Derivative Of Polynomial Research Articles

Optimal Approximation of Biquartic Polynomials by Bicubic Splines

Recently an unexpected approximation property between polynomials of degree three and four was revealed within the framework of two-part approximation models in 2-norm, Chebyshev norm and Holladay seminorm. Namely, it was proved that if a two-component cubic Hermite spline’s first derivative at the shared knot is computed from the first derivative of a quartic polynomial, then the spline is a clamped spline of classC2and also the best approximant to the polynomial.Although it was known that a 2 × 2 component uniform bicubic Hermite spline is a clamped spline of classC2if the derivatives at the shared knots are given by the first derivatives of a biquartic polynomial, the optimality of such approximation remained an open question.The goal of this paper is to resolve this problem. Unlike the spline curves, in the case of spline surfaces it is insufficient to suppose that the grid should be uniform and the spline derivatives computed from a biquartic polynomial. We show that the biquartic polynomial coefficients have to satisfy some additional constraints to achieve optimal approximation by bicubic splines.

A Fixed Point Approach to the Stability of a Mean Value Type Functional Equation

We prove the generalized Hyers–Ulam stability of a mean value type functional equation f ( x ) − g ( y ) = ( x − y ) h ( x + y ) by applying a method originated from fixed point theory.

Numerical Simulation and Convergence Analysis of Fractional Optimization Problems With Right-Sided Caputo Fractional Derivative

This paper presents an efficient approximation schemes for the numerical solution of a fractional variational problem (FVP) and fractional optimal control problem (FOCP). As basis function for the trial solution, we employ the shifted Jacobi orthonormal polynomial. We state and derive a new operational matrix of right-sided Caputo fractional derivative of such polynomial. The new methodology of the present schemes is based on the derived operational matrix with the help of the Gauss–Lobatto quadrature formula and the Lagrange multiplier technique. Accordingly, the aforementioned problems are reduced into systems of algebraic equations. The error bound for the operational matrix of right-sided Caputo derivative is analyzed. In addition, the convergence of the proposed approaches is also included. The results obtained through numerical procedures and comparing our method with other methods demonstrate the high accuracy and powerful of the present approach.

Nikol’skii–Stechkin-Type Inequalities for the Increments of Trigonometric Polynomials in Metric Spaces

In the spaces LΨ [0, 2𝜋] with the metric $$ \rho \left(f,0\right)\varPsi =\frac{1}{2\pi }{\int}_0^{2\uppi}\varPsi \left(|f(x)|\right) dx $$ , where is a function of Ψ the modulus-of-continuity type, we investigate an analog of the Nikol’skii–Stechkin inequalities for the increments and derivatives of trigonometric polynomials.

Exact Constant in Dzyadyk’s Inequality for the Derivative of an Algebraic Polynomial

For natural k and n ≥ 2k, we determine the exact constant c(n, k) in Dzyadyk’s inequality $$ {\left\Vert {P}_n^{\prime }{\varphi}_n^{1-k}\right\Vert}_{C\left[-1,1\right]}\le c\left(n,k\right)n{\left\Vert {P}_n{\varphi}_n^{-k}\right\Vert}_{C\left[-1,1\right]} $$ for the derivative $$ {P}_n^{\prime } $$ of an algebraic polynomial P n of degree ≤ n, where $$ {\varphi}_n(x):= \sqrt{n^{-2}+1-{x}^2}. $$ Namely, $$ c\left(n,k\right)={\left(1+k\frac{\sqrt{1+{n}^2}-1}{n}\right)}^2-k. $$

Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions

Abstract A recursion formula for derivatives of Chebyshev polynomials is replaced by an explicit formula. Similar formulae are derived for scaled Fibonacci numbers.

Critical Values and Moduli of the Derivative of a Complex Polynomial at its Zeros

Critical Values and Moduli of the Derivative of a Complex Polynomial at its Zeros

Investigation of magneto-hemodynamic flow in a semi-porous channel using orthonormal Bernstein polynomials

In this paper, the problem of the magneto-hemodynamic laminar viscous flow of a conducting physiological fluid in a semi-porous channel under a transverse magnetic field is investigated numerically. Using a Berman’s similarity transformation, the two-dimensional momentum conservation partial differential equations can be written as a system of nonlinear ordinary differential equations incorporating Lorentizian magneto-hydrodynamic body force terms. A new computational method based on the operational matrix of derivative of orthonormal Bernstein polynomials for solving the resulting differential systems is introduced. Moreover, by using the residual correction process, two types of error estimates are provided and reported to show the strength of the proposed method. Graphical and tabular results are presented to investigate the influence of the Hartmann number (Ha) and the transpiration Reynolds number (Re on velocity profiles in the channel. The results are compared with those obtained by previous works to confirm the accuracy and efficiency of the proposed scheme.

