Convolutionele

Convolutionele is a term used in some mathematical and computational literature to denote a generalized convolution operator. It refers to a family of linear operators that extend classical convolution to non-uniform, adaptive, and domain-specific kernels, enabling smoothing, filtering, and feature extraction on diverse data spaces.

In the standard setting, a kernel k(x,y) defines an integral operator (C_k f)(x) = ∫ k(x,y) f(y) dy. If k(x,y) = k(x−y) and the domain is translation-invariant, this reduces to ordinary convolution. Convolutionele kernels may depend on x, y, or both, allowing spatially varying filters, anisotropy, and manifold or graph domains. The operators are linear and continuous under appropriate function spaces; spectral analysis can be applied via eigenfunction decomposition or generalized Fourier transforms if a suitable symmetry group exists.

Applications include image and signal processing with adaptive smoothing, medical imaging, computer graphics, geophysical data analysis, and signal processing on irregular domains such as graphs and meshes. In machine learning, convolutionele concepts underpin generalized graph convolutions and non-stationary filtering in neural networks, offering a framework for locality and weight sharing beyond traditional convolution.

Related concepts include convolution, integral operators, graph convolution, and kernel methods, with the convolutionele framework often viewed as a broad generalization of these ideas to non-uniform or domain-specific settings.