Multivariate Approximation by Overactivated and Spiked Multivariate Convolutions as Positive Linear Operators

This study quantitatively approximates the unit operator using three types of multivariate, overactivated, and spiked convolution operators. The core of these operators is a multivariate “cusp” kernel, which acts as a novel, compact-support activation function derived from a constructed S-shaped finite-length arc. This arc is itself formed by combining two general sigmoid activation functions. The operators are multivariate positive linear ones. The research initially establishes their basic convergence properties. It then explores simultaneous and iterated approximations, utilizing inequalities and the multivariate modulus of continuity of the function being approximated.