Culegere de probleme de Analiz˘a numeric˘a

112 Operatori liniari s¸i pozitivi
de exemplu prin induct¸ie.
(Bmf) (r) (x) = m[r]
m r
xk =
m−r
pm−r,k(x)f (r) (xk)
n=0
k +θkr
0 < θk < 1
m
k k +r
,
m m
xk ∈
(am aplicat formula de medie)
Notăm
C(m,r) = m[r]
= 1−
mr 1
1−
m
2
... 1−
m
r−1
m
f (r) (x)−(Bmf) (r) m−r
(x) = pm−r,k(x)(f (r) (x)−f (r) (xk))+
m−r
k=0
k=0
m−r
+[1−c(m,r)]
k=0
pm−r,k(x)f (r) (xk)
pm−r,k(x)|f (r) (xk)| ≤ Mr(f) = sup |f
x∈[0,1]
(r) (x)|
(1−a1)...(1−ar−1) ≥ 1−(a1 +···+ar−1)
dacăa1,...,ar−1 ≤ 1 de acelas¸i semn
Putem scrie
C(m,r) ≥ 1−
1+2+···+(r −1)
m
= 1−
|f (r) (x)−(Bmf) (r) m−r
(x)| ≤ pm−r,k(x)|f (r) (x)−f (r) (xk)|
Fie
k=0
S
Fm = {k||x−xk| ≤ δ}
Jm = {k||x−xk| > δ}
r(r −1)
m
+ r(r −1)
2m Mr(f)