MuntzCompletenessAndCoefficientDivergence

Müntz Completeness and Coefficient Divergence

The connection runs through the Müntz–Szász theorem and is genuine, not metaphorical.

The Müntz–Szász completeness theorem

For a sequence of exponents $0=\lambda_0 < \lambda_1 < \lambda_2 < \cdots$, the Müntz system $\lbrace u^{\lambda_k}\rbrace_{k\ge 0}$ is dense in $C[0,1]$ (uniform norm, with constants included) iff

$$\sum_{k=1}^{\infty}\frac{1}{\lambda_k};=;\infty.$$

This is the Müntz–Szász condition (Müntz 1914; Szász 1916). The corresponding statement for $L^2[0,1]$ replaces $1/\lambda_k$ by $\lambda_k/(1+\lambda_k^2)$, but for the rough-Heston exponents $\lambda_k = k\mu$ with $\mu = H+\tfrac12 \in (\tfrac12,1)$, both sums diverge: $\sum 1/(k\mu) = \infty$. So the Müntz system $\lbrace u^{k\mu}\rbrace_{k\ge 0}$ is complete in $C[0,1]$ and in $L^2[0,1]$.

What divergence of $\sum 1/\lambda_k$ means for the coefficients

The Riccati series for $h(v,t)$ in the rough-Heston construction is the expansion

$$h(v,t);=;\sum_{k=0}^{\infty} a_k(v),t^{k\mu}$$

in the Müntz basis $\lbrace t^{k\mu}\rbrace$. Müntz–Szász completeness says: the closure of $\mathrm{span}\lbrace t^{k\mu}\rbrace$ in $C[0,T]$ (or $L^2[0,T]$) is the full space iff $\sum 1/(k\mu)=\infty$. Equivalently, no nontrivial continuous (or $L^2$) function on $[0,T]$ is orthogonal to every $t^{k\mu}$.

The divergence of $\sum 1/\lambda_k$ is what guarantees the infinite-dimensional Müntz system is rich enough to represent arbitrary functions on the interval. If $\sum 1/\lambda_k$ were finite, the Müntz closure would be a proper subspace, and a function $h\in C[0,T]$ could fail to be approximable by the Riccati partial sums even if every coefficient $a_k$ is computed correctly — the system itself would be incomplete, and the truncation error would not vanish.

How divergence of the coefficients fits in

Now the coefficients $a_k(v)$ — solutions of the Γ-ratio Puiseux recurrence — typically grow in $k$. The Riccati series has finite radius of convergence in $t^\mu$ (controlled by the nearest singularity of $h(\cdot,T)$ on the positive axis, which for rough Heston is the moment-explosion time or a Riccati blow-up), so $|a_k|$ grows geometrically: $|a_k|\sim C,r^{-k}$ for some $r>0$ determined by the singularity location.

The growth of $|a_k|\to\infty$ and the divergence of $\sum 1/\lambda_k$ are two faces of one fact: the Müntz exponents $\lambda_k = k\mu$ increase only linearly in $k$ (with slope $\mu<1$), not faster. Linear growth of $\lambda_k$ is exactly what makes $\sum 1/\lambda_k$ diverge logarithmically — the harmonic-series tail. If the exponents were $\lambda_k = k^2$ (quadratic, lacunary), then $\sum 1/k^2 < \infty$: the system would be incomplete (missing functions in its closure), and one would have far smaller coefficient growth because the gaps between successive basis functions $t^{k^2}$ grow super-linearly. Lacunary Müntz systems have better coefficient behavior but worse approximation behavior.

So the fact that the Riccati coefficients diverge to infinity in $k$ is the price paid for completeness of the Müntz system: dense representation of arbitrary functions on $[0,T]$ requires basis functions packed densely enough that $\sum 1/\lambda_k=\infty$, which forces $\lambda_k = O(k)$, which forces — by the singularity structure of $h$ — coefficient growth.

The Padé acceleration is precisely the right response

This is why Padé summation in the variable $z = t^\mu$ is the correct device. The Padé–Müntz approximant is

$$h(v,t);\approx;\frac{P_L(v,z)}{Q_M(v,z)}\Big|_{z=t^\mu}\quad\text{with } z=t^\mu,$$

a rational function in $z=t^\mu$ constructed to match the first $L+M+1$ Riccati coefficients. The Padé approximant converts a series whose coefficients diverge geometrically into a rational function whose values are finite, by encoding the singularity of the Borel-summed analytic continuation of the Riccati series in the poles of the denominator $Q_M$. This is de Montessus de Ballore convergence: when $h(v,\cdot)$ is meromorphic in $z$ with finitely many poles inside a disk, the diagonal Padé approximants converge geometrically inside that disk despite the divergent coefficient growth of the Taylor / Riccati expansion.

The interplay is sharp:

• Müntz–Szász divergence $\sum 1/\lambda_k=\infty$ ⇒ the basis is complete, the Riccati expansion captures every $C[0,T]$ function in principle.
• $\lambda_k = O(k)$ (linear growth, forced by completeness) ⇒ the Riccati coefficients $a_k$ grow geometrically, the series has finite radius of convergence in $z=t^\mu$.
• Padé–Müntz acceleration ⇒ Hankel resummation at a single point converts the divergent coefficient sequence into a convergent rational approximation by exposing the meromorphic structure in $z$.

This is exactly what the framework paper pade_muntz_fractional_riccati.tex establishes (with the closure properties of $\lbrace t^{k\mu}\rbrace$ under $D^\mu, I^r$, multiplication, and the Hankel system $H_M q = -b$), and the resurgence / Borel–Padé / Riemann–Hilbert section identifies the same structure abstractly: a divergent series whose Borel transform is meromorphic, whose Hankel determinants encode the singularity structure of the Borel sum, and whose Padé approximants effect the resummation. The Müntz system's completeness is the non-asymptotic underpinning that lets this work for any target function on $[0,T]$, not just for special analytic ones.

One-paragraph summary

Coefficient divergence and Müntz–Szász completeness are dual: completeness of the basis $\lbrace t^{k\mu}\rbrace$ requires $\sum 1/(k\mu)=\infty$, which requires exponents linear in $k$, which forces the Riccati / Borel coefficients to grow geometrically against the singularity-controlled radius of convergence. The Padé–Müntz approximant is the resummation device that uses Hankel determinants to convert the divergent coefficients into a meromorphic rational function in $z=t^\mu$ whose poles are the analytic-continuation singularities of $h(v,\cdot)$. So divergence of coefficients is not a defect — it is the signature, in coefficient space, of a basis dense enough to approximate arbitrary functions, and the Padé acceleration is the unique device that simultaneously inherits Müntz completeness and Borel–Padé summation of the divergent expansion.