Sentence examples for the proof of the approximation from inspiring English sources

We only give the proof of the approximation order.

The following fixed-point theorem is fundamental for the proof of the best approximation theorem.

One can now easily prove on the lines of the proof of the above theorem that the approximation results are similar to those of Theorems 2.11-2.12 due to Hussain and Jungck [7] in the setting of a Hausdorff locally convex space.

In the next section we recall some concepts and facts associated to Carnot Carathéodory spaces and give the proof of the modified A-harmonic approximation lemma for vector fields.

In fact, the proof of the convergence of the Carathéodory approximation represents an alternative to the procedure for establishing the existence and uniqueness of the solution to delay DPSDEs.

In this paper we introduce a family of weakly contractive maps on the space (B(S)) of bounded real valued functions and use it to show that a fundamental step in the proof of the well-known Stone-Weierstrass approximation theorem can be achieved via Rakotch's fixed point theorem for weakly contractive maps.

In Section 2 we introduce a family of weakly contractive maps on the space (B(S)) of bounded real functions and show that Rakotch's theorem can be used to prove a result of Zemanek [8] about the existence of square roots in some Banach subalgebras of (B(S)), a result which is a fundamental step in the proof of the well-known Stone-Weierstrass approximation theorem [9].

For the proof of the main result, we adopt an approximation scheme following the well-known mollify technique.

Remark 2. Note that, in contrast to the stochastic approximation approach for the proof of the convergence in the complete observation case, we used a different approach to prove the convergence of the learning with partial observations since it is difficult to apply the stochastic approximation in the partial observation case.

□ A priori estimates for the discrete approximation values and increments are basic tools in the proof of the convergence result.

The proof of the pecking order?