KosinusTerminen

KosinusTerminen is a term used in trigonometric series to denote the cosine components in a Fourier-like expansion of a periodic function. In the standard Fourier series for a real-valued function f with period 2π, the expansion is f(x) = a0/2 + sum from n=1 to ∞ of [an cos(nx) + bn sin(nx)]. The cosine terms are the set of terms an cos(nx) for n ≥ 1, together with the constant a0/2 forming the even part of the series.

The cosine coefficients an are determined by integrating f against cos(nx) over a full period. A common normalization is an = (1/π) ∫_{-π}^{π} f(x) cos(nx) dx, though conventions can vary by interval or factor. The cosine terms capture the even component of f: if f is an even function, all sine coefficients bn vanish, and the Fourier series consists solely of cosine terms.

Applications and context: KosinusTerminen appear in harmonic analysis and signal processing, where trigonometric series decompose periodic signals into simpler oscillatory components. They are central to solving partial differential equations with periodic boundary conditions and to numerical methods for function approximation. A closely related concept is the discrete cosine transform (DCT), which uses a finite set of cosine basis functions to represent data, notably in image and video compression.

Interpretation: Each cosine term an cos(nx) represents the amplitude of the nth cosine basis component in the overall reconstruction. The collection of all such terms forms the cosine portion of the Fourier representation, reflecting the even-symmetric content of the function. See also Fourier series, cosine transform, and DCT.