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Results tagged with polynomials
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user 171320
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.
1
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211
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On the degree elevation needed to bring Bernstein coefficients to [0, 1]
It is nothing I could find so far in the papers on Bernstein polynomials. …
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1
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1
answer
150
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Coefficient bounds for polynomials that map the unit interval to itself
While studying the "coin-flipping degree" problem I have come across the following conjecture. It gives bounds on the power coefficients of a polynomial that maps the unit interval to itself. If tru …
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3
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1
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227
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Explicit bounds on the difference between Bernstein polynomials
For the polynomials $2x(1-x)$ and $2x^2(1-x)$, the left-hand side of $(1)$ appears to converge to 0 at the rate $O(1/n^2)$. … References
Butzer, Paul (1953) "Linear Combinations of Bernstein Polynomials". Canadian Journal of Mathematics, 5, 559-567. doi:10.4153/CJM-1953-063-7, MR0058023, Zbl 0051.05002. …
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3
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Explicit and fast error bounds for polynomial approximation
After analyzing the proof of Güntürk and Li (2021), Theorem 2.4, I believe I found explicit error bounds for the Micchelli–Felbecker polynomials (iterated Bernstein polynomials) when $f(x)$ has a given … The following is Python code I used to calculate the error bounds for the iterated Bernstein polynomials. …
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6
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Using the Lorentz operators to build polynomials that converge to a continuous function
Find explicit bounds, with no hidden constants, on the approximation error for the "Lorentz operator" $Q_{n,r}(f)$ (described below), and for the polynomials $(f_n)$ and $(g_n)$ formed with it. … They used the Lorentz operators to build a family of polynomials $(g_n)$ that converge from below to $f$ and satisfy the following: $(g_{2n}−g_{n})$ is a polynomial with non-negative Bernstein coefficients …
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17
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Explicit and fast error bounds for polynomial approximation
Polynomials with faster convergence than Bernstein polynomials
As is known since Voronovskaya (1932), the Bernstein polynomials converge uniformly to $f$, in general, at a rate no faster than $O(1/n)$, … (See also a related question by Luis Mendo on ordinary Bernstein polynomials.) …
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4
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Explicit and fast error bounds for approximating continuous functions
Find explicit bounds, with no hidden constants, on the error in approximating $f$ with the following polynomials: The polynomials are similar to Chebyshev interpolants, but evaluate $f$ at rational values … (See also a related question by Luis Mendo on ordinary Bernstein polynomials.) …
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Explicit and fast error bounds for approximating continuous functions
Here are some results on certain polynomials. … "Linear combinations of two Bernstein polynomials", Mathematical Foundations of Computing, 2022.
[^5]: Xie, L. Uniform approximation by combinations of Bernstein polynomials. Approx. …
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2
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223
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Bounds on the coin-flipping degree
Powers, V., Reznick, B., "A new bound for Pólya's Theorem with applications to polynomials positive on polyhedra, Journal of Pure and Applied Algebra 164 (24 October 2001). …
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Bounds on the coin-flipping degree
An example is the family of degree-2 polynomials $r\lambda-r\lambda^2$, where $r$ is a rational number greater than 0 and less than 4. …
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3
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A conjecture on consistent monotone sequences of polynomials in Bernstein form
They relate to polynomials that achieve a better convergence rate than Bernstein polynomials (namely $O(1/n^{r/2})$ rather than $O(1/n)$), such as linear combinations (Butzer 1953) and iterated Boolean … sums (Micchelli 1973) of Bernstein polynomials. …
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Concave and other bounded functions: Series representation and converging polynomials
Prove or disprove: Given that $f:[0,1]\to (0,1]$ is convex, the polynomials $(g_n) = (B_n(f) - \max_{0\le\lambda\le 1}|B_n(f)(\lambda)-f(\lambda)|)$ (where $n\ge 1$ is an integer power of 2) are in Bernstein … The same is true for the polynomials $(g_n) = (B_n(f) - |B_n(f)(1/2)-f(1/2)|)$, if $f$ is also symmetric about 1/2.
References
Keane, M. S., and O'Brien, G. …
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