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About 10,000 papers for “over-approximation”

Complexity of Weighted Approximation over ...

Grzegorz W. Wasilkowski, Henryk Wozniakowski · 2000

We study approximation of univariate functions defined over the reals. We assume that the rth derivative of a function is bounded in a weighted L p norm with a weight /. Approximation algorithms use the values of a function and its derivatives up to order r \\Gamma 1. The worst case error of an algorithm is defined in a weighted L q norm with a weight ae. We study the worst case (information) complexity of the weighted approximation problem, which is equal to the minimal number of function and derivative evaluations needed to obtain error ". We provide necessary and sufficient conditions in terms of the weights / and ae, as well as the parameters r; p and q for the weighted approximation problem to have finite complexity. We also provide conditions which guarantee that the complexity of weighted approximation is of the same order as the complexity of the classical approximation problem over a finite interval. Such necessary and sufficient conditions are also provided for a weighted in...

Cited by 0

Strong approximation over function fields

Qile Chen, Yi Zhu · Journal of Algebraic Geometry · 2018

By studying <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper A Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-curves on varieties, we propose a geometric approach to the strong approximation problem over function fields of complex curves. We prove that strong approximation holds for smooth, low degree affine complete intersections with smooth boundary at infinity.

Cited by 2

Artin approximation over Banach spaces

Guillaume Rond · Communications in Algebra · 2020

We give examples showing that the usual Artin approximation theorems valid for convergent series over a field are no longer true for convergent series over a commutative Banach algebra. In particular, we construct an example of a commutative integral Banach algebra R such that the ring of formal power series over R is not flat over the ring of convergent power series over R.

Cited by 0Open access

Arithmetic Fujita approximation over adelic curves

Chunhui Liu · arXiv (Cornell University) · 2024

In this paper, we will prove an analogue of Fujita's approximation theorem under the framework of Arakelov theory over adelic curves, which proves a conjecture of Huayi Chen and Atsushi Moriwaki.

Cited by 0Open access

Improved sparse approximation over quasiincoherent dictionaries

Joel A. Tropp, Anna C. Gilbert, S. Muthukrishnan et al. · 2004

This paper discusses a new greedy algorithm for solving the sparse approximation problem over quasiincoherent dictionaries. These dictionaries consist of waveforms that are uncorrelated "on average," and they provide a natural generalization of incoherent dictionaries. The algorithm provides strong guarantees on the quality of the approximations it produces, unlike most other methods for sparse approximation. Moreover, very efficient implementations are possible via approximate nearest-neighbor data structures.

Cited by 74

Effective Diophantine Approximation over Number Fields

Hughes, N · Figshare · 2024

Classical Diophantine approximation as first explored by Dirichlet quantifies the density of the rationals in the real line by examining how many rational numbers approximate a given real number well. This subfield of number theory has inspired many great contributions to the literature, including Minkowski’s Geometry of Numbers and Liouville’s proof of the existence of transcendental numbers. Since then, the study of Diophantine approximation has been facilitated by a number of other seemingly unrelated fields, the most successful of these arguably being the study of dynamical systems on spaces of lattices. This thesis is concerned with the counting of Diophantine approximates to linear forms by rational numbers in number fields. In it, we obtain error terms for existing asymptotic results using mean value formulas for unimodular lattices and the ergodic property of certain flows on the space comprised of these lattices. We first prove an effective ergodic theorem that associates the rate of convergence of an ergodic flow to its moments and, using existing mean value results due to Siegel and Rogers which calculate the integrals of powers of functions on the space of lattices, we provide error terms to the asymptotic count of rational approximates to real linear forms of dimension higher than 1 × 1 and describe their distribution. This work is based on an argument first given by Athreya, Ghosh and Tseng. Secondly, we describe a moment formula for lattices in number fields, generalising Rogers’ result, and apply this to obtain counting results for approximation by algebraic numbers in a given number field to linear forms, answering a question posed recently by Alam and Ghosh. We conclude by proving a new second moment formula for the primitive Siegel transform of indicator functions on a space of two dimensional lattices associated to a number field, from which we deduce appropriate error rates for totally real number fields.

Cited by 0Open access

Tracking Matrix Approximation over Distributed Sliding Windows

Haida Zhang, Zengfeng Huang, Zhewei Wei et al. · 2017

In many modern applications, input data is represented as matrices and often arrives continuously. The ability to summarize and approximate data matrices in streaming fashion has become a common requirement in many emerging environments. In these applications, input data is usually generated at multiple distributed sites and simply centralizing all data is often infeasible. Therefore, novel algorithmic techniques are required. Furthermore, in most of these applications, queries must be answered solely based on the recently observed data points (e.g., data collected over the last hour/day/month), which makes the problem even more challenging. In this paper, we propose to study the problem of tracking matrix approximations over distributed sliding windows. In this problem, there are m distributed sites each observing a stream of d-dimensional data points. The goal is to continuously track a small matrix B as an approximation to Aw, the matrix consists of data points in the union of the streams which arrived during the last W time units. The quality of the approximation is measured by the covariance error kAT wAw? BTBk = kAk2 F [1], and the primary goal is to minimize communication, while providing provable error guarantee. We propose novel communication-efficient algorithms for this problem. Our sampling-based algorithms continuously track a weighted sample of rows according to their squared norms, which generalize and simplify the sampling techniques in [2]. We also propose deterministic tracking algorithms that require only one-way communication and provide better error guarantee. All algorithms have provable guarantees, and extensive experimental studies on real and synthetic datasets validate our theoretical claims and demonstrate the efficiency of these algorithms.

Cited by 8

Forms over number fields and weak approximation

C. M. SKINNER · Compositio Mathematica · 1997

Let $K$ be a number field, and let $X \subseteq P^{s-1}_K$ be a smooth complete intersection defined over $K$ . In this paper, weak approximation is shown to hold for $X$ provided $s$ exceeds some function of the degree and codimension of $X$ . This is a corollary of a more general result about the number of integral points on certain affine varieties in homogeneously expanding regions. This general result is established via a suitable adaptation of the Hardy-Littlewood Circle Method.

Cited by 29Open access

Best Approximation-Preserving Operators over Hardy Space

Fahreddin. G. Abdullayev, V. V. Savchuk, Maryna V. Savchuk · arXiv (Cornell University) · 2022

Let $T_n$ be the linear Hadamard convolution operator acting over Hardy space $H^q$, $1\le q\le\infty$. We call $T_n$ a best approximation-preserving operator (BAP operator) if $T_n(e_n)=e_n$, where $e_n(z):=z^n,$ and if $\|T_n(f)\|_q\le E_n(f)_q$ for all $f\in H^q$, where $E_n(f)_q$ is the best approximation by algebraic polynomials of degree a most $n-1$ in $H^q$ space. We give necessary and sufficient conditions for $T_n$ to be a BAP operator over $H^\infty$. We apply this result to establish an exact lower bound for the best approximation of bounded holomorphic functions. In particular, we show that the Landau-type inequality $\left|\widehat f_n\right|+c\left|\widehat f_N\right|\le E_n(f)_\infty$, where $c&gt;0$ and $n

Cited by 0Open access