Chebyshev polynomial approximation to approximate partial differential equations

Theorem 1: if V ∈ '.K[a, b], then for all ε > 0 there exists a polynomial P(s) such that
∀5 ∈ [a, b ] | V (5) - P(5) ∣≤ ε .

Remark 1. The above theorem is known as the Weierstrass theorem. It states that any
continuous function can be approximated with a certain degree of accuracy by using a
polynomial. Although very important theoretically, this theorem is of little practical use
since it does not give any indication of what polynomial is the most appropriate to use, or
even what order polynomial is needed to achieve a certain degree of accuracy.

The error made by using a polynomial of order n to approximate the function
given in Theorem 1 can be easily calculated as:

n

V⁽ⁿ⁺¹⁾(ε)∏(5-5i)
i=0

The objective of using such an efficient polynomial consists in choosing a set of nodes 5_i

n

so as to make the term ∏(5 -5_i) as small as possible (Judd, 1998). One possibility is to
i=0

approximate the function V at the n-evenly spaced nodes. However, it is well known that
in general, even for smooth functions, polynomials of this type do not produce very good
approximations.² Therefore, we suggest approximating the function over the interval
[a,b], at the Chebyshev nodes defined as:

2i+1

si = cos(-------π ), i = 0,1,..., n

ⁱ 2n + 2

Our approach can be justified by appealing to Rivlin’s theorem, stating that Chebyshev
node polynomial interpolants are nearly optimal polynomial approximants (Rivlin, 1990),

²A classic example is Runge's function (Rivlin, 1990).

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