An entire function and its derivatives sharing a polynomial

In this paper, we study the growth of solutions of a first-order linear differential equation and that of a second-order linear differential equation. From this we obtain some uniqueness theorems of a nonconstant entire function and its first derivative having the same fixed points with the same multiplicities. The results in this paper also improve some known results. Some examples show that the results in this paper are best possible.     

Criterion of polynomial increase of a function and its derivatives

ДОкАжАНО, ЧтО Дль тОгО, ЧтОБы Дльr РАж ДИФФЕРЕНцИРУЕМОИ НА пРОМЕжУткЕ [А, + ∞) ФУНкцИИf сУЩЕстВОВА л тАкОИ МНОгОЧлЕН (1) $$P(x) = \mathop \Sigma \limits_{\kappa = 0}^{r - 1} a_k x^k ,$$ , ЧтО (2) $$\mathop {\lim }\limits_{x \to + \infty } (f(x) - P(x))^{(k)} = 0,k = 0,1,...,r - 1,$$ , НЕОБхОДИМО И ДОстАтО ЧНО, ЧтОБы схОДИлсь ИН тЕгРАл (3) $$\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{r - 1} }^{ + \infty } {f^{(r)} (t)dt.}$$ ЕслИ ЁтОт ИНтЕгРАл сх ОДИтсь, тО Дль кОЁФФИц ИЕНтОВ МНОгОЧлЕНА (1) ИМЕУт МЕс тО ФОРМУлы $$\begin{gathered} a_{r - m} = \frac{1}{{(r - m)!}}\left( {\mathop \Sigma \limits_{j = 1}^m \frac{{( - 1)^{m - j} f^{(r - j)} (x_0 )}}{{(m - j)!}}} \right.x_0^{m - j} + \hfill \\ + ( - 1)^{m - 1} \left. {\mathop \Sigma \limits_{l = 0}^{m - 1} \frac{{x_0^l }}{{l!}}\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{m - l - 1} }^{ + \infty } {f^{(r)} (t_{m - 1} )dt_{m - 1} } } \right),m = 1,2,...,r. \hfill \\ \end{gathered}$$ ДОстАтОЧНыМ, НО НЕ НЕОБхОДИМыМ Усл ОВИЕМ схОДИМОстИ кРА тНОгО ИНтЕгРАлА (3) ьВльЕтсь схОДИМОсть ИНтЕгРАл А \(\int\limits_a^{ + \infty } {x^{r - 1} f^{(r)} (x)dx}\)      

On the zeros of a polynomial and its derivatives

If is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk, then the Gauss-Lucas theorem states that all zeros of lie in the same disk. We study the following question: what is the maximum distance from the arithmetic mean of all zeros of to a nearest zero of ? We obtain bounds for this distance depending on degree. We also show that this distance is equal to for polynomials of degree 3 and polynomials with real zeros.

     

On the approximation of derivatives of the interpolation Hermite polynomial on a triangle

Bounds on the deviation of the directional derivatives of the Hermite polynomial in the directions of a triangle sides are obtained; it is proved that these bounds are sharp. As a consequence, bounds on the deviations of the partial derivatives up to the third order inclusive are obtained.     

Simultaneouse approximation to a differentiable function and its derivatives by Lagrange interpolating polynomials

This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima-tion:then|f~(k)(x)-P_n~(k)(f,x)|=O(1)△_n~(q-k)(x)ωwhere P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodesX_nUY_n(see the definition of the next).     

Successive polynomial spline function approximation

The use of successive polynomial spline approximation is established as a method of improving the accuracy of estimates of derivatives of periodic functions approximated by interpolating odd order splines defined on a uniformly spaced set of data points. For the various configurations possible with this multiple-approximation method, bounds for the leading error terms are explicitly given. In particular, for the quintic spline, the variety of approximation sequences is described in detail.     

On arrangements of roots for a real hyperbolic polynomial and its derivatives

In this paper we count the number ?_n^(0,k), k?n−1, of connected components in the space Δ_n^(0,k) of all real degree n polynomials which a) have all their roots real and simple; and b) have no common root with their kth derivatives. In this case, we show that the only restriction on the arrangement of the roots of such a polynomial together with the roots of its kth derivative comes from the standard Rolle's theorem. On the other hand, we pose the general question of counting all possible root arrangements for a polynomial p(x) together with all its nonvanishing derivatives under the assumption that the roots of p(x) are real. Already the first nontrivial case n=4 shows that the obvious restrictions coming from the standard Rolle's theorem are insufficient. We prove a generalized Rolle's theorem which gives an additional restriction on root arrangements for polynomials.     

