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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Exact MRI Fourier samples from a two-point object

Statement

Consider a two-point object in $T^{2} = [0, 1)^{2}$ with locations $x_{0} = (1 / 4, 0), x_{1} = (1 / 2, 1 / 4),$ and weights $c_{0} = 1, c_{1} = 2$ . At the two type-3 frequency samples $ξ_{0} = (0, 0), ξ_{1} = (1 / 3, 1 / 5),$ the exact nonuniform Fourier samples are $G_{0} = 3, G_{1} = e^{- 2 π i / 12} + 2 e^{- 2 π i (13 / 60)} .$ These are finite Fourier samples of a point-source model of an MRI object; no acquisition or reconstruction claim is made.

Facts & Assumptions

Given: The two locations, weights and frequency samples stated above.

[F1]

Type-1, type-2, and type-3 nonuniform Fourier transforms: a type-3 sample is $G_{ℓ} = \sum_{j} c_{j} e^{- 2 π i ξ_{ℓ} \cdot x_{j}}$ for finite real source and frequency lists.

[F2]

The complex exponential by its power series: its power series defines every displayed value, and substituting $z = 0$ gives $e^{0} = 1$ .

Proof

1.1F1F2givenalgebra

For $ξ_{0} = (0, 0)$ , both dot products are zero, so [F1] gives $G_{0} = 1 \cdot e^{0} + 2 \cdot e^{0} = 3$ .

2.1F1F2givenalgebra∎

For $ξ_{1} = (1 / 3, 1 / 5)$ , the dot products are $ξ_{1} \cdot x_{0} = 1 / 12$ and $ξ_{1} \cdot x_{1} = 1 / 6 + 1 / 20 = 13 / 60$ . Substituting these and the weights in [F1] yields $G_{1} = e^{- 2 π i / 12} + 2 e^{- 2 π i (13 / 60)}$ , as claimed.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Leslie Greengard and June-Yub Lee, Accelerating the Nonuniform Fast Fourier Transform (standard reference, not scraped)