Strict Total Positivity from Spectral Darboux and Toeplitz Smoothing Mechanisms

Abstract: We prove two strict total-positivity results by isolating two strictification mechanisms. The first is a spectral Darboux mechanism: an induction converts positivity and ordered endpoint asymptotics for a one-dimensional spectral family into positive Wronskians and hence into strict total positivity. As an application, the modified-Bessel kernel $K(x,s)=I_s(x)$, $x&gt;0$, $s\ge 0$, is strictly totally positive of infinite order. This proves the real-order determinant positivity asked for by Buchstaber and Glutsyuk after their nonnegative-integer-order theorem. The second mechanism is discrete Toeplitz smoothing: every two-sided Polya-frequency sequence is a pointwise limit of totally positive Polya-frequency sequences. This gives a product-topology answer to Question 12.2 of Belton, Guillot, Khare, and Putinar. The density statement is in the product topology on $\mathbb{R}<sup>{\mathbb{Z}}$; no uniform, weighted, or norm-density assertion is made.

• The paper demonstrates that the modified-Bessel kernel is strictly totally positive for real orders using a spectral Darboux mechanism.
• It introduces a discrete Toeplitz smoothing method using a discrete Gaussian to show that strictly totally positive PF sequences are dense among all PF sequences.
• The techniques bypass classical Hilbert space methods, opening new pathways in spectral analysis, Chebyshev-system theory, and interpolation.

Strict Total Positivity via Spectral Darboux and Toeplitz Smoothing Mechanisms

Introduction and Context

The study of strict total positivity (STP) of kernels remains a central topic in analysis, numerical linear algebra, and the theory of special functions. This paper, "Strict Total Positivity from Spectral Darboux and Toeplitz Smoothing Mechanisms" (2607.02778), isolates two strictification mechanisms leading to new STP results: a spectral Darboux mechanism for spectral families of solutions to differential equations and a discrete smoothing approach based on Toeplitz kernels. Notably, the work resolves two explicit conjectures in the recent literature: (1) the strict total positivity of the modified-Bessel kernel $(x,s)\mapsto I_s(x)$ for $x>0$, $s\ge0$ for real $s$, and (2) the product-topology density of strictly totally positive Pólya-frequency sequences within the space of all Pólya-frequency sequences.

Strict Total Positivity of the Real-Order Modified-Bessel Kernel

Analytical Mechanism and Main Theorem

The strict total positivity result for the modified-Bessel kernel answers a question raised by Buchstaber and Glutsyuk, who previously established this property for integer order $s\in\mathbb{Z}_{\ge0}$ [Buchstaber & Glutsyuk, 2019]. The generalization to real order presents considerable technical obstacles, as standard Hilbert space arguments and classical total-positivity results for Bessel-type kernels do not extend directly. The mechanism introduced in the paper leverages the Darboux–Crum transform, a fundamental tool in Sturm–Liouville theory. By employing a careful induction utilizing endpoint asymptotics and positivity for spectral families, the author establishes a spectral Darboux mechanism that produces positive Wronskians for all finite collections of $I_{s_j}(x)$ for $0\le s_1<\cdots<s_m$ and $x>0$. Consequently, all minors of these kernels are strictly positive, so the kernel is $STP_\infty$.

Strong Claim: For all $m\ge1$, $x>0$0, and $x>0$1,

$x>0$2

i.e., the kernel $x>0$3 is strictly totally positive of infinite order on $x>0$4.

Technical Innovations and Implications

The proof avoids dependence on Hilbert space techniques, instead relying solely on analytic properties—positivity, spectral parameterization, and asymptotic control at endpoints—for the Bessel family. By interpreting the Bessel parameter as the spectral parameter in a Sturm–Liouville family, and combining this with a Chebyshev-system perspective (ensured by positive Wronskians), the paper demonstrates a general strictification approach applicable to wide classes of parameterized spectral families, contingent on suitable endpoint behavior propagation under Darboux transforms.

This result ensures that the collection $x>0$5 forms an oriented extended Chebyshev system on $x>0$6 for all real $x>0$7, with positive iterated Wronskians. This resolves a longstanding question in interpolation theory for special functions and creates new avenues for Chebyshev-system constructions via spectral analytic tools.

Density of Strictly Totally Positive Pólya-Frequency Sequences

Product-Topology Density Result

The second main theorem addresses a question posed by Belton, Guillot, Khare, and Putinar (2023) regarding the density of strictly totally positive Pólya-frequency (PF) sequences among all PF sequences. The affirmative answer provided here uses a discrete Toeplitz smoothing—namely, convolution with the discrete Gaussian $x>0$8, $x>0$9—whose Toeplitz kernel is shown to be $s\ge0$0. Thus, for any two-sided PF sequence $s\ge0$1, the smoothed sequence $s\ge0$2 is strictly totally positive PF, and as $s\ge0$3, converges pointwise to $s\ge0$4.

Strong Claim: Strictly totally positive PF sequences are dense in the space of all PF sequences under the product topology, i.e., for every PF sequence, there exists a sequence of $s\ge0$5 PF sequences converging pointwise.

Technical Approach and Applications

A direct Vandermonde-type proof is provided for the STP property of the discrete Gaussian kernel, extending the classical arguments for the continuous Gaussian kernel and harnessing exponential decay and Laurent-series representations of PF sequences. The proof handles geometric exceptions (rank-one geometric cases) via a separate multiplicative Gaussian tilt.

Practically, this density result supplies an explicit, analytically robust device for strictification: smoothing by the discrete Gaussian. The mechanism is analytic and entirely constructive, with implications for the approximation theory of totally positive matrices and for transforms in moment problems and related positive kernel approximations.

The product-topology (coordinatewise) nature of the density result addresses the specific form posed in [Belton et al., 2023]; norm-density, weighted convergence, and tail conditions are not treated and are left as open questions requiring further hypotheses.

Broader Implications and Open Directions

The proposed strictification mechanisms have ramifications for total positivity theory in several domains. The spectral Darboux approach isolates robust criteria—positivity, spectral ordering, and endpoint asymptotics—to guarantee propagation of STP through analytic continuations of parameter families for second-order ODEs. This suggests potential for new classes of STP kernels arising from integrable systems, special function theory, and spectral transformations.

On the discrete side, Toeplitz smoothing with strictly totally positive kernels provides an effective, analytic pathway for constructing strictly totally positive approximants of arbitrary PF sequences. This is of interest for signal processing, stochastic processes, and combinatorial matrices.

A precise open question remains: Under which general conditions do the ordered endpoint asymptotics required in the spectral Darboux strictification mechanism persist after repeated Darboux transformations for families of positive solutions? Resolution of this would lead to generalizations and further systematization of spectral mechanisms for strict total positivity.

Conclusion

This work conclusively settles two open conjectures in strict total positivity theory by exposing precise structural mechanisms—one continuous and spectral, one discrete and Toeplitz—to generate strictly totally positive kernels and sequences from broader positive structures. These mechanisms facilitate new connections between Chebyshev-system theory, spectral analysis, and analytic approximation, and suggest fruitful directions for the extraction of strict total positivity in broader analytic and algebraic settings.