On parametric polynomial circle approximation

In the paper, the uniform approximation of a circle arc (or a whole circle) by a parametric polynomial curve is considered. The approximant is obtained in a closed form. It depends on a parameter that should satisfy a particular equation, and it takes only a couple of tangent method steps to compute it. For low degree curves, the parameter is provided exactly. The distance between a circle arc and its approximant asymptotically decreases faster than exponentially as a function of polynomial degree. Additionally, it is shown that the approximant could be applied for a fast evaluation of trigonometric functions too.     

On the convergence of hybrid polynomial approximation to higher derivatives of rational curves

In this paper, we derive the bounds on the magnitude of l  th (l=2,3)$(l=2,3)$ order derivatives of rational Bézier curves, estimate the error, in the _L∞$L_{\infty }$ norm sense, for the hybrid polynomial approximation of the l  th (l=1,2,3)$(l=1,2,3)$ order derivatives of rational Bézier curves. We then prove that when the hybrid polynomial approximation converges to a given rational Bézier curve, the l  th (l=1,2,3)$(l=1,2,3)$ derivatives of the hybrid polynomial approximation curve also uniformly converge to the corresponding derivatives of the rational curve. These results are useful for designing simpler algorithms for computing tangent vector, curvature vector and torsion vector of rational Bézier curves.     

Hybrid polynomial approximation to higher derivatives of rational curves

In this paper, we extend the results published in JCAM volume 214 pp. 163-174 in 2008. Based on the bound estimates of higher derivatives of both Bernstein basis functions and rational Bézier curves, we prove that for any given rational Bézier curve, if the convergence condition of the corresponding hybrid polynomial approximation is satisfied, then not only the l-th (l=1,2,3) derivatives of its hybrid polynomial approximation curve uniformly converge to the corresponding derivatives of the rational Bézier curve, but also this conclusion is tenable in the case of any order derivative. This result can expand the area of applications of hybrid polynomial approximation to rational curves in geometric design and geometric computation.     

On best error bounds for approximation by piecewise polynomial functions

Summary An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.The work presented in this paper was supported by the ERDA Mathematics and Computing Laboratory, Courant Institute of Mathematical Sciences, New York University, under Contract E(11-1)-3077 with the Energy Research and Development Administration.     

On simultaneous approximation to a differentiable function and its derivative by inverse Pal-Type interpolation polynomials

Let ξ_{n −1} < ξ_{n −2} < ξ_{n − 2} < ... < ξ₁ be the zeros of the the (n−1)-th Legendre polynomial P_n−1(x) and −1=x_n<x_n−1<...<x₁=1, the zeros of the polynomial . By the theory of the inverse Pal-Type interpolation, for a function f(x)∈C _[−1,1] ¹ , there exists a unique polynomial R_n(x) of degree 2n−2 (if n is even) satisfying conditions R_n(f, ξ_k) = f (ε_k) (1 ⩽ k ⩽ n −1); R¹ _n(f,x_k)=f¹(x_k)(1≤k≤n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {R_n(f, x)} (n is even) and the main result of this paper is that if f∈C _[1,1] ^r , r≥2, n≥r+2, and n is even then |R¹ _n(f,x)−f¹(x)|=0(1)|W_n(x)|h(x)·n^3−r·E_2n−r−3(f^(r)) holds uniformly for all x∈[−1,1], where .     

On weighted polynomial approximation with monotone weights

We construct an even weight monotone on the right half line such that the logarithmic integral of the largest -convex minorant of converges and the polynomials are dense in .     

On polynomial approximation of solutions of differential-operator equations

Polynomial approximations are constructed for the solutions of differential equations of the first and second order in a Banach space for which the Cauchy problem is stated correctly.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 440–442, March, 1993.     

亚纯函数及其微分多项式的唯一性

研究了具有两个或三个公共小函数的亚纯函数及其微分多项式的唯一性问题,推广了邱氵金弟等人的有关结果.例子表明了定理所给的条件是必要的.