15approximaatio

15approximaatio is a term used in certain mathematical and computational contexts to denote a finite-term approximation technique. The name blends the numeral 15 with the Finnish word approximatio (approximation), reflecting a convention in which exactly 15 terms are used to represent a target function or data set.

Definition: In 15approximaatio, a target function f defined on a domain D is approximated by a linear combination of 15 basis functions φ_i, i = 1,...,15: f_approx(x) = sum_{i=1}^{15} a_i φ_i(x). The coefficients a_i are chosen to minimize an error measure, typically the least-squares error with respect to observed data or an integral norm over D.

Construction and basis: The basis {φ_i} can be any suitable functions (polynomials, trigonometric, wavelets, or problem-specific kernels). The 15-term constraint distinguishes it from methods that use variable or more flexible model orders. In practice, the basis may be orthogonalized to simplify coefficient computation.

Applications: The method appears in contexts such as data smoothing, simple curve fitting, and educational demonstrations of approximation principles. It is especially used where a compact, interpretable model is preferred and where the number of terms is fixed for standardization.

Advantages and limitations: Benefits include fixed complexity, straightforward interpretation, and fast computation for small models. Limitations include potential underfitting for complex functions, sensitivity to the choice of basis, and reduced flexibility compared with adaptive or high-term models.

See also: Least squares approximation, polynomial approximation, Fourier and wavelet approximations, best m-term approximation.