Knowlpedia

Proposition

Holomorphic realization of weighted analytic coefficients

The factorial-binomial coefficient bound gives a convergent radial series and a common complex parameter radius.

Statement

Let FF have finite . For each 0<r<R0<r<R, its radial series and parameter Taylor series give a jointly holomorphic extension near ∣Y∣≤r|Y|\le r and η∈I\eta\in I, with any sufficiently small parameter radius strictly below

ρ(1−r/R).\rho(1-r/R).

At interval endpoints use the same Taylor series to extend from the one-sided data.

Estimate

Dropping the polynomial denominators from the weights and writing q=r/R<1q=r/R<1 gives

∑α≥0sup⁡I∣∂ηβFα∣rα≤∥F∥R,ρρ−ββ!∑α≥0(α+ββ)qα=∥F∥R,ρρ−ββ!(1−q)β+1.\sum_{\alpha\ge0}\sup_I|\partial_\eta^\beta F_\alpha|r^\alpha \le\|F\|_{R,\rho}\rho^{-\beta}\beta! \sum_{\alpha\ge0}\binom{\alpha+\beta}{\beta}q^\alpha =\frac{\|F\|_{R,\rho}\rho^{-\beta}\beta!}{(1-q)^{\beta+1}}.

The generating identity follows by differentiating the geometric series β\beta times. Taylor's remainder tends to zero at smaller parameter distances. Absolute convergence gives a holomorphic power series, and overlapping local extensions agree by analytic uniqueness. A strict loss of radius leaves room for derivative estimates of every fixed order.