Classical Analysis and ODEs
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Showing new listings for Monday, 29 June 2026
- [1] arXiv:2606.27594 [pdf, html, other]
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Title: Reduction of Multiple Orthogonal Polynomials to Standard Orthogonal PolynomialsComments: 39 pages, 2 figuresSubjects: Classical Analysis and ODEs (math.CA)
In this article, we derive explicit formulae expressing multiple orthogonal polynomials in terms of standard orthogonal polynomials. We treat both the real-line and unit-circle settings: multiple orthogonal polynomials on the real line (MOPRL) are reduced to orthogonal polynomials on the real line (OPRL), while multiple orthogonal polynomials on the unit circle (MOPUC) are reduced to orthogonal polynomials on the unit circle (OPUC). These formulae also yield corresponding reductions of the Christoffel--Darboux kernels, from the MOPRL kernel to the OPRL kernel and from the MOPUC kernel to the OPUC kernel. In particular, they recover Zinn-Justin's kernel for the external-source random matrix model [arXiv:cond-mat/9703033] and Baik's kernel reduction formula in the spiked source model [arXiv:0809.3970]. We also apply our general results to concrete examples: in the real-line setting, we obtain an explicit expression for the multiple Hermite Christoffel--Darboux kernel in terms of classical Hermite polynomials, while in the unit-circle setting, we use arc-indicator weights to exhibit resonance-type zero escape phenomena for type II MOPUC.
- [2] arXiv:2606.28292 [pdf, html, other]
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Title: On the Bourgain--Brezis--Mironescu spaces over Carleson tentsComments: 25 pages, 2 figures. Comments welcome!Subjects: Classical Analysis and ODEs (math.CA)
We introduce Carleson analogs of the Bourgain--Brezis--Mironescu spaces $B$ and $B_0$ by measuring mean oscillation over upper Carleson tents. For these spaces, denoted by $B_{\mathcal C}^p$ and $B_{\mathcal C,0}^p$, we prove two types of structural results. First, we show that they contain several natural classes of functions, including BMO/VMO--Carleson spaces, tent-space potential classes, and fractional Sobolev classes. Second, motivated by Zhu's structural theorem for BMO spaces induced by the Bergman metric, we establish decompositions of $B_{\mathcal C}^p$ and $B_{\mathcal C,0}^p$ into bounded-oscillation and bounded-average components. We then revisit the Bourgain--Brezis--Mironescu rigidity phenomenon in the Carleson setting. Although the direct rigidity statement fails for $B_{\mathcal C,0}^p$, we introduce a natural $B_{\mathcal C}^p$-trace and prove that the rigidity theorem survives at the level of traces.
New submissions (showing 2 of 2 entries)
- [3] arXiv:2606.28271 (cross-list from math-ph) [pdf, other]
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Title: Which Saddles Contribute? The South-East Rule for Multidimensional IntegralsSubjects: Mathematical Physics (math-ph); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Classical Analysis and ODEs (math.CA)
In this paper, we introduce and demonstrate a simple geometric algorithm to determine which critical points, both complex as well as real, contribute to the asymptotic evaluation of multiple integrals with exponential integrands of the form $e^{ikf(\boldsymbol{x})}$ over $\mathbb R^d$, for finite $d\ge 1$ and $f$ is analytic. In so doing, the algorithm removes the need to compute the flows of $-\text{Re} (i\nabla f)$ in $\mathbb C^d$ that is required to identify such relevant critical points in Picard-Lefschetz approaches to the derivation of such asymptotic expansions. By contrast, our algorithm relies on the combination of three simple features: the values of $f$ at all the critical points plotted in the complex Borel plane, the concept of adjacency between such points derived from algebraic resurgence/hyperasymptotic approaches and the new result here of a geometric "South-East" rule. The algorithm incorporates functions $f$ that remain bounded or unbounded on $\mathbb R^d$. We illustrate this new approach with both pedagogical and advanced examples, and draw conclusions as to its importance for resolving issues associated with Wick rotations and its implications for path integrals. This is a significant step towards a systematic way of identifying instanton contributions in real-time path integrals.
Cross submissions (showing 1 of 1 entries)
- [4] arXiv:2512.19840 (replaced) [pdf, html, other]
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Title: Quantum Mechanics on Lie Groups: I. Noncommutative Fourier TransformsComments: 45 pages, 3 figures. v2: Typos fixed, clarifications added, longer sections 4 and 5. Matches published version up to proofs in main textSubjects: Quantum Physics (quant-ph); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA)
Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.
- [5] arXiv:2605.15580 (replaced) [pdf, html, other]
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Title: A Weyl-type theorem for Diophantine approximations driven by LCA groups and applicationsSubjects: Dynamical Systems (math.DS); Classical Analysis and ODEs (math.CA); Number Theory (math.NT)
We investigate actions of locally compact Abelian (LCA) groups on the torus $\mathbb{T}^n$, motivated by their close connection with Diophantine approximation. While Kronecker's theorem yields a classical density result, we prove a stronger equidistribution theorem of Weyl type: every such action admits a decomposition into uniquely ergodic subsystems. The proof of this result is based on a characterization of unique ergodicity for actions of amenable groups on compact metric spaces. As consequences, we establish several foundational results for LCA groups, including the Bohr orthogonality of characters along arbitrary Folner sequences, a Bohr mean formula for almost periodic functions, and a Wiener-type theorem on LCA groups characterizing the discrete part of a Borel probability measure through its Fourier transform. An application to numerical analysis is also discussed.
- [6] arXiv:2606.27285 (replaced) [pdf, html, other]
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Title: Recovering Governing Equations from Solution Data: Identifiability Bounds for Linear and Nonlinear ODEsSubjects: Machine Learning (cs.LG); Information Theory (cs.IT); Classical Analysis and ODEs (math.CA); Dynamical Systems (math.DS)
Learning governing equations from observed solution data is a fundamental challenge in scientific machine learning, yet the theoretical conditions under which a ground-truth ODE can be uniquely and stably identified from multiple solution observations remain largely undeveloped, and no quantitative analysis of the sample complexity of such learning tasks exists in the literature. To address this gap, we introduce the Hausdorff distance on solution sets as the natural metric for comparing differential equations, since it captures the worst-case separation between two equations over all admissible initial conditions and thus encodes the minimax structure of the identification problem. We establish identifiability bounds for governing ODEs across a wide class of structure equations--ranging from linear ODEs to nonlinear classes with Lipschitz (Hölder)-continuous vector fields--characterizing precisely when two distinct equations can be distinguished from solution data. Using this metric, we derive metric entropy estimates for the relevant ODE classes and analyze sample complexity bounds, quantifying how many solution observations are needed to reliably recover the governing equation.