Results 141 to 150 of about 186,989 (178)

Approximate Polynomial GCD by Approximate Syzygies

Mathematics in Computer Science, 2019
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Approximation by Polynomials

2009
This chapter introduces some of the essentials of approximation theory, in particular approximating functions by “nice” ones such as polynomials. In general, the intention of approximation theory is to replace some complicated function with a new function, one that is easier to work with, at the price of some (hopefully small) difference between the ...
Kenneth R. Davidson, Allan P. Donsig
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Simultaneous approximation by algebraic polynomials

Constructive Approximation, 1996
Some estimates for simultaneous polynomial approximation of a function and its derivatives are obtained. These estimates are exact in a certain sense. In particular, the following result is derived as a corollary: For \(f\in C^r[-1,1]\), \(m\in\mathbb{N}\), and any \(n\geq\max\{m+ r-1,2r+1\}\), an algebraic polynomial \(P_n\) of degree \(\leq n ...
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Approximation by Polynomials

1991
The classical Weierstrass approximation theorem states that for every continuous function f on [0, 1] and for any n ≥ 0, there is a real-valued polynomial function φ n such that |f(x) - φ n (x){ ≤ 2-n for all x ∈ [0, 1], In this chapter we investigate the polynomial-time version of the Weierstrass approximation theorem: Is the sequence |φ n ...
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Approximation by Homogeneous Polynomials

Constructive Approximation, 2006
Let \(K\subset\mathbb{R}^d\) be the boundary of a convex domain symmetric to the origin. The conjecture that any continuous even function can be uniformly approximated by homogeneous polynomials of even degree on K is proven in the following cases: (a) if d = 2; (b) if K is twice continuously differentiable and has positive curvature in every point; or
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Simultaneous Approximation by Polynomial Projection Operators

Mathematische Nachrichten, 1996
AbstractPointwise estimates are obtained for simultaneous approximation of a function f and its derivatives by means of an arbitrary sequence of bounded projection operators with some extra condition (1.3) (we do not require the operators to be linear) which map C[‐1,1] into polynomials of degree n, augmented by the interpolation of f at some points ...
Xie, T. F., Zhou, S. P.
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Approximation by spherical polynomials

1984
Let \(C(\sigma)=C^ 0(\sigma)\) be the space of all continuous functions on the unit sphere \(\sigma\) in \(R^ n\), \(C^{\ell}(\sigma)\) the space of all functions of the class \(C^{\ell}\) on \(\sigma\) with the norm \[ \| f\|_{C^{\ell}(\sigma)}=\| f\|_{C(\sigma)}+\max_{\mu \in \sigma}\max_{\vec s}| \partial^{\ell} f(\mu)/\partial s^{\ell}| \] and \(H^
Nikol'skij, S. M., Lizorkin, P. I.
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Approximation by Bernstein-Chlodowsky polynomials

2002
In the paper the weighted approximation of continuous functions by Bernstein-Chlodowsky polynomials and their generalizations are presented. The Bernstein-Chlodowsky polynomials are defined by \[ (B_n f)(x)= \sum^n_{k=0} f\Biggl({k\over n} b_n\Biggr){n\choose k} \Biggl({x\over b_n}\Biggr)^k\Biggl(1- {x\over b_n}\Biggr)^{n-k},\tag{1} \] where \(0\leq x ...
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Approximation in the Mean by Polynomials

The Annals of Mathematics, 1991
Let \(\mu\) be a positive measure with compact support in the complex plane and let \(t\in[1,\infty)\). Denote by \(P^ t(\mu)\) the closure in \(L^ t(\mu)\) of the polynomials in one complex variable. The paper deals with the description of \(P^ t(\mu)\). The main results are the following: There exists a Borel partition \(\{\Delta_ i\}^ \infty_{i=0}\)
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Approximation of analytic functions by polynomials ?close? to polynomials of best approximation

Ukrainian Mathematical Journal, 1992
See the review in Zbl 0758.30034.
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