Successive Remainders of the Newton Series

Abstract

If $f$ is analytic in the open unit disc $D$ and $\lambda$ is a sequence of points in $D$ converging to 0, then $f$ admits the Newton series expansion $f(z) = f({\lambda _1}) + \sum \nolimits _{n = 1}^\infty {\Delta _\lambda ^nf({\lambda _{n + 1}})(z - {\lambda _1})(z - {\lambda _2}) \cdots (z - {\lambda _n})}$, where $\Delta _\lambda ^nf(z)$ is the $n$th divided difference of $f$ with respect to the sequence $\lambda$. The Newton series reduces to the Maclaurin series in case ${\lambda _n} \equiv 0$. The present paper investigates relationships between the behavior of zeros of the normalized remainders $\Delta _\lambda ^kf(z) = \Delta _\lambda ^kf({\lambda _{k + 1}}) + \sum \nolimits _{n = k + 1}^\infty {\Delta _\lambda ^nf({\lambda _{n + 1}})(z - {\lambda _{k + 1}}) \cdots (z - {\lambda _n})}$ of the Newton series and zeros of the normalized remainders $\sum \nolimits _{n = k}^\infty {{a_n}{z^{n - k}}}$ of the Maclaurin series for $f$. Let ${C_\lambda }$ be the supremum of numbers $c > 0$ such that if $f$ is analytic in $D$ and each of $\Delta _\lambda ^kf(z),\;0 \leqslant k < \infty$, has a zero in $|z| \leqslant c$, then $f \equiv 0$. The corresponding constant for the Maclaurin series (${C_\lambda }$, where ${\lambda _n} \equiv 0$) is called the Whittaker constant for remainders and is denoted by $W$. We prove that ${C_\lambda } \geqslant W$, for all $\lambda$, and, moreover, ${C_\lambda } = W$ if $\lambda \in {l_1}$. In obtaining this result, we prove that functions $f$ analytic in $D$ have expansions of the form $f(z) = \sum \nolimits _{n = 0}^\infty {\Delta _\lambda ^nf({z_n}){C_n}(z)}$, where $|{z_n}| \leqslant W$, for all $n$, and ${C_n}(z)$ is a polynomial of degree $n$ determined by the conditions $\Delta _\lambda ^j{C_k}({z_j}) = {\delta _{jk}}$.