This is a note for the following paper:
F. Cucker, S. Smale, On the mathematical foundations of learning, Bulletin of The American Mathematical Society, 39(1), 1-49, 2001.
Page 6, Remark 2
In addition, 
, the error above specializes to the error mentioned in that discussion, and the regression function 
of 
coincides with 
execpt for a set of measure zero in 
.
Note:
For a given 
, we have
For the regression function, we may have
%20%3D%20%5Cint_%7BY%7Dyd%5Crho(y%7Cx)%20%3D%201%5Crho(1%7Cx)%20%2B%200%5Crho(0%7Cx)%2C)
where %20%3D%201)
for 
and %20%3D%200)
for 
. Hence, we find that
%20%3D%20%5Cleft%5C%7B%5Cbegin%7Baligned%7D%20%26%201%20%5Cquad%20x%5Cin%20T%2C%20%5C%5C%20%26%200%20%5Cquad%20x%5Cnotin%20T%2C%20%5Cend%7Baligned%7D%5Cright.)
which coincides with .
Page 10, line 6
Thus, 
is a vector space of dimension
.)
Note:
Obviously, the conclusion is correct for 
. We employ second mathematical induction to illustrate the result. Suppose the result is correct for 
to 
, let us verify the case 
. For the additional dimension, we can let 
. Then, the number of possible ways should be
%20%2B%20%5Cleft(%5Cbegin%7Baligned%7D%20%26%20n-1%2Bd%20-%201%20%5C%5C%20%26%20%5Cqquad%20n-1%20%5Cend%7Baligned%7D%5Cright)%20%2B%20%5Ccdots%20%2B%20%5Cleft(%5Cbegin%7Baligned%7D%20%26%20n-1%20%5C%5C%20%26%20n-1%20%5Cend%7Baligned%7D%5Cright).)
Written the above formula in a concise manner, we obtain
%20%3D%20%5Csum_%7Bi%3D0%7D%5E%7Bd%7D%5Cleft(%5Cbegin%7Baligned%7D%20%26%20n-1%2Bi%20%5C%5C%20%26%20%5Cquad%5Cquad%20i%20%5Cend%7Baligned%7D%5Cright)%20%3D%20%5Cleft(%5Cbegin%7Baligned%7D%20%26%20n%20%2B%20d%20%5C%5C%20%26%20%5Cquad%20d%20%5Cend%7Baligned%7D%5Cright)%20%3D%20%5Cleft(%5Cbegin%7Baligned%7D%20%26%20n%20%2B%20d%20%5C%5C%20%26%20%5Cquad%20n%20%5Cend%7Baligned%7D%5Cright).)
These calculations verify the desired conclusions.
Page 21, The proof of Proposition 7
Proposition 7 follows from Lemma 8 by applying the same argument used to prove Theorem B from Proposition 3
Note:
Let )
and consider 
such that the disks 
centered at 
and with radius 
cover 
. Then for every 
, we have 
. Employing Lemma 8, we find that
-%5Cmathcal%7BE%7D_%7B%5Cmathcal%7BH%7D%2Cz%7D(f)%7D%7B%5Cmathcal%7BE%7D_%7B%5Cmathcal%7BH%7D%7D(f)%2B%5Cepsilon%7D%20%5Cgeq%203%5Calpha%5Cright%5C%7D%20%5C%5C%20%5Cleq%20%26%20%5Ctext%7BProb%7D_%7Bz%5Cin%20Z%5Em%7D%5Cleft%5C%7B%20%5Cfrac%7B%20%5Cmathcal%7BE%7D_%7B%5Cmathcal%7BH%7D%7D(f_j)-%5Cmathcal%7BE%7D_%7B%5Cmathcal%7BH%7D%2Cz%7D(f_j)%7D%7B%5Cmathcal%7BE%7D_%7B%5Cmathcal%7BH%7D%7D(f_j)%2B%5Cepsilon%7D%20%5Cgeq%203%5Calpha%5Cright%5C%7D%20%5Cleq%20e%5E%7B-%5Cfrac%7B%5Calpha%5E2%20m%5Cepsilon%7D%7B8M%5E2%7D%7D.%20%5Cend%7Balign%7D)
Proposition 7 has been proved.
Page 27, Proof of Theorem 3
First note that by replacing 
by 
we can reduce the problem in both part (1) and (2) to the case 
Note:
Since 
is equivalent to 
with 
. From the proof, especially the formula of 
, we know that
holds true when 
. Replacing with 
, we obtain
Finally, we arrive at
with 
. Similarly, we can deduce the estimation (2). Here, the result (1) is slightly different from the statment in Theorem 3. It may be a small mistake.
