On Zeros of Interpolating Polynomials
Roger Barnard, Wayne T. Ford, Hsing Y. Wang · SIAM Journal on Mathematical Analysis · 1986
Polynomials to be used in interpolation of digital signals are called interpolating polynomials. They may require modification to assure convergence of their reciprocals on the unit circle. This paper concerns discrete time windowing, which consists of scaled truncation of a series such as \[ p_N (z)\mathop = \limits^\Delta 1 + \sum\limits_{m = 1}^\infty {(z^m + z^{ - m} )} {\operatorname { sinc }}\frac{{m\pi }} {N},\quad {\operatorname {sinc }}x\mathop = \limits^\Delta \frac{{\sin x}} {x}, \] where $N > 1$, to obtain an expression of the form $p_{N,L}^ * (z)\mathop = \limits^\Delta z^{L - 1} \left( {1 + \sum\limits_{m = 1}^{L - 1} {\left( {z^m + z^{ - m} } \right)c_m {\operatorname { sinc }}\frac{{m\pi }} {N}} } \right).$ We delete the asterisk to write $P_{N,L} $ when each $c_m = 1$. The zeros of $P_{N,L} $ are shown to have unit modulus for $L \leqq N$. Examples are given to show that little can be said of the zeros of $P_{N,L} $ for $L > N$. Conditions are found to define real sequences of the form, $\{ {c_m :1 \leqq m < \infty } \}$, so that $P_{N,L}^ * $ has no zero of unit modulus. Several standard discrete time windows are shown to define real sequences which are special cases of the conditions developed.