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Degree of Simultaneous Coconvex Polynomial Approximation [PDF]

Results in Mathematics, 1998
Let \(f\in C^1[-1,1]\) change its convexity \(s\)-times at the points \(y_j\in (-1,1)\) \((j-1, \dots,s)\). Then \(f\) is approximated by polynomials \(p_n\), which are coconvex with \(f\), i.e., \(p_n\) changes its convexity exactly at the same points \(y_j\) \((j=1, \dots,s)\).
Kopotun, K., Leviatan, D.
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Approximate Degree, Weight, and Indistinguishability

ACM Transactions on Computation Theory, 2022
We prove that the OR function on {-1,1\} n can be pointwise approximated with error ε by a polynomial of degree O ( k ) and weight 2 O ( n log (1/ε)/k)
Xuangui Huang, Emanuele Viola
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Approximate Degree in Classical and Quantum Computing

Foundations and Trends® in Theoretical Computer Science, 2022
The approximate degree of a Boolean function f captures how well f can be approximated pointwise by low-degree polynomials. This monograph surveys what is known about approximate degree and illustrates its applications in theoretical computer science.
Mark Bun, Justin Thaler
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An Approximate Minimum Degree Ordering Algorithm

SIAM Journal on Matrix Analysis and Applications, 1996
An approximative minimum degree ordering algorithm (AMD) based on the symmetric analogue of the degree bounds in the unsymmetric-pattern multifrontal method is described. The analysis of the performance and accuracy on a set of test matrices show that AMD is typically much faster compared with other established codes that compute minimum degree ...
Davis, Timothy A., Amestoy, Patrick, Duff, Iain S.   +2 more
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Approximating PageRank from In-Degree

2008
PageRank is a key element in the success of search engines, allowing to rank the most important hits in the top screen of results. One key aspect that distinguishes PageRank from other prestige measures such as in-degree is its global nature. From the information provider perspective, this makes it difficult or impossible to predict how their pages ...
Santo Fortunato, Marián Boguñá, Alessandro Flammini, Filippo Menczer   +3 more
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Degree of Approximation by Rational Functions with Prescribed Numerator Degree

Canadian Journal of Mathematics, 1994
AbstractWe prove a Jackson type theorem for rational functions with prescribed numerator degree: For continuous functions f: [—1,1] —> ℝ with ℓ sign changes in (—1,1), there exists a real rational function Rℓ,n(x) with numerator degree ℓ and denominator degree at most n, that changes sign exactly where f does, and such thatHere C is independent of f,
Leviatan, D., Lubinsky, D. S.
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Approximation representations for reals and their wtt‐degrees

Mathematical Logic Quarterly, 2004
AbstractWe study the approximation properties of computably enumerable reals. We deal with a natural notion of approximation representation and study their wtt‐degrees. Also, we show that a single representation may correspond to a quite diverse variety of reals. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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Degree Reduction Approximations

1990
The majority of CAD systems provide some form of parametric curve and surface representation but the precise form of this representation varies considerably. Some systems use simple parametric polynomial curves and surfaces, others provide some form of rational parametric representation.
M. A. Lachance, P. A. Sherar, R. J. Goult   +2 more
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On the Degree of Approximation in Multivariate Weighted Approximation

2002
Let s ≥ 1 be an integer, f ∈ L P (R s ) for some p, 1 ≤ p ∞ or be a continuous function on R S vanishing at infinity. We consider the degree of approxima-tion of f by expressions of the form exp \( ( - {\text{ }}\sum\limits_{k = 1}^s {{Q_k}\left( {{x_k}} \right)} )P\left( {{x_1},...,{x_s}} \right) \) where each exp(—Q k (·)) is a Freud type weight ...
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Degree of Approximation to Functions in a Normed Space

Journal of Computational Analysis and Applications, 2000
Let \(A\) be a lower triangular nonnegative matrix and let \(\{t_n\}\) be the sequence of \(A\)-transforms of the sequence of partial sums of the Fourier series of a \(2\pi\)-periodic function \(f\). Here, the authors study the degree of approximation of \(f\) in the Hölder metric by the sequence \(\{t_n\}\).
Mittal, M. L., Rhoades, B. E.
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