Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

### Appendix: Relation between Sinc and Lagrange Interpolation

Lagrange interpolation is a well known, classical technique for interpolation [#!Hildebrand!#]. It is also called Waring-Lagrange interpolation, since Waring actually published it 16 years before Lagrange [#!Meijering02!#, p. 323]. Given a set of n+1 known samples f(xk), , the problem is to find the unique order n polynomial y(x) which interpolates the samples. The solution can be expressed as a linear combination of elementary nth order polynomials: (17)

where (18)

From the numerator of the above definition, we see that lk(x) is an order n polynomial having zeros at all of the samples except the kth. The denominator is simply the constant which normalizes its value to 1 at xk. Thus, we have (19)

In other words, the polynomial lk is the kth basis polynomial for constructing a polynomial interpolation of order n over the n+1 sample points xk.

In the case of an infinite number of equally spaced samples, with spacing , the Lagrangian basis polynomials converge to shifts of the sinc function, i.e., (20)

where A simple argument is based on the fact that any analytic function is determined by its zeros and its value at one point. Since is zero on all the integers except 0, and since , it must coincide with the infinite-order Lagrangian basis polynomial for the sample at x=0 which also has its zeros on the nonzero integers and equals 1 at x=0.

The equivalence of sinc interpolation to Lagrange interpolation was apparently first published by the mathematician Borel in 1899, and has been rediscovered many times since [#!Meijering02!#, p. 325].

A direct proof can be based on the equivalance between Lagrange interpolation and windowed-sinc interpolation using a scaled binomial window'' [#!KootsookosAndWilliamson95!#,#!VesaT!#]. That is, for a fractional sample delay of D samples, multiply the shifted-by-D, sampled, sinc function by a binomial window and normalize by [#!VesaT!#] which scales the interpolating filter to have a unit L2 norm, to obtain the Nth-order Lagrange interpolating filter Since the binomial window converges to the Gaussian window as , and since the window gets wider and wider, approaching a unit constant in the limit, the convergence of Lagrange to sinc interpolation can be seen.

A more recent alternate proof appears in [#!Yekta09!#].

Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search