A Proof of Theorem

Eq. 7 implies that gradient operator is Below we supplement the Lemma A.1 used to prove Theorem 1. ( j 1) Jacobian matrix, the second equality is due to the induction hypothesis, and the third equality is an adoption of chain rule. Then by induction, we can conclude the proof. For a sake of clarity, we first introduce few notations in algebra and real analysis. Definition B.2. (Differential Operator) Suppose a compact set Definition B.4. (F ourier Transform) Given real-valued function Definition B.5. (Convolution) Given two real-valued functions Before we prove Theorem 2, we enumerate the following results as our key mathematical tools: First of all, we note the following well-known result without a proof. Lemma B.2. (Stone-W eierstrass Theorem) Suppose A C ( X, R) is a unital sub-algebra which separates points in X .