Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/9510214

Compact Jacobi matrices: from Stieltjes to Krein and M(a,b)

Walter Van Assche
View original source ↗Read source document ↗Export metadata ↓

Abstract

In a note at the end of his paper {\it Recherches sur les fractions continues}, Stieltjes gave a necessary and sufficient condition when a continued fraction is represented by a meromorphic function. This result is related to the study of compact Jacobi matrices. We indicate how this notion was developped and used since Stieltjes, with special attention to the results by M. G. Krein. We also pay attention to the perturbation of a constant Jacobi matrix by a compact Jacobi matrix, work which basically started with Blumenthal in 1889 and which now is known as the theory for the class $M(a,b)$.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Walter Van Assche. 1995-10-09. Compact Jacobi matrices: from Stieltjes to Krein and M(a,b). https://arxiv.org/abs/math/9510214

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications

Let $p$ be an odd prime. We prove the extension estimate $R_{S_j}^*(2\to r)\lesssim_r 1$ for every nonzero-radius sphere $S_j\subseteq\mathbb{F}_p^4$ and every $r\geq \, 34/11$, uniformly in $p$ and $j$. This improves the Stein--Tomas exponent $10/3$ established by Iosevich and Koh (2008). We also formulate a localized spherical restriction/extension conjecture that predicts the sharp dependence of the restriction norm on the size of the physical support. This conjecture implies the spherical extension estimates $R_{S_j}^*(2\to r)\lesssim_r 1$ for every $r>3$, and yields almost-every-pin distance estimates at the conjectured Erdős--Falconer exponent in four dimensions, up to an arbitrarily small power loss in the set-size hypothesis. Using the same method, we improve the bounds supplied by Fourier decay and Plancherel at intermediate support scales and derive new almost-every-pin distance estimates in $\mathbb{F}_p^4$.

math.CA↗
arXiv2026

Dimension-free estimates for discrete maximal functions over cubes in $\mathbb Z^d$

In this short note, we establish dimension-free $\ell^p(\mathbb Z^d)$ bounds, for all $p\in(1,\infty]$, for the discrete Hardy--Littlewood maximal functions associated with cubes in $\mathbb Z^d$, answering a question that had been open for a while. The key idea is to prove dimension-free bounds for the $\ell^p(\mathbb Z^d)$ norms of the differences of the corresponding averages. This follows from an ad hoc interpretation of the associated discrete multipliers as a special continuous family of multipliers to which basic fractional integration and complex interpolation can be applied. The same method also yields an elementary proof of Bourgain's dimension-free $L^p(\mathbb R^d)$ bounds for the Hardy--Littlewood maximal function associated with cubes in $\mathbb R^d$.

math.CA↗
arXiv2026

Establishing the Polynomial Wolff Axioms for $δ$-Separated $δ$-Tubes With #o-minimality

We establish the full version of a conjecture of Guth and Zahl, giving a lower bound for the volume of a semialgebraic set that has a large intersection with a collection of $δ$-separated $δ$-tubes. Our proof uses o-minimal methods to simplify the proof of Katz and Rogers, who proved the conjecture up to a small factor. We also establish that the constants depend polynomially on the complexity of the semialgebraic set, and more generally in the #o-minimal setting.

math.CA↗