On a family of approximation operators

AbstractStarting from a defining differential equation (∂∂t) W(λ, t, u) = (λ(u − t)p(t)) W(λ, t, u) of the kernel of an exponential operator Sλ(ƒ, t) = ∫−∞∞ W(λ, t, u)ƒ(u) du with normalization ∫−∞∞ W(λ, t, u) du = 1, we determine Sλ for various p(t) including; for example, p(t) a quadratic polynomial, all the known exponential operators are recovered and some new ones are constructed. It is shown that all the exponential operators are approximation operators. Further approximation properties of these operators are discussed. For example, functions satisfying ∥ Sλ(ƒ, t) − ƒ(t)∥ = O(λ−α) are characterized. Several results of C. P. May are also improved