Multivariate Perturbed Hyperbolic Tangent-Activated Singular Integral Approximation

Here we study the quantitative multivariate approximation of perturbed hyperbolic tangent-activated singular integral operators to the unit operator. The engaged neural network activation function is both parametrized and deformed, and the related kernel is a density function on R N . We exhibit uniform and L p , p ≥ 1 approximations via Jackson-type inequalities involving the first L p modulus of smoothness, 1 ≤ p ≤ ∞ . The differentiability of our multivariate functions is covered extensively in our approximations. We continue by detailing the global smoothness preservation results of our operators. We conclude the paper with the simultaneous approximation and the simultaneous global smoothness preservation by our multivariate perturbed activated singular integrals.