If n + 1 pairs (x k, y k), (k = 0, 1, ..., n) of real or complex numbers are given, where {xk} are distinct, then there exists exactly one polynomial Pn(x) of degree (at most n) such that

(1)

One way of obtaining Pn(x) is through the Lagrange interpolation formula [Davis (1975)]:

(2)

where

(3)

When the interpolation points {xk} are equidistant, that is xk = x0 + kh [Henrici (1964)], using s = x — x0/h (3) can be reduced to the form

If yk = f(xk), with f(x) an (n + l)-differentiable function in an interval [a, b] and with xk [a, b], then for all x [a, b],

(4)

for some ξ, x0 ≤ ξ ≤ xn, [Davis (1975)]. If, furthermore, |f(n+1)| is bounded on [a, b] by a constant M and if {xk} 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 z0, z1, ..., zn 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)]:

(5)

with

If {zk} are the (n + 1) roots of zn+1 − 1=0, that is zk = e2 πik/n+1, then (2) takes the form:

with

where K is a constant and En = 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

1. Davis, P. J. (1975). Interpolation and Approximation. Ch. 2 & 3. Dover, N.Y.
2. Gaier, D. (1987). Lectures on Complex Approximation. Ch. 2; Â§1 and Â§4. BirkhÃ¤user. Boston. 1987. DOI: 10.1137/1032019
3. Henrici, P. K. (1964). Elements of Numerical Analysis. Ch. 9 & 10. John Wiley. N.Y.