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Lagrange Interpolation
Lagrange interpolation is a well known, classical technique for
interpolation [177]. It is also called Waring-Lagrange
interpolation, since Waring actually published it 16 years before
Lagrange [291, p. 323]. More generically, the term
polynomial interpolation normally refers to Lagrange interpolation.
In the first-order case, it reduces to linear interpolation.
Given a set of known samples ,
, the
problem is to find the unique order polynomial which
interpolates the samples.I.1The solution can be expressed as a linear combination of elementary
th order polynomials:
where
From the numerator of the above definition, we see that is an
order polynomial having zeros at all of the samples except the
th. The denominator is simply the constant which normalizes its
value to at . Thus, we have
In other words, the polynomial is the th basis polynomial
for constructing a polynomial interpolation of order over the
sample points . It is an order polynomial having zeros
at all of the samples except the th, where it is 1. An example of
a set of eight basis functions for randomly selected
interpolation points is shown in Fig. I.1.
Figure I.1:
Example Lagrange basis functions
in the eighth-order case for randomly selected interpolation points
(marked by dotted lines). The unit-amplitude points are marked by
dashed lines.
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