### Fourier cosine transform

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.

## Contents

## Definition

The **Fourier sine transform** of $f\; (t)$, sometimes denoted by either $\{\backslash hat\; f\}^s$ or $\{\backslash mathcal\; F\}\_s\; (f)$, is

- $2\; \backslash int\backslash limits\_\{-\backslash infty\}^\backslash infty\; f(t)\backslash sin\backslash ,\{2\backslash pi\; \backslash nu\; t\}\; \backslash ,dt.$

If $t$ means time, then $\backslash nu$ 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.,

- $\{\backslash hat\; f\}^s(\backslash nu)\; =\; -\; \{\backslash hat\; f\}^s(-\backslash nu)$ for all $\backslash nu$.

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 $L^2$ norm of $\backslash frac\; 1\; \{\backslash sqrt2\}$.

The **Fourier cosine transform** of $f\; (t)$, sometimes denoted by either $\{\backslash hat\; f\}^c$ or $\{\backslash mathcal\; F\}\_c\; (f)$, is

- $2\; \backslash int\backslash limits\_\{-\backslash infty\}^\backslash infty\; f(t)\backslash cos\backslash ,\{2\backslash pi\; \backslash nu\; t\}\; \backslash ,dt.$

It is necessarily an even function of $\backslash nu$, i.e., $\{\backslash hat\; f\}^s(\backslash nu)\; =\; \{\backslash hat\; f\}^s(-\backslash nu)$ for all $\backslash nu$.

Some authors^{[1]} only define the cosine transform for even functions of $t$, in which case its sine transform is zero. Since cosine is also even, a simpler formula can be used, $4\; \backslash int\backslash limits\_0^\backslash infty\; f(t)\backslash cos\backslash ,\{2\backslash pi\; \backslash nu\; t\}\; \backslash ,dt.$ Similarly, if $f$ is an odd function, then the cosine transform is zero and the sine transform can be simplified to $4\; \backslash int\backslash limits\_0^\backslash infty\; f(t)\backslash sin\backslash ,\{2\backslash pi\; \backslash nu\; t\}\; \backslash ,dt.$

## Fourier inversion

The original function $f(t)$ can be recovered from its transforms 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^{[2]}

- $f(t)\; =\; \backslash int\; \_0^\backslash infty\; \{\backslash hat\; f\}^c\; \backslash cos\; (2\backslash pi\; \backslash nu\; t)\; d\backslash nu\; +\; \backslash int\; \_0^\backslash infty\; \{\backslash hat\; f\}^s\; \backslash sin\; (2\backslash pi\; \backslash nu\; t)\; d\backslash nu,$

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

- $\backslash frac\backslash pi2\; (f(x+0)+f(x-0))\; =\; \backslash int\; \_0^\backslash infty\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash cos\; \backslash omega\; (t-x)\; f(t)\; dt\; d\backslash omega,$

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.

If the original function $f$ is an even function, then the sine transform is zero; if $f$ is an odd function, then the cosine transform is zero. In either case, the inversion formula simplifies.

## Relation with complex exponentials

The form of the Fourier transform used more often today is

- $$

\hat f(\nu) = \int\limits_{-\infty}^\infty f(t) e^{-2\pi i\nu t}\,dt.

Expanding the integrand by means of Euler's formula results in

- $=\; \backslash int\backslash limits\_\{-\backslash infty\}^\backslash infty\; f(t)(\backslash cos\backslash ,\{2\backslash pi\backslash nu\; t\}\; -\; i\backslash ,\backslash sin\{2\backslash pi\backslash nu\; t\})\backslash ,dt,$

which may be written as the sum of two integrals

- $=\; \backslash int\backslash limits\_\{-\backslash infty\}^\backslash infty\; f(t)\backslash cos\backslash ,\{2\backslash pi\; \backslash nu\; t\}\; \backslash ,dt\; -\; i\; \backslash int\backslash limits\_\{-\backslash infty\}^\backslash infty\; f(t)\backslash sin\backslash ,\{2\backslash pi\; \backslash nu\; t\}\backslash ,dt,$

- $=\; \backslash frac\; 12\; \{\backslash hat\; f\}^c\; (\backslash nu)\; -\; \backslash frac\; i2\; \{\backslash hat\; f\}^s\; (\backslash nu).$

## See also

## References

- Whittaker, Edmund, and James Watson,
*A Course in Modern Analysis*, Fourth Edition, Cambridge Univ. Press, 1927, pp. 189, 211