koefficiensek

koefficiensek is a term used in mathematical literature to denote a set of generalized coefficients that quantify the contribution of basis elements in a non-orthogonal expansion of a vector or function. The koefficiensek extend ordinary coefficients by incorporating the weights provided by a dual basis, so that a function f can be represented as f = sum_i c_i phi_i with c_i determined by a corresponding dual functional.

In a concrete framework, let V be a finite-dimensional vector space over R or C with a basis {phi_i} and a dual basis {psi_i} in V*. The koefficiensek c_i of f are defined by c_i = psi_i(f). When the basis is not orthogonal, these coefficients need not coincide with simple inner products, but together with the dual system they yield a stable expansion. The koefficiensek vector C = (c_i) may be stored in a koefficiensek matrix for problems where multiple vectors are expanded with respect to a common basis.

Relation to standard coefficients: If {phi_i} is an orthonormal basis relative to an inner product, then c_i = <f, phi_i>, and the koefficiensek reduce to standard projection coefficients.

Applications: The concept appears in signal processing with non-orthogonal dictionaries, in Galerkin and other numerical methods using non-orthogonal bases, and in sparse representation and data compression, where koefficiensek quantify the influence of each basis element on the reconstruction.

Etymology and usage: The word is a coined term in some texts, serving as a generalized label for expansion coefficients. Its exact definition can vary between authors, so consulting the source is recommended.