Neural network architectures, with connection weights determined using Legendre Memory Unit equations, are trained while optionally keeping the determined weights fixed. Networks may use spiking or non-spiking activation functions, may be stacked or recurrently coupled with other neural network architectures, and may be implemented in software and hardware. Embodiments of the invention provide systems for pattern classification, data representation, and signal processing, that compute using orthogonal polynomial basis functions that span sliding windows of time.