One such method was the fast Fourier transform.

A direct evaluation of either summation ( above ) requires operations for an output sequence of length N. An indirect method, using transforms, can take advantage of the efficiency of the fast Fourier transform ( FFT ) to achieve much better performance.

Laplace also recognised that Joseph Fourier's method of Fourier series for solving the diffusion equation could only apply to a limited region of space as the solutions were periodic.

* Fourier transform spectroscopy is an efficient method for processing spectra data obtained using interferometers.

The two-dimensional images taken at different rotations are converted into a three-dimensional model of the density of electrons within the crystal using the mathematical method of Fourier transforms, combined with chemical data known for the sample.

As derived below, the electron density within the crystal and the diffraction patterns are related by a simple mathematical method, the Fourier transform, which allows the density to be calculated relatively easily from the patterns.

Since Bragg's law does not interpret the relative intensities of the reflections, however, it is generally inadequate to solve for the arrangement of atoms within the unit-cell ; for that, a Fourier transform method must be carried out.

Formally stated, the FFT is a method for computing the discrete Fourier transform of a sampled signal.

He introduced a class of closed orthogonal sets, developed asymptotic Liouville – Steklov method for orthogonal polynomials, proved theorems on generalized Fourier series, and developed an approximation technique later named Steklov function.

It was the key, for example, to Karatsuba's fast multiplication method, the quicksort and mergesort algorithms, the Strassen algorithm for matrix multiplication, and fast Fourier transforms.

The spectral density is usually estimated using Fourier transform methods, but other techniques such as Welch's method and the maximum entropy method can also be used.

Inspired by the work of on Morse theory, found another proof, using Deligne's l-adic Fourier transform, which allowed him to simplify Deligne's proof by avoiding the use of the method of Hadamard and de la Vallée Poussin.

This method, which has advantages over the fast Fourier algorithm, especially for images, is treated in The Hartley Transform ( 1986 ), in U. S. Patent 4, 646, 256 ( 1987, now in the public domain ), and in over 200 technical papers by various authors that were stimulated by the discovery.

In practice, the difficulties associated with the Gibbs phenomenon can be ameliorated by using a smoother method of Fourier series summation, such as Fejér summation or Riesz summation, or by using sigma-approximation.

* Fourier – Hankel spectral method, first introduced by Braun and Duvall, was originally used to search for wave absorption by sunspots.

* Fourier transform Coulomb method ( FTC )

* Fourier – Motzkin elimination ( FME method ), a mathematical algorithm for eliminating variables from a system of linear inequalities

This method is, in fact, a simple version of the discrete Fourier transform ( DFT ).

The method is based on the Fourier shift theorem.

The advantage of this method is that the discrete Fourier transform and its inverse can be performed using the fast Fourier transform, which is much faster than correlation for large images.

* The spectral method, which represents functions as a sum of particular basis functions, for example using a Fourier series.

Fourier profilometry is a method for measuring profiles using distortions in periodic patterns.

The bilinear transform ( also known as Tustin's method ) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.

He invented the operator method for solving linear differential equations, which resembles current Laplace transform methods ( see inverse Laplace transform, also known as the " Bromwich integral ").

This method, utilizing the Hilbert transform to phase shift the baseband audio, can be done at low cost with digital circuitry.

Inverse transform sampling ( also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, golden rule, etc.

A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random variate U drawn from the uniform distribution on the unit interval ( 0, 1 ), the variate

* A method applicable for particles without spin is to do a polar decomposition of the wavefunction and transform Schrödinger's equation into two coupled equations: the continuity equation from above and the Hamilton – Jacobi equation.

The Box – Muller transform ( by George Edward Pelham Box and Mervin Edgar Muller 1958 ) is a pseudo-random number sampling method for generating pairs of independent, standard, normally distributed ( zero expectation, unit variance ) random numbers, given a source of uniformly distributed random numbers.

Quality function deployment ( QFD ) is a “ method to transform user demands into design quality, to deploy the functions forming quality, and to deploy methods for achieving the design quality into subsystems and component parts, and ultimately to specific elements of the manufacturing process .”, as described by Dr. Yoji Akao, who originally developed QFD in Japan in 1966, when the author combined his work in quality assurance and quality control points with function deployment used in value engineering.

Techniques for solving these include the inverse scattering transform and the method of Lax pairs, the Yang-Baxter equation and quantum groups.

* Bilinear transform, a method of transforming from the S to Z domain in control theory and signal processing

A frequent criticism of coding method is that it seeks to transform qualitative data into quantitative data, thereby draining the data of its variety, richness, and individual character.

Two German chemists, Robert Bunsen and Gustav Kirchhoff, discovered caesium in 1860 by the newly developed method of flame spectroscopy.

The transit method reveals the radius of a planet, and it has the benefit that it sometimes allows a planet's atmosphere to be investigated through spectroscopy.

This ratio of large HDL to total HDL particles varies widely and is measured only by more sophisticated lipoprotein assays using either electrophoresis ( the original method developed in the 1970s ) or newer NMR spectroscopy methods ( See also: NMR and spectroscopy ), developed in the 1990s.

This was a pioneering use of the method of X-ray spectroscopy in physics, using Bragg's diffraction law to determine the X-ray wavelengths.

Moseley's method in early X-ray spectroscopy was able to sort out the above chemical problems promptly, some of which had occupied chemists for a number of years.

German chemists Robert Bunsen and Gustav Kirchhoff discovered rubidium in 1861 by the newly developed method of flame spectroscopy.

Nuclear magnetic resonance ( NMR ) spectroscopy is a widely used resonance method and ultrafast laser methods are also now possible in the infrared and visible spectral regions.

* Auger spectroscopy is a method used to study surfaces of materials on a micro-scale.

A less common method is infrared spectroscopy, which detects differences in the bond

Both used the newly developed method of flame spectroscopy, in which thallium produces a notable green spectral line.

After the publication of the improved method of flame spectroscopy by Robert Bunsen and Gustav Kirchhoff and the discovery of caesium and rubidium in the years 1859 to 1860, flame spectroscopy became an approved method to determine the composition of minerals and chemical products.

For comparison, the nearest competing method in terms of structures analyzed is nuclear magnetic resonance ( NMR ) spectroscopy, which has resolved 9561 chemical structures.

* Energy-dispersive X-ray spectroscopy, a method used to determine the energy spectrum of X-ray radiation

However, CD spectroscopy is a quick method that does not require large amounts of proteins or extensive data processing.

In 1865 English amateur astronomer William Huggins used his visual spectroscopy method to examine the nebula showing it, like other nebula he had examined, was made up of " luminous gas ".

In 1885, Edward Pickering began to supervise photographic spectroscopy at Harvard College Observatory, using the objective prism method.

This method is used in flame emission spectroscopy, and it was also the method used by Anders Jonas Ångström when he discovered the phenomenon of discrete emission lines in 1850s.