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Another remark:
If f is in C[0,1] , f:[0,1]-->R is derivable on [0,1], b is fixed in
(0,1),
and the (uni-lateral) derivatives f"(b-) , f"(b+) exist, then
following asymptotic formula is valid:
lim_{n-->infty}A_n(b)*( (S_nf)(b) -f(b)-
- B_n(b)*(f"(b+) - f"(b-)) ) = (f"(b+)+f"(b-))/4 ,
where S_nf is defined as above (see my posted message)
and
A_n(b)= n^2/( {nb}(1-{nb}) ) ,
B_n(b)= {nb}(1-{nb})(1-2{nb})/(4n^2) .
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Likewise, if f is in C[0,1] and the derivatives f'(b+), f'(b-) exist,
then
lim_{n-->infty}C_n(b)*( (S_nf)(b) - f(b) )= ( f'(b+)-f'(b-) )/2
with
C_n(b):=A_n(b)/n= n/({nb}(1-{nb}) ) .
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