On a basic proof in interpolation: usage of the Fundamental Algebra Theorem

Hi,

I tried to post that message yesterday and upon verification it did
not appear on the forum so I am trying to post it again. Sorry if it
appears twice.

This message concerns a proof given for an identity in Interpolation
(Approximation) theory given in

B.I. Kvasov, "Method of Shape-Preserving Spline Approximation", World
Scientific, Singapore, 2000, p. 9, Lemma 1.1.

My main concern and interrogation is on the use of the Fundamental
Theorem of Algebra in the proof of that Lemma 1.1 on p. 9 of the book
by Kvasov. My goal is not to criticize Dr. Kvasov book and proofs
there in but it is rather to understand better the use of the
Fundamental Theorem of Algebra he made in his proof.

The Lemma states that for any polynomial P_k(x) of degree k <= N we
have that


-- N
\
P_k (x) = / P_k(x_j) l_j(x)
-- j=0

where l_j(x) are the Lagrange coefficient polynomials defined as
follow:

l_j(x) = w_N(x)/((x-x_j) w'_N(x_j)) with w_N(x) = (x-x_o)(x-x_1) ...
(x-x_N) as it can be found in any standard Num Analysis textbook.

The proof given starts by indicating that we only need to prove the
identity for monomials i.e 1, x,x^2,x^3, etc. This is correct as any
polynomial can be represented by a series of monomials with decomposed
in a serie of monomials with appropriate coefficients. The proof
continues where a polynomial of degree N is formed as follow:


-- N
\
F_k,N(x) = x^k - / (x_j)^k l_j(x) for 0<= k <= N.
-- j=0

Then the proof states that polynomial F_k,N(x) as N+1 roots which are
the interpolation nodes i.e x_i for i=0,...,N. This statement is true
in the sense that there are N+1 value of x that makes F_k,N(x) = 0 but
let us recall here that F_k,N(x) is of degree N and that by the
Fundamental Theorem of Algebra a polynomial of degree N cannot have
more that N roots (see for instance, G.H. Hardy, " A Course of Pure
Mathematics", Cambridge University Press, 10th ed, 2006, p. 88). It is
true that for any of the interpolation node x_i, F_k,N(x) will be
equal to zero because for any j different from i we have that by
definition of the l_j(x) that l_j(x_i) = 0 but for j=i we have that
l_i(x_i) = 1. Then we have:

F_k,N(x_i) = (x_i)^k - (x_i)^k l_i(x_i)
= (x_i)^k (1 - l_i(x_i)) and with l_1(x_i) = 1 we have
= (x_i)^k (1 - 1) = 0

Therefore F_k,N(x_i) = 0 and this is valid for all interpolation nodes
x_i, i = 0 to N.
One can say as given in the book of Kvasov that F_k,N(x) has N+1
roots. In his book, Kvasov adds to complete the proof that by the
Fundamental Theorem of Algebra and because F_k,N(x) has N+1 roots,
F_k,N(x) must be identically equal to zero. This where I have an
interrogation on the use of the Fundamental Theorem of Algebra in
manner Kvasov completes the proof as it is difficult for me to put
side-by-side what the Fundamental Theorem of Algebra states that the
number of roots for a polynomial of degree N, P_N(x) = 0 which is at
most N. The converse is also true that if a polynomial of degree N,
P_N(x), has N roots then P_N(x) = 0. How can we justify using the
Fundamental Theorem of Algebra for a polynomial of degree N that has N
+1 roots to state that it is identically equal to zero?

In their book "Analysis of Numerical Methods",Dover Publications, New
York, 1994, Isaacson and Keller make reference in their proof for
Lemma 1 on p. 189, to the "identically vanishing polynomial" as the
only polynomial of degree at most N with more that N roots. Is this
what Kvasov is in fact referring to in his proof of his Lemma 1.1 and
is this related directly or indirectly related to the Fundamental
Theorem of Algebra ?

Also, in their book "Analysis of Numerical Methods",Dover
Publications, New York, 1994, Isaacson and Keller on p. 192 indicates
that the identity

-- N
\
P_k (x) = / P_k(x_j) l_j(x)
-- j=0

can be shown using the Remainder Theorem for interpolation
polynomials.

To summarize, did Kvasov made the correct usage of the Fundamental
Theorem of Algebra
in his proof of the above identity ? Did I missed something here ?

Any comment to clarify the above would be much appreciated.

Regards

Rene



.

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