content_image/326348.jpg content_image/326347.jpg content_image/326349.jpg content_image/326350.jpg content_image/326351.jpg
kernel-based discretized version of η with grid G$_{n}$ , that is, 

where for i = 1 , . . . , d 

<img src='content_image/326348.jpg'>

<img src='content_image/326347.jpg'>

K : [0 , ∞ ) → [0 , 1] is the continuous and strictly decreasing kernel function sat- isfying condition (6) and h$_{n}$ , n ∈ N , is some positive sequence. We have already seen in Lemma 3.1 that g$_{i,n}$ ( t ) ∈ [0 , 1], t ∈ [0 , 1] $^{k}$, n ∈ N . Furthermore we have the following result. 

Lemma 3.3. Choose t ∈ [0 , 1] $^{k}$. There is a sequence i ( n ) , n ∈ N , such that t ∈ ⋂ n ∈ $_{N}$N ( s$_{i}$$_{(}$$_{n}$$_{)}$$_{,n}$ ) . Define g$_{i}$$_{(}$$_{n}$$_{)}$$_{,n}$ and ε$_{n}$ as above, n ∈ N . Then 

if ε$_{n}$ →$_{n}$$_{→∞}$ 0 , h$_{n}$ →$_{n}$$_{→∞}$ 0 , ε$_{n}$/h$_{n}$ →$_{n}$$_{→∞}$ ∞ . 

<img src='content_image/326349.jpg'>

Proof. Let t ∈ [0 , 1] k and choose a sequence i ( n ), n ∈ N , as above. Put for simplicity s$_{i}$$_{(}$$_{n}$$_{)}$$_{,n}$ =: s$_{i,n}$ and g$_{i}$$_{(}$$_{n}$$_{)}$$_{,n}$ =: g$_{i,n}$ . We have 

<img src='content_image/326350.jpg'>

From t ∈ N ( s$_{i,n}$ ) we conclude ‖ t − s$_{i,n}$ ‖ ≤ ε$_{n}$ . Hence, we have due to (1) and the properties of the kernel function K 

<img src='content_image/326351.jpg'>

