Untitled - Cdm.unimo.it

Results in Approximation Theory 95
We now collect some result on the problem of best approximation. We begin with the
celebrated theorem by Jackson (see for instance feinerman and newman(1974)).
Theorem 6.1.2 (Jackson) - Let I =]−1,1[, then it is possible to find a constant C > 0
such that, for any f ∈ C0 ( Ī), we have
(6.1.6) f − Ψ∞,n(f)∞ ≤ C sup
|x1−x2| 0 such that, for any f ∈ Ck ( Ī), we have
k 1
(6.1.7) f − Ψ∞,n(f)∞ ≤ C sup
n
|x1−x2| 0. Then, for any ǫ > 0, we can find n ∈ N and pn ∈ Pn such that
(6.1.8) |f(x) − pn(x)| e −δx < ǫ, ∀x ∈ Ī.
Theorem 6.1.5 - Let I = R and let f ∈ C 0 (R) satisfy limx→±∞ f(x)e −δx2
for a given δ > 0. Then, for any ǫ > 0, we can find n ∈ N and pn ∈ Pn such that
(6.1.9) |f(x) − pn(x)| e −δx2
< ǫ, ∀x ∈ R.
= 0,