Multidimensional Variable Tauberian Theorems

• Several variable Tauberian theorems are a set of principles that deduce asymptotic behavior of multivariate sums, integrals, and transforms based on analytic boundary information.
• They extend classical one-variable Tauberian methods to multidimensional settings, underpinning applications in harmonic analysis, PDEs, and operator theory.
• The framework employs regularizing transforms, precise scaling estimates, and polynomial corrections to characterize asymptotic and convergence properties in complex functional spaces.

Several variable Tauberian theorems provide a systematic framework for deducing asymptotic properties of multivariate sums, integrals, transforms, and operator families from analytic or regularizing information. In modern analysis, these theorems connect multidimensional distribution theory, vector-valued function spaces, operator theory, non-Archimedean harmonic analysis, and probabilistic models via generalized Tauberian or Abelian correspondences. Central to these developments is the extension of classical one-variable Tauberian principles to multidimensional distributions, transforms, and functional settings as found in harmonic analysis, partial differential equations, multivariate regular variation, and operator theory.

1. Foundations: Multidimensional and Vector-Valued Tauberian Theorems

A principal advance is the characterization of weak asymptotic properties of tempered distributions $f \in \mathcal{S}'(\mathbb{R}^n,E)$ (possibly Banach-space valued) via boundary behavior of regularizing transforms, notably those of the form

$$M^\mathbf{f}_\phi(x,y) = (f \ast \phi_y)(x), \qquad \phi_y(t) = y^{-n}\phi(t/y),$$

where the kernel $\phi$ may be either wavelet-type (vanishing zeroth moment) or non-wavelet-type. The main Tauberian hypothesis requires precise scaling estimates, for instance,

$$\limsup_{h \to 0^+} \frac{1}{h^\alpha L(h)} \sup_{|x|^2 + y^2 = 1,\,y>0} \| M^\mathbf{f}_\phi(x_0 + h x, h y)\|_E < \infty,$$

which is necessary and sufficient for $f$ to exhibit weak-asymptotic boundedness of degree $\alpha$ with respect to a slowly varying function $L$ at $x_0$ (Pilipović et al., 2010).

When the relevant scaling index $\alpha$ is not an integer, $f$ displays homogeneous asymptotic behavior, while for integer $\alpha$, polynomial and associated homogeneous corrections must be incorporated. The precise decomposition is given by

$$f(x_0 + ht) = P(ht) + h^\alpha L(h) g(t) + o(h^\alpha L(h)),$$

where $P$ is a uniquely determined polynomial and $g$ is (associate) homogeneous (Pilipović et al., 2010, Pilipovic et al., 2013).

2. Regularizing Transforms, Wavelets, and Operator Extensions

The theory accommodates both classical (non-wavelet) and wavelet transforms. For non-wavelet-type $\phi$ (zeroth moment nonzero), the distribution $f$ and its asymptotics can be recovered directly from the regularizing transform as $y \to 0^+$. For wavelet-type kernels with vanishing moments up to order $k$, the polynomial part of $f$ is not detected by the transform, and a polynomial correction must be isolated (Pilipović et al., 2010).

Tauberian theorems extend to operator-theoretic frameworks, notably in quantum harmonic analysis (QHA), where convolution and translation are interpreted in terms of operator adjoint actions. Fundamental results (e.g., the limit function version of Wiener’s Tauberian theorem) characterize classes of functions or operators (such as slowly oscillating operators, $SO(\mathcal{H})$) in terms of asymptotic properties of their regularized convolutions or limit operators. It is shown that $$SO(\mathcal{H})=\mathcal{C}_1(\mathcal{H})+B_0(\mathcal{H})$$, which contains but strictly extends the class of compact operators (Fulsche et al., 14 May 2024).

3. Tauberian Theorems for Integrals and Probability Measures

Tauberian theory for integrals of the form

$J(r) = \int_0^\infty K(t/r)d\mu(t),$

with $\mu$ a Radon measure and $K$ a kernel, is abstracted using limit sets of measures. For appropriate scaling functions $V(r) = r^{p(r)}$, the set of all possible normalized weak limits of the rescaled measures, denoted $\operatorname{Fr}[\mu]$, governs the asymptotics of $J(r)$. The core result states

$$\mathcal{L}(J, \infty) = \{ \int_0^\infty K(u) dv(u) : v \in \operatorname{Fr}[\mu] \},$$

thus reducing asymptotic statements for integrals to properties of the tail measure (Grishin et al., 2012). This framework includes multivariate generalizations, such as the multivariate Abel–Tauberian theorems for regularly varying measures on $\mathbb{R}_+^p$, which equate the regular variation of the original measure $U$ with the asymptotics of its Laplace transform: $$\frac{1}{t} \hat{U}\left(\frac{\lambda_1}{b_1(t)},\ldots, \frac{\lambda_p}{b_p(t)}\right) \to \hat{U}_\infty(\boldsymbol{\lambda}),$$ where $b_i(t)$ are scaling functions corresponding to different directions and $U_t$ are rescaled measures (Resnick et al., 2014). Such results have direct applications in the asymptotic analysis of in- and out-degree distributions in preferential attachment networks.

