The Bernstein polynomial of degree n for f e C[0, 1]is given by´kB,(x) = E (; ) f (;) +*c(1 – xy"-*,|kk=0where (") denotes n!/k!(n – k)!.а.Find B3(x) for the functionsii.f (x) = 1 i. f (x) = x%3D:) - ()kb. Show that for each k < n,kk – 1Use part (b) and the fact, from (ii) in part (a), that .cп1 = EC)*(1 –x)"-k, for eachkk=0to show that, for f (x) = x²,("")*+*1x + -x.nB„(x) =

Given- The Berstein polynomial of degree n for f ∈ C0, 1 is given by Bnx = ∑k = 0nnk fknxk1 - xn-k,…

The Bernstein polynomial of degree n for f e C[0, 1] is given by B„(x) = k r*(1 – x)"-k, k k=0 where (") denotes n!/k!(n – k)!. а. Find B3(x) for the functions ii. f (x) = 1 i. f (x) = x - 1 b. Show that for each k < n, k – 1 Use part (b) and the fact, from (ii) in part (a), that .c r*(1 – x)"-k, for each n, k k=0 to show that, for f (x) = x², ("금)- 1 x + -x. n 1 В, (х) — n

The Bernstein polynomial of degree n for f e C[0, 1] is given by B„(x) = k r(1 – x)"-k, k k=0 where (") denotes n!/k!(n – k)!. а. Find B3(x) for the functions ii. f (x) = 1 i. f (x) = x - 1 b. Show that for each k < n, k – 1 Use part (b) and the fact, from (ii) in part (a), that .c r(1 – x)"-k, for each n, k k=0 to show that, for f (x) = x², ("금)- 1 x + -x. n 1 В, (х) — n

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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The Bernstein polynomial of degree n for f e C[0, 1]
is given by
´k
B,(x) = E (; ) f (;) +*c(1 – xy"-*,
|
k
k=0
where (") denotes n!/k!(n – k)!.
а.
Find B3(x) for the functions
ii.
f (x) = 1 i. f (x) = x
%3D
:) - ()
k
b. Show that for each k < n,
k
k – 1
Use part (b) and the fact, from (ii) in part (a), that .c
п
1 = EC)*(1 –x)"-k, for each
k
k=0
to show that, for f (x) = x²,
("")*+*
1
x + -x.
n
B„(x) =

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