Generalized Derivatives in Hilbert Spaces

In this paper, we present a novel approach to derivatives of scalar fields in Hilbert spaces, which extends the concept of Lp-derivatives introduced by Calderón and Zygmund for the case of p=2. We investigate the properties of these generalized derivatives, provide characterizations, and derive explicit formulas to compute them. As a consequence, we establish a necessary and sufficient condition for the existence of the best local approximation in Hilbert spaces by employing these new concepts of derivatives. Furthermore, we show that the local best approximation can be expressed as a generalized Taylor polynomial involving the generalized derivatives. This result provides a practical tool to determine when the best local approximation exists and presents an explicit formula for computing it based on the Hilbert partial derivatives.