Turán type converse Markov inequalities in [formula omitted] on a generalized Erőd class of convex domains

P. Turán was the first to derive lower estimations on the uniform norm of the derivatives of polynomials p of uniform norm 1 on the disk D:={z∈C:|z|≤1} and the interval I:=[−1,1], under the normalization condition that the zeros of the polynomial p in question all lie in D or I, resp. Namely, in 1939 he proved that with n:=degp tending to infinity, the precise growth order of the minimal possible derivative norm is n for D and n for I.Already the same year J. Erőd considered the problem on other domains. In his most general formulation, he extended Turán’s order n result on D to a certain general class of piecewise smooth convex domains. Finally, a decade ago the growth order of the minimal possible norm of the derivative was proved to be n for all compact convex domains.Turán himself gave comments about the above oscillation question in Lq norm on D. Nevertheless, till recently results were known only for D, I and so-called R-circular domains. Continuing our recent work, also here we investigate the Turán-Erőd problem on general classes of domains.

Differentiability of Polynomials over Reals

Summary In this article, we formalize in the Mizar system [3] the notion of the derivative of polynomials over the field of real numbers [4]. To define it, we use the derivative of functions between reals and reals [9].

Orthogonal fast spherical Bessel transform on uniform grid

We propose an algorithm for the orthogonal fast discrete spherical Bessel transform on a uniform grid. Our approach is based upon the spherical Bessel transform factorization into the two subsequent orthogonal transforms, namely the fast Fourier transform and the orthogonal transform founded on the derivatives of the discrete Legendre orthogonal polynomials. The method utility is illustrated by its implementation for the problem of a two-atomic molecule in a time-dependent external field simulating the one utilized in the attosecond streaking technique.

A new operational matrix of Caputo fractional derivatives of Fermat polynomials: an application for solving the Bagley-Torvik equation

Herein, an innovative operational matrix of fractional-order derivatives (sensu Caputo) of Fermat polynomials is presented. This matrix is used for solving the fractional Bagley-Torvik equation with the aid of tau spectral method. The basic approach of this algorithm depends on converting the fractional differential equation with its initial (boundary) conditions into a system of algebraic equations in the unknown expansion coefficients. The convergence and error analysis of the suggested expansion are carefully discussed in detail based on introducing some new inequalities, including the modified Bessel function of the first kind. The developed algorithm is tested via exhibiting some numerical examples with comparisons. The obtained numerical results ensure that the proposed approximate solutions are accurate and comparable to the analytical ones.

Asymptotic Markov inequality on Jordan arcs

Markov's inequality for the derivative of algebraic polynomials is considered on -smooth Jordan arcs. The asymptotically best estimate is given for the th derivative for all . The best constant is related to the behaviour around the endpoints of the arc of the normal derivative of the Green's function of the complementary domain. The result is deduced from the asymptotically sharp Bernstein inequality for the th derivative at inner points of a Jordan arc, which is derived from a recent result of Kalmykov and Nagy on the Bernstein inequality on analytic arcs. In the course of the proof we shall also need to reduce the analyticity condition in this last result to -smoothness. Bibliography: 21 titles.

Solving fractional optimal control problems within a Chebyshev–Legendre operational technique

ABSTRACTIn this manuscript, we report a new operational technique for approximating the numerical solution of fractional optimal control (FOC) problems. The operational matrix of the Caputo fractional derivative of the orthonormal Chebyshev polynomial and the Legendre–Gauss quadrature formula are used, and then the Lagrange multiplier scheme is employed for reducing such problems into those consisting of systems of easily solvable algebraic equations. We compare the approximate solutions achieved using our approach with the exact solutions and with those presented in other techniques and we show the accuracy and applicability of the new numerical approach, through two numerical examples.

Some applications of Dubinin's lemma to rational functions with prescribed poles

By developing a new technique some results for the rational functions with prescribed poles and restricted zeros have been obtained. The estimated results strengthen many well known inequalities concerning the derivative and polar derivative of polynomials.

Local recovery of lithospheric stress tensor from GOCE gravitational tensor

The sublithospheric stress due to mantle convection can be computed from gravity data and propagated through the lithosphere by solving the boundary-value problem of elasticity for the Earth's lithosphere. In this case, a full tensor of stress can be computed at any point inside this elastic layer. Here, we present mathematical foundations for recovering such a tensor from gravitational tensor measured at satellite altitudes. The mathematical relations will be much simpler in this way than the case of using gravity data as no derivative of spherical harmonics (