PIderivaten

PIderivaten is a theoretical class of differential operators designed to generalize the classical derivatives by incorporating a pi-centered integral transform. The term combines a geometric intuition, drawn from the constant pi and circular symmetry, with a nonlocal view of change, aiming to capture both local and near-field variations of a function.

Definition and conceptually, the PI-derivative of order n, applied to a suitably smooth function f, is obtained by a linear combination of weighted local changes around a point x, where the weights form a kernel influenced by pi. The operator is constructed to be linear and to recover the standard n-th derivative in a local limit when the kernel becomes highly concentrated at x. In general, the PI-derivative is nonlocal, meaning it integrates information from a neighborhood of x rather than relying solely on an infinitesimal interval.

Key properties include linearity, nonlocality, and a dependence on a scale parameter that controls the neighborhood size. The kernel is typically chosen to be symmetric, positive, and to sum to one in a normalization, ensuring stable behavior under translations and modest boundary handling. Numerical implementations rely on quadrature or fast convolution techniques, with attention paid to edge effects and discretization error.

Applications of PIderivaten appear in numerical differentiation for irregular or noisy data, geometric modeling and computer graphics, regularization of partial differential equations, and signal processing contexts where nonlocal information improves robustness. The concept sits within a broader family of nonlocal derivatives and kernel-based operators, linked to fractional calculus and integral transforms.

See also: derivative, fractional derivative, nonlocal operator, kernel method, integral transform.