Preliminary Data A two-dimensional Fourier transform (2D-FT) is computed numerically or carried out in two stages, both involving `standard', one-dimensional Fourier transforms. However, the second stage Fourier transform is not the inverse Fourier transform (which would result in the original function that was transformed at the first stage), but a Fourier transform in a second variable- which is `shifted' in value- relative to that involved in the result of the first Fourier transform. Such 2D-FT analysis is a very powerful method for three-dimensional reconstruction of polymer and biopolymer structures by two-dimensional Nuclear Magnetic Resonance (2D-NMR, [1]) of solutions for molecular weights () of the dissolved polymers up to about 50,000 . For larger biopolymers or polymers, more complex methods have been developed to obtain the desired resolution needed for the 3D-reconstruction of higher molecular structures, e.g. for , methods that can also be utilized in vivo. The 2D-FT method is also widely utilized in optical spectroscopy, such as 2D-FT NIR hyperspectral imaging, or in MRI imaging for research and clinical, diagnostic applications in Medicine.

A more precise mathematical definition of the `double' Fourier transform involved is specified next, and a precise example follows the definition.

Definition 0.1   A 2D-FT, or two-dimensional Fourier transform, is a standard Fourier transformation of a function of two variables, , carried first in the first variable , followed by the Fourier transform in the second variable of the resulting function . (For further specific details and example for 2D-FT Imaging v. URLs provided in the following recent Bibliography).

Example 0.1 A 2D Fourier transformation and phase correction is applied to a set of 2D NMR (FID) signals yielding a real 2D-FT NMR `spectrum' (collection of 1D FT-NMR spectra) represented by a matrix whose elements are

with taking all possible single-quantum frequency values and with the summation carried out over all discrete, double quantum frequencies .

Example 0.2 2D-FT STEM Images (obtained at Cornell University) of electron distributions in a high-temperature cuprate superconductor `paracrystal' reveal both the domains (or `location') and the local symmetry of the “pseudo-gap” in the electron-pair correlation band responsible for the high-temperature superconductivity effect (a definite possibility for the next Nobel (?) iff the mathematical physics treatment is also developed to include also such results).
Remark: So far there have been three Nobel prizes awarded for 2D-FT NMR/MRI during 1992-2003, and an additional, earlier Nobel prize for 2D-FT of X-ray data (`CAT scans'); recently the advanced possibilities of 2D-FT techniques in Chemistry, Physiology and Medicine received very significant recognition.

Bibliography

1

Kurt Wütrich: 1986, NMR of Proteins and Nucleic Acids., J. Wiley and Sons: New York, Chichester, Brisbane, Toronto, Singapore. (Nobel Laureate in 2002 for 2D-FT NMR Studies of Structure and Function of Biological Macromolecules); 2D-FT NMR Instrument Image Example: a JPG color image of a 2D-FT NMR Imaging `monster' Instrument

2

Richard R. Ernst. 1992. Nuclear Magnetic Resonance Fourier Transform (2D-FT) Spectroscopy. Nobel Lecture, on December 9, 1992.

3

Peter Mansfield. 2003. Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI.

4

D. Benett. 2007. PhD Thesis. Worcester Polytechnic Institute. (lots of 2D-FT images of mathematical, brain scans.) PDF of 2D-FT Imaging Applications to MRI in Medical Research.

5

Paul Lauterbur. 2003. Nobel Laureate in Physiology and Medicine for (2D and 3D) MRI.

6

Jean Jeener. 1971. Two-dimensional Fourier Transform NMR, presented at an Ampere International Summer School, Basko Polje, unpublished. A verbatim quote follows from Richard R. Ernst's Nobel Laureate Lecture delivered on December 2nd, 1992, ``A new approach to measure two-dimensional (2D) spectra has been proposed by Jean Jeener at an Ampere Summer School in Basko Polje, Yugoslavia, 1971 ([6]). He suggested a 2D Fourier transform experiment consisting of two pulses with a variable time between the pulses and the time variable measuring the time elapsed after the second pulse as shown in Fig. 6 that expands the principles of Fig. 1. Measuring the response of the two-pulse sequence and Fourier-transformation with respect to both time variables produces a two-dimensional spectrum of the desired form (62,63). This two-pulse experiment by Jean Jeener is the forefather of a whole class of experiments (8,63) that can also easily be expanded to multidimensional spectroscopy.''