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Results tagged with polynomials
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user 89429
Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
2
votes
Minimum degree of a nonnegative polynomial uniformly approximating two constant values on tw...
Concerning the nonnegativity constraint, obviously, it cannot be satisfied by odd approximating polynomials. … Yuditskii, Polynomials of the best uniform approximation to sgn(x) on two intervals, J. Anal. Math. 114 (2011), 285-315. …
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6
votes
Accepted
Real vs complex norm for polynomials
Let $K=[-1,1]$ and let $\Omega$ be its complement in $\mathbb C$. By the Bernstein's lemma (or Bernstein-Walsh lemma), see Theorem 5.5.7 in T. Ransford, Potential theory in the complex plane, Cambridg …
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4
votes
Accepted
A bounded polynomial having bounded coefficients: several variables
Bernstein, On certain elementary extremal properties of
polynomials in several variables, Doklady Akad. Nauk SSSR (N.S.) 59
(1948), 833-836. … Kellogg, On bounded polynomials in several variables, Math. Z.
27 (1928), no. 1, 5-64. …
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15
votes
Accepted
Find the maximum of $|a_{p}|$, if $a_0+a_1x+\dots+a_nx^n:[-1,1]\mapsto [-1,1]$
Analytic theory of polynomials. London
Mathematical Society Monographs. New Series, 26. The Clarendon Press,
Oxford University Press, Oxford, 2002. … The proof of the first assertion consists in considering a one-parameter family of polynomials constructed from $f$ and $T_{n}$ and depending on a parameter $\theta\in(-1,1)$, and then using Descartes' …
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3
votes
Bounds on polynomial values
.$$
About the Legendre polynomials discussed in the comments, it is known that the sum $\sum_{k=0}^{n}P_{k}^{2}(x)$, that is the reciprocal of the so-called Christoffel function, behaves, for $n$ large …
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1
vote
Accepted
Reference request: maximal ratio of different norms of polynomials
For the interval $[-1,1]$, the extremal polynomial is
$$\frac{P_{n}(x)-P_{n+1}(x)}{1-x},$$
where $P_{n}$ is the Legendre polynomials of degree $n$ (with normalization $\int P_{n}^{2}(x)dx=2/(2n+1)$). …
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7
votes
Accepted
Half spaces free of roots of a given polynomial
The property that the zeros of the derivative of a polynomial $P$ lie in the convex hull of the zeros of $P$ is usually called the Gauss-Lucas theorem.
About question 2), the algebra of entire functi …
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3
votes
Accepted
Do Zernike polynomials form an orthogonal basis of $L^2 ( \mathbb{D} )$?
The (complex) Zernike polynomials $V_{n}^{l}(x,y)$, of total degree $n$, with
$|l|\leq n,~n-|l|\text{ even}$, are such that
$$
V_{n}^{l}(x,y)=R_{n}^{l}(\rho)e^{il\varphi},\quad \text{with }x=\rho\cos\varphi … Some details about the Zernike polynomials can be found in Appendix VII of
M. Born, E. …
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4
votes
Accepted
L1 analog of Bernstein's inequality
Erdelyi, Polynomials and Polynomial inequalities, Graduate Texts in Mathematics 161, Springer
should be a good source for your question. … There is also a weighted analog of the above inequality, see Theorem A.4.16., which holds for generalized polynomials, see (A.4.1) for a definition. …
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2
votes
Generalisation of Chebyshev series to arbitrary sets
The series of Chebyshev polynomials corresponding to $[-1,1]$ are then replaced with series of Faber polynomials of the set $K$. … Liesen, Properties and examples of Faber-Walsh polynomials. Comput. Meth. Funct. Theory 17 (2017), 151-177. …
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1
vote
Accepted
For every table of interpolating nodes, there is a positive continuous function whose interp...
Yes, it follows from the following result of S. Bernstein (Quelques remarques sur l'interpolation, Math. Ann. 79 (1918), 1-12):
For an arbitrary scheme $X=\cup_{n} A_{n}$ of points in $[-1,1]$,
…
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3
votes
Does every positive continuous function have a non-negative interpolating polynomial of ever...
The interpolating polynomials $p_{n}$ are seeked in the form
\begin{align*}
p_{n}(x) & =a_{p}^{2}(x)+x(1-x)b_{p-1}^{2}(x),\quad n=2p+1,\\
p_{n}(x) & =xa_{p}^{2}(x)+(1-x)b_{p}^{2}(x),\quad n=2p,
\end{align … *}
with $a_{p}$ and $b_{p}$ polynomials of degree $p$, and $b_{p-1}$ a polynomial of degree $p-1$, which are classical representations for non-negative polynomials in $[0,1]$. …
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21
votes
1
answer
3k
views
Density of polynomials in $C^k(\overline\Omega)$
Then, the polynomials are dense in $C^k(\Omega)$, see e.g. Treves, Topological vector spaces, distributions and kernels. … .$$
Is it true that the polynomials are dense in $C^k(\overline\Omega)$ ? …
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