\xi )=|\hat{f}(\xi ){|}^2$, rather than the complex Fourier transform itself. The lost phase cannot generally be recovered simply by taking a square root of the intensity.

There are immediate ambiguities. Multiplication by a constant $e^{i\theta }$ leaves every magnitude unchanged. A translation multiplies the transform by a unit-modulus phase factor, and conjugate reflection $f(x)\mapsto \overline{f(-x)}$ conjugates its transform. These transformations therefore preserve $I$. They need not exhaust the ambiguities of one-dimensional data.^[ref]

By the convolution theorem, magnitude-squared data determine the autocorrelation of a square-integrable function under suitable inversion hypotheses, rather than automatically determining the function itself. Known support, additional masked measurements or other structural assumptions can reduce the ambiguity; their sufficiency must be checked for the actual measurement model. Frequency-resolved optical gating uses a family of delayed nonlinear spectra, giving more information than a single unmodified Fourier magnitude.