4. Multidimensional Laplace and Fourier Analysis: Distributions, Ultradistributions, and Non-Archimedean Settings

Several variable Tauberian theorems are established for Laplace transforms of ultradistributions supported in convex cones. In such frameworks, boundedness in Gelfand–Shilov spaces $S^*[T]$ is fully characterized by uniform growth bounds on the Laplace transform $\mathcal{L}\{f;z\}$ in corresponding tube domains. Under Tauberian-type conditions on the boundary behavior,

$\lim_{r\to 0^+} p(1/r)(-\mathcal{L}\{f; riy\})$

exists for $y$ in some subcone $C'$, and additional growth bounds imply that $f$ admits a quasiasymptotic expansion of the form $f(\lambda \cdot)\sim p(\lambda) g(\cdot)$ as $\lambda\to\infty$ (Neyt et al., 2019).

Non-Archimedean (p, q)-adic Tauberian theorems introduce a Banach algebraic Fourier theory on $C(\mathbb{Z}_p,\mathbb{C}_q)$, establishing that the density of the span of translates of a function’s (or measure’s) Fourier–Stieltjes transform is equivalent to the nonvanishing of the Radon–Nikodym derivative at all points of $\mathbb{Z}_p$. This unifies Tauberian equivalences between the invertibility (nowhere vanishing) of an element, denseness of its translates, and convolution invertibility in the non-Archimedean context (Siegel, 2022).

5. Applications in Operator Theory, Stochastic Games, and PDEs

Tauberian theorems for general iterations of operators (e.g., $\Psi:[0,1]\times X\to X$ nonexpansive) quantify the transfer of asymptotic convergence properties under classical Cesàro and Abel evaluations to much more general weighted games, including sequences defined by dynamic programming equations for stochastic games: $v_\theta = \Psi(\theta_1, v_{\hat{\theta}}),$ with $\hat{\theta}$ the shifted normalized evaluation (Ziliotto, 2016). Specific quantified decay results for bounded vector-valued sequences and their applications to the Katznelson–Tzafriri theorem and Ritt operators are presented, showing how decay rates such as $\|T^n(I-T)\| = O(m_{\log}^{-1}(cn))$ can be derived from smoothness or boundary properties of associated holomorphic functions (Pritchard et al., 21 Feb 2025).

Further, several variable Tauberian theory ensures that quantitative stabilization of solutions to Cauchy or parabolic problems corresponds precisely to Tauberian-type scaling behavior of the initial data (Pilipovic et al., 2013, Pilipović et al., 2010).

6. Arithmetic, Asymptotic Series, and Multivariable Summability

Tauberian theorems in arithmetic and combinatorial applications deduce asymptotics for partial sums $\sum_{\lambda_n \leq x} a_n$ from analytic properties of associated Dirichlet or Mellin series under precise hypotheses about pole order, analytic continuation, and growth (Pierce et al., 22 Apr 2025). Explicitly, in the presence of a pole of order $m$ at $s=\alpha$, one obtains

$$\sum_{\lambda_n \leq x} a_n = [g(\alpha)/\Gamma(m)] x^\alpha (\log x)^{m-1} + o(x^\alpha (\log x)^{m-1}),$$

with explicit (and optimally sharp) remainder terms if more detailed growth is assumed.

Generalizations include multidimensional Euler–Maclaurin expansions for multi-index sums and series with singularities, where the error terms are controlled via analytic smoothness and decay (Bringmann et al., 2019). In the context of analytic germs and $k$-summability, Tauberian theorems classify when formal power series in several complex variables, $f(x)$, are convergent based on their summability properties with respect to several non-equivalent analytic germs $P_j(x)$ and scales $k_j^{-1}$, leveraging monomialization and multisummability frameworks (Carrillo et al., 2019).

7. Operator Compactness, Function Spaces, and Quantum Harmonic Analysis

Operator Tauberian theorems extend classical results to characterize compactness and essential spectrum of localization and Toeplitz operators in quantum harmonic analysis. A localization operator $$A_f^{(\phi_1,\phi_2)} = f \star (\phi_2\otimes\phi_1)$$ is compact on $L^2(\mathbb{R}^d)$ if and only if, for a nonzero Schwartz window $\Phi$, the short-time Fourier transform $V_\Phi(f-A)$ vanishes at spatial infinity—precisely,

$$\lim_{|x| \to \infty} \sup_{|\omega| \leq R} |V_\Phi(f-A)(x,\omega)| = 0 \quad \forall R>0,$$

for some constant $A$ (Luef et al., 2020). These results ground compactness and tail behavior in time–frequency localization and have applications to operator theory in Gabor and Bargmann–Fock spaces, pseudo-differential calculus, and quantization schemes.

8. Outlook and Open Problems

Several variable Tauberian theorems continue to expand into new analytic, algebraic, and geometric settings. Extensions involve:

• Full multidimensional analogues in regular variation and heavy-tailed random structures, yielding results on joint extremal dependence in networks (Resnick et al., 2014).
• Refinement of quantitative and uniform Tauberian theorems in Banach algebras and operator modules, with applications to stability and spectrum (Fulsche et al., 14 May 2024).
• Non-classical boundary behaviors (e.g., local pseudofunctions or pseudomeasures) in Laplace and power series contexts, relaxing uniformity or analyticity to sharper minimal distributional control (Debruyne et al., 2016).
• Direct connections between weighted and unweighted asymptotic expansions via Mellin or modified Mellin transforms, with applications to orthorecursive expansions (Cloitre, 11 May 2025).

Research directions suggest further abstraction and application in microlocal analysis, analytic number theory, harmonic analysis on groups and quantum structures, stochastic games, as well as summability theory and o-minimality. Fundamental challenges persist in characterizing the sharpness of remainder terms, identifying minimal Tauberian hypotheses under various forms of regularity or boundary singularity, and systematizing the interplay between global analytic properties and local asymptotic behavior in multivariate settings.