If n + 1 pairs (x_{ k}, y_{ k}), (k = 0, 1, ..., n) of real or complex numbers are given, where {x_{k}} are distinct, then there exists exactly one polynomial Pn(x) of degree (at most n) such that
One way of obtaining P_{n}(x) is through the Lagrange interpolation formula [Davis (1975)]:
where
When the interpolation points {x_{k}} are equidistant, that is x_{k} = x_{0} + kh [Henrici (1964)], using s = x — x_{0}/h (3) can be reduced to the form
If y_{k} = f(x_{k}), with f(x) an (n + l)-differentiable function in an interval [a, b] and with x_{k} [a, b], then for all x [a, b],
for some ξ, x_{0} ≤ ξ ≤ x_{n}, [Davis (1975)]. If, furthermore, |f^{(n+1)}| is bounded on [a, b] by a constant M and if {x_{k}} are the zeros of the Chebyshev Polynomial of degree n + 1 defined on [a, b], (4) gives
The Lagrange formula for trigonometric interpolation is obtained from (2) with
If the interpolation points are complex numbers z_{0}, z_{1}, ..., z_{n} and lie in a domain D bounded by a piecewise smooth contour γ, and if f is analytic in D and continuous in its closure = D γ, then the Lagrange formula has the form [Gaier (1987)]:
with
If {z_{k}} are the (n + 1) roots of z^{n+1} − 1=0, that is z_{k} = e^{2 πik/n+1}, then (2) takes the form:
with
where K is a constant and E_{n} = infimum { maximum |f(z) − p(z)|; z }, the infimum is taken over all polynomials of degree n and the maximum over the closed domain .
REFERENCES
Davis, P. J. (1975). Interpolation and Approximation. Ch. 2 & 3. Dover, N.Y.
Gaier, D. (1987). Lectures on Complex Approximation. Ch. 2; §1 and §4. Birkhäuser. Boston. 1987.
Henrici, P. K. (1964). Elements of Numerical Analysis. Ch. 9 & 10. John Wiley. N.Y.
References
- Davis, P. J. (1975). Interpolation and Approximation. Ch. 2 & 3. Dover, N.Y.
- Gaier, D. (1987). Lectures on Complex Approximation. Ch. 2; Â§1 and Â§4. BirkhÃ¤user. Boston. 1987. DOI: 10.1137/1032019
- Henrici, P. K. (1964). Elements of Numerical Analysis. Ch. 9 & 10. John Wiley. N.Y.