Causality structure and the Weierstrass theorem

AbstractIn a recent paper P. M. Prenter has shown that the Weierstrass theorem can be lifted up to a real separable Hilbert space H. In this paper H is equipped with an identity resolving orthoprojector chain. The Weierstrass type result of Prenter, namely, if ƒ is any continuous function on H, then there exists a finite order approximating polynomic operator on every compact K ⊂ H, is sharpened by the extension: if ƒ is strictly causal (strictly anticausal) then the polynomic approximation can also be strictly causal (strictly anticausal). Other extensions in the same spirit are developed and the results are interpreted in the setting of Volterra operators on L2