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Appendix: Relation between Sinc and Lagrange Interpolation

Lagrange interpolation is a well known, classical technique for interpolation [7]. It is also called Waring-Lagrange interpolation, since Waring actually published it 16 years before Lagrange [11, p. 323]. Given a set of n+1 known samples f(xk), $k=0,1,2,\ldots,n$, 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 nth order polynomials:

\begin{displaymath}
y(x) = \sum_{k=0}^n l_k(x)f(x_k)
\end{displaymath} (17)

where
\begin{displaymath}
l_k(x) \isdef { (x - x_0) \cdots (x - x_{k-1}) (x - x_{k+1}) \cdots (x - x_n)
\over (x_k - x_0) \cdots (x_k - x_{k-1}) (x_k - x_{k+1}) \cdots (x_k - x_n) }
\end{displaymath} (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
\begin{displaymath}
l_k(x_j) = \delta_{kj} \isdef \left\{\begin{array}{ll}
1, & j=k \\
0, & j\neq k \\
\end{array} \right.
\end{displaymath} (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 $x_{k+1}-x_k = \Delta$, the Lagrangian basis polynomials converge to shifts of the sinc function, i.e.,

\begin{displaymath}
l_k(x) = \mbox{sinc}\left(x-k\Delta\over\Delta\right), \quad k=\ldots,-2,-1,0,1,2,\ldots
\end{displaymath} (20)

where

\begin{displaymath}
\mbox{sinc}(x) \isdef {\sin(\pi x)\over \pi x}
\end{displaymath}

A simple argument is based on the fact that any analytic function is determined by its zeros and its value at one point. Since $\mbox{sinc}(x)$ 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 [11, p. 325].

A direct proof can be based on the equivalance between Lagrange interpolation and windowed-sinc interpolation using a ``scaled binomial window'' [9,23]. That is, for a fractional sample delay of D samples, multiply the shifted-by-D, sampled, sinc function

\begin{displaymath}
h_s(n) = \mbox{sinc}(n-D) = \frac{\sin[\pi(n-D)]}{\pi(n-D)}
\end{displaymath}

by a binomial window

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

and normalize by [23]

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

which scales the interpolating filter to have a unit L2 norm, to obtain the Nth-order Lagrange interpolating filter

\begin{displaymath}
h_D(n)=C(D)w(n)h_s(n), \quad n=0,1,2,\ldots,N
\end{displaymath}

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.

A more recent alternate proof appears in [24].


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