Coppersmith-rivlin Type Inequalities and the Order of Vanishing of Polynomials at 1
نویسنده
چکیده
for every n ≥ 2, where Fn denotes the set of all polynomials of degree at most n with coefficients from {−1, 0, 1}. Littlewood considered minimization problems of this variety on the unit disk. His most famous, now solved, conjecture was that the L1 norm of an element f ∈ Fn on the unit circle grows at least as fast as c logN , where N is the number of non-zero coefficients in f and c > 0 is an absolute constant.
منابع مشابه
Some new families of definite polynomials and the composition conjectures
The planar polynomial vector fields with a center at the origin can be written as an scalar differential equation, for example Abel equation. If the coefficients of an Abel equation satisfy the composition condition, then the Abel equation has a center at the origin. Also the composition condition is sufficient for vanishing the first order moments of the coefficients. The composition conjectur...
متن کاملA fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations
In this paper we introduce a type of fractional-order polynomials based on the classical Chebyshev polynomials of the second kind (FCSs). Also we construct the operational matrix of fractional derivative of order $ gamma $ in the Caputo for FCSs and show that this matrix with the Tau method are utilized to reduce the solution of some fractional-order differential equations.
متن کاملSome compact generalization of inequalities for polynomials with prescribed zeros
Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$. In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2leq rRleq k^2$, $k^2 leq rRleq R^2$ and for $Rleq r leq k$. Our results refine and generalize certain well-known polynomial inequalities.
متن کاملRate of Growth of Polynomials Not Vanishing inside a Circle
A well known result due to Ankeny and Rivlin [1] states that if p(z) = ∑n v=0 avz v is a polynomial of degree n satisfying p(z) 6= 0 for |z| < 1 then for R ≥ 1 max |z|=R |p(z)| ≤ R n + 1 2 max |z|=1 |p(z)|. It was proposed by late Professor R.P. Boas, Jr. to obtain an inequality analogous to this inequality for polynomials having no zeros in |z| < K, K > 0. In this paper, we obtain some results...
متن کاملPseudo-boolean Functions and the Multiplicity of the Zeros of Polynomials
Let p : {−1, 1} → R be a polynomial in variables x1, x2, . . . , xn. Since x2j = 1 we can restrict our consideration to multi-linear polynomials in which each variable appears with degree at most 1. Using the observation that every p : {−1, 1} → R symmetric multilinear polynomial can be represented by a polynomial P : {0, 1, . . . , n} → R of a single variable as p(x) = P (|x|) , x = (x1, x2, ....
متن کامل