Approximation by Overactivated and Spiked Convolutions as Positive Linear Operators

In this work, the author studied the quantitative approximation to the unit operator of three kinds of overactivated and spiked convolution type-operators. These operators have as a kernel a cusp coming from a constructed S-shaped finite-length arc, serving as a new activation function of compact support. This is derived from the composition of two general sigmoid activation functions with domain all reals. Our operators are positive linear ones and are treated as such. Initially we establish the basic convergence, then we move on to simultaneous and iterated approximations, all via inequalities and involving the modulus of continuity of the approximated univariate function.