Kosinustransformation

Kosinustransformation is a linear transform that uses cosine functions as its basis to convert a function or data sequence into a spectrum of coefficients. It is discussed in both continuous and discrete forms and is closely related to the family of cosine transforms used in signal processing and numerical analysis.

In the continuous form, defined on an interval [0, L], the forward Kosinus transform is F(k) = ∫_0^L f(x) cos(π k x / L) dx for k = 0, 1, ..., K. The inverse reconstruction uses f(x) = (2/L) ∑_{k=0}^K F(k) cos(π k x / L), with normalization adjusted for energy preservation. The kernel is orthogonal on [0, L] with respect to suitable weight.

In the discrete setting, for a sequence x_n with n = 0, ..., N−1, several conventions exist. A common variant (analogous to the discrete cosine transform Type II) is X_k = ∑_{n=0}^{N−1} x_n cos(π k (n+1/2)/N), k = 0, ..., N−1, with the inverse combining X_k with the same cosine basis. Different normalizations yield unitary, orthonormal, or energy-preserving forms.

Because the basis functions are real and bounded, the transform is widely used for spectral analysis of real signals, image compression, and as a tool for solving partial differential equations with cosine-symmetric boundary conditions. Algorithms such as the fast cosine transform enable efficient computation, comparable in speed to the fast Fourier transform.

In practical literature, the term Kosinustransformation is sometimes used interchangeably with the discrete cosine transform or with cosine transform variants, depending on regional naming conventions. See also Fourier transform, Discrete Cosine Transform, and spectral methods.