8 Analytic interpolation and exact Taylor radii
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The threshold-free Mahler theorem gives continuous interpolation. Analytic interpolation needs a stronger estimate, and the factorial bounds provide it directly. The ordinary Taylor radius is determined by the coefficient growth, including infinite exact subsequences.
Theorem 8.1: Taylor coefficient bounds and radii
For \(j=0,1,2,3\), the dyadic interpolation has an ordinary power series
Its Taylor series converges for \(|x|_2<2\). At addresses zero and one,
At address two,
The radius at address three is at least two; its exact radius is not asserted here. The stated exact radii at addresses zero, one, and two hold at every center in \(\mathbb Z_2\).
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Proof of Theorem 8.1
Put \(a_m=\Delta_1^mC_j(0)\) and expand the falling factorial in the Newton series. The ordinary coefficients are
Since \(s(m,\ell)\) is integral and \(v_2(a_m/m!)\geq m+\beta_j\), the coefficient sum converges and has valuation at least \(\ell+\beta_j\). The sum exchange is justified on \(\mathbb Z_2\) by these uniform geometric coefficient bounds. The ordinary power series agrees there with the Newton interpolation and converges whenever \(|x|_2<2\).
For \(j=0,1\), the term \(m=\ell\) in (8.3) has exact valuation \(\ell\), since \(s(\ell,\ell)=1\) and the Newton valuation is exact. Every later term has valuation at least \(\ell+1\). Thus \(v_2(d_{j,\ell})=\ell\), and the coefficient radius is exactly two. At a center \(x_0\in\mathbb Z_2\), the coefficient of \(y^\ell\) in the translated series is
Its initial term again has exact valuation \(\ell\), and all later terms have at least one more bit. The exact radius is therefore two at every integral center.
For address two, the stronger factorial filtration gives the lower bound in (8.2). At each \(\ell=2^r\geq4\), Theorem 6.3 makes the \(m=\ell\) term in (8.3) have valuation \(2\ell+\ell/2=5\ell/2\). Every \(m>\ell\) term has strictly greater lower bound \(2m+\lfloor m/2\rfloor\). Hence
The universal lower bound and this unbounded exact subsequence prove that the coefficient radius is exactly \(2^{5/2}\). Translation by an integral center preserves the same unique leading coefficients at these degrees in (8.4), proving the exact radius at every such center.
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Remark 8.2: Boundary interpretation
For addresses zero and one, at an argument of absolute value two every positive-degree Taylor term has valuation zero, so that ordinary Taylor series diverges. For address two, over a complete extension admitting arguments of absolute value \(2^{5/2}\), the power-of-two-degree terms have valuation zero and fail to tend to zero. These statements concern the Taylor expansions just constructed; they do not prohibit a different analytic continuation. On \(\mathbb Q_2\)-valued arguments, open-disc notation must be read with its discrete value group. The coefficient radius is the precise invariant in the theorem.