Sine and cosine transforms
In mathematics, the Fourier sine and cosine transforms are forms of the Fourier integral transform that do not use complex numbers. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics.
The Fourier sine transform of f (t), sometimes denoted by either or , is
If t means time, then ν is frequency in cycles per unit time, but in the abstract, they can be any pair of variables which are dual to each other.
This transform is necessarily an odd function of frequency, i.e. for all ν:
The numerical factors in the Fourier transforms are defined uniquely only by their product. Here, in order that the Fourier inversion formula not have any numerical factor, the factor of 2 appears because the sine function has L2 norm of
The Fourier cosine transform of f (t), sometimes denoted by either or , is
It is necessarily an even function of frequency, i.e. for all ν:
Similarly, if f is an odd function, then the cosine transform is zero and the sine transform can be simplified to
Other authors also define the cosine transform as
and sine as
The original function f can be recovered from its transform under the usual hypotheses, that f and both of its transforms should be absolutely integrable. For more details on the different hypotheses, see Fourier inversion theorem.
The inversion formula is
which has the advantage that all frequencies are positive and all quantities are real. If the numerical factor 2 is left out of the definitions of the transforms, then the inversion formula is usually written as an integral over both negative and positive frequencies.
Using the addition formula for cosine, this is sometimes rewritten as
where f (x + 0) denotes the one-sided limit of f as x approaches zero from above, and f (x − 0) denotes the one-sided limit of f as x approaches zero from below.
Relation with complex exponentials
The form of the Fourier transform used more often today is
Using standard methods of numerical evaluation for Fourier integrals, such as Gaussian or tanh-sinh quadrature, is likely to lead to completely incorrect results, as the quadrature sum is (for most integrands of interest) highly ill-conditioned. Special numerical methods which exploit the structure of the oscillation are required, an example of which is Ooura's method for Fourier integrals This method attempts to evaluate the integrand at locations which asymptotically approach the zeros of the oscillation (either the sine or cosine), quickly reducing the magnitude of positive and negative terms which are summed.
- Whittaker, Edmund, and James Watson, A Course in Modern Analysis, Fourth Edition, Cambridge Univ. Press, 1927, pp. 189, 211
- "Highlights in the History of the Fourier Transform". pulse.embs.org. Retrieved 2018-10-08.
- Mary L. Boas, Mathematical Methods in the Physical Sciences, 2nd Ed, John Wiley & Sons Inc, 1983. ISBN 0-471-04409-1
- "Fourier Transform, Cosine and Sine Transforms". cnyack.homestead.com. Retrieved 2018-10-08.
- Poincaré, Henri (1895). Theorie analytique de la propagation de chaleur. Paris: G. Carré. pp. 108ff.
- Takuya Ooura, Masatake Mori, A robust double exponential formula for Fourier-type integrals, Journal of computational and applied mathematics 112.1-2 (1999): 229-241.