An extremal problem for algebraic polynomials with zero mean value on an interval

LetP _n ^O(h) be the set of algebraic polynomials of degreen with real coefficients and with zero mean value (with weighth) on the interval [?1, 1]: $$\smallint _{ - 1}^1 h(x)p_n (x) dx = 0;$$ hereh is a function which is summable, nonnegative, and nonzero on a set of positive measure on [?1, 1]. We study the problem of the least possible value $$i_n (h) = \inf \{ \mu (p_n ):p_n \in \mathcal{P}_n^0 \} $$ of the measure μ(P _n)=mes{x∈[?1,1]:P _n(x)≥0} of the set of points of the interval at which the polynomialp _n∈P _n ^O is nonnegative. We find the exact value ofi _n(h) under certain restrictions on the weighth. In particular, the Jacobi weight $$h^{(\alpha ,\beta )} (x) = (1 - x)^\alpha (1 + x)^\beta $$ satisfies these restrictions provided that ?1<α, β≤0.     

One-sided weighted integral approximation of characteristic functions of intervals by polynomials on a closed interval

We consider the problem of one-sided weighted integral approximation on the interval [?1, 1] to the characteristic functions of intervals (a, 1] ? (?1, 1] and (a, b) ? (?1, 1) by algebraic polynomials. In the case of half-intervals, the problem is solved completely. We construct an example to illustrate the difficulties arising in the case of an open interval.     

Algebraic polynomials least deviating from zero in measure on a segment

We investigate the problem of algebraic polynomials with given leading coefficients that deviate least from zero on the segment [–1, 1] with respect to a measure, or, more precisely, with respect to the functional μ(f) = mes{x ∈ [–1, 1]: ∣f (x)∣ ≥ 1}. We also discuss an analogous problem with respect to the integral functionals ∫_–1¹ φ (∣f (x)∣) dx for functions φ that are defined, nonnegative, and nondecreasing on the semiaxis [0, +∞).     

Certificates for nonnegativity of polynomials with zeros on compact semialgebraic sets

We prove a criterion for an element of a commutative ring to be contained in an archimedean subsemiring. It can be used to investigate the question whether nonnegativity of a polynomial on a compact semialgebraic set can be certified in a certain way. In case of (strict) positivity instead of nonnegativity, our criterion simplifies to classical results of Stone, Kadison, Krivine, Handelman, Schmüdgen et al. As an application of our result, we give a new proof of the following result of Handelman: If an odd power of a real polynomial in several variables has only nonnegative coefficients, then so do all sufficiently high powers.Partially supported by the DFG project 214371 “Darstellung positiver Polynome”.     

Algebraic polynomials with integer coefficients deviating little from zero on an interval

Translated from Matematicheskie Zametki, Vol. 50, No. 3, pp. 58–67, September, 1991.     

Stability of a conditional Cauchy equation on a set of measure zero

In this paper, we prove the Hyers–Ulam stability theorem when \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy $$|f(x + y) - g(x) - h(y)| \leq \epsilon$$ in a set \({\Gamma \subset \mathbb{R}^{2}}\) of measure \({m(\Gamma) = 0}\) , which refines a previous result in Chung (Aequat Math 83:313–320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy the Pexider equation $$f(x + y) - g(x) - h(y) = 0$$ in \({\Gamma}\) , then the equation holds for all \({x, y \in \mathbb{R}}\) . Using our method of construction of the set, we can find a set \({\Gamma \subset \mathbb{R}^{2n}}\) of 2n-dimensional measure 0 and obtain the above result for the functions \({f, g, h : \mathbb{R}^{n} \to \mathbb{C}}\) .     

Quadratic functional equations in a set of Lebesgue measure zero

Let R$\mathbb{R}$ be the set of real numbers, Y   a Banach space and f:R→Y$f:\mathbb{R}\to Y$. We prove the Hyers–Ulam stability theorem for the quadratic functional inequality

‖f(x+y)+f(x−y)−2f(x)−2f(y)‖≤?$\Vert f(x+y)+f(x-y)-2f(x)-2f(y)\Vert \le ?$

for all (x,y)∈Ω$(x,y)\in \Omega$, where Ω⊂R²$\Omega \subset {\mathbb{R}}^2$ is of Lebesgue measure 0. Using the same method we dealt with the stability of two more functional equations in a set of Lebesgue measure 0.     

An inequality for polynomials with a prescribed zero

Let p_n(z)=∑_(k-0)~n a_kz~k be a polynomial of degree n such that |p_n(z)|≤M for |z|≤1. It is well.known that for 0≤u相似文献   

Algebra of bounded polynomials on a set Zariski closed at infinity cannot be finitely generated

We prove that if a set S⊂Rⁿ$S\subset {\mathbb{R}}^n$ is Zariski closed at infinity, then the algebra of polynomials bounded on S cannot be finitely generated. It is a new proof of a fact already known to Plaumann and Scheiderer (2012) [1]. On the way we show that if the ring R[_ζ1,…,_ζk]⊂R[X]$\mathbb{R}[{\zeta }_1,\dots ,{\zeta }_k]\subset \mathbb{R}[X]$ contains the ideal (_ζ1,…,_ζk)R[X]$({\zeta }_1,\dots ,{\zeta }_k)\mathbb{R}[X]$, then the mapping (_ζ1,…,_ζk):Rⁿ→R^k$({\zeta }_1,\dots ,{\zeta }_k):{\mathbb{R}}^n\to {\mathbb{R}}^k$ is finite.