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 $f(x_k)$, $k=0,1,2,\ldots,n$, 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 $n$th order polynomials:

$$y(x) = \sum_{k=0}^n l_k(x)f(x_k)$$ (36)

where

$$l_k(x) \isdef { (x - x_0) \cdots (x - x_{k-1}) (x - x_{k+1})... ...0) \cdots (x_k - x_{k-1}) (x_k - x_{k+1}) \cdots (x_k - x_n) }$$ (37)

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

$$l_k(x_j) = \delta_{kj} \isdef \left\{\begin{array}{ll} 1, & j=k \\ 0, & j\neq k \\ \end{array} \right.$$ (38)

In other words, the polynomial $l_k$ is the $k$th basis polynomial for constructing a polynomial interpolation of order $n$ over the $n+1$ sample points $x_k$.

In the case of an infinite number of equally spaced samples, with spacing $x_{k+1}-x_k = \Delta$, the Lagrangian basis polynomials converge to shifts of the sinc function, i.e.,

$$l_k(x) = \mbox{sinc}\left(x-k\Delta\over\Delta\right), \quad k=\ldots,-2,-1,0,1,2,\ldots$$ (39)

where

$$\mbox{sinc}(x) \isdef {\sin(\pi x)\over \pi x}$$ (40)

A simple argument is based on the fact that any analytic function is determined by its zeros and its value at one point. Since $\sin(\pi x)$ is zero on all the integers except $0$, and since $\mbox{sinc}(0)=1$, 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 ``binomial window'' [#!KootsookosAndWilliamson95!#,#!VesaT!#]. That is, for a fractional sample delay of $D$ samples, multiply the shifted-by-$D$, sampled, sinc function

$$h_s(n) = \mbox{sinc}(n-D) = \frac{\sin[\pi(n-D)]}{\pi(n-D)}$$ (41)

by a binomial window

$$w(n) = \left(\begin{array}{c}N\\ n\end{array}\right), \quad n=0,1,2,\ldots N$$ (42)

and normalize by [#!VesaT!#]

$$C(D) = (-1)^N\frac{\pi(N+1)}{\sin(\pi D)}\left(\begin{array}{c}D\\ N+1\end{array}\right),$$ (43)

which normalizes the interpolating filter to have a unit $L_2$ norm, to obtain the $N$th-order Lagrange interpolating filter

$$h_D(n)=C(D)w(n)h_s(n), \quad n=0,1,2,\ldots,N$$ (44)

Since the binomial window converges to the Gaussian window as $N\to\infty$, 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.