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Asked Sep 29, 2019
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Hello, kindly assist me with the solution to Q4. I will appreciate it if you provide a very detailed solution, thanks

Problem 7.14. Let (æ1,...,am+1) be a sequence of pairwise distinct scalars in R and let
(B1,...,Bm+1) be any sequence of scalars in R, not necessarily distinct.
(1) Prove that there is a unique polynomial P of degree at most m such that
P(a4) Bi, 1i <m+1.
Hint. Remember Vandermonde!
(2) Let Li(X) be the polynomial of degree m given by
(X - a1)(X- a1-1)(X -a441) (X -am+1)
(a-a1)(a- ai-1)(a-ai+1)
1 im
L&(X)
(a4-am+1)
The polynomials L'(X)
Lagrange polynomial interpolants. Prove that
are known as
Li(aj) 6 1 i,j <m+1
Prove that
Bm+1 Lm+1(X)
Р(X) — BiL1(X) +
is the unique polynomial of degree at most m such that
P(a4) Bi, 1< i <m+1
(3) Prove that L1(X),..., Lm+1(X) are linearly independent, and that they form a basis
of all polynomials of degree at most m
How is 1 (the constant polynomial 1) expressed
over the basis (L1(X),..., Lm+1(X) )?
Give the expression of every polynomial P(X) of degree at most m over the basis
(L1(X), ...,m+1(X)
(4) Prove that the dual basis (Li, ..., L1)of the basis (L1(X),..., Lm+1(X)) consists
of the linear forms L given by
L;(P) P(a)
for every polynomial P of degree at most m; this is simply evaluation at a
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Problem 7.14. Let (æ1,...,am+1) be a sequence of pairwise distinct scalars in R and let (B1,...,Bm+1) be any sequence of scalars in R, not necessarily distinct. (1) Prove that there is a unique polynomial P of degree at most m such that P(a4) Bi, 1i <m+1. Hint. Remember Vandermonde! (2) Let Li(X) be the polynomial of degree m given by (X - a1)(X- a1-1)(X -a441) (X -am+1) (a-a1)(a- ai-1)(a-ai+1) 1 im L&(X) (a4-am+1) The polynomials L'(X) Lagrange polynomial interpolants. Prove that are known as Li(aj) 6 1 i,j <m+1 Prove that Bm+1 Lm+1(X) Р(X) — BiL1(X) + is the unique polynomial of degree at most m such that P(a4) Bi, 1< i <m+1 (3) Prove that L1(X),..., Lm+1(X) are linearly independent, and that they form a basis of all polynomials of degree at most m How is 1 (the constant polynomial 1) expressed over the basis (L1(X),..., Lm+1(X) )? Give the expression of every polynomial P(X) of degree at most m over the basis (L1(X), ...,m+1(X) (4) Prove that the dual basis (Li, ..., L1)of the basis (L1(X),..., Lm+1(X)) consists of the linear forms L given by L;(P) P(a) for every polynomial P of degree at most m; this is simply evaluation at a

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Expert Answer

Step 1

Basis and Dual basis are given as,

and (4.5.
(4.L.
0 if i
j
.1)
if i= j
m+1
So, LL
1 if i = j
[0 if i
Since, L, ()1 if i = j
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and (4.5. (4.L. 0 if i j .1) if i= j m+1 So, LL 1 if i = j [0 if i Since, L, ()1 if i = j

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Step 2

Hence, equation (1) can also be written as

L(L),(a)
= L
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L(L),(a) = L

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Step 3

Above equation can also be writt...

(P) P(a)
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(P) P(a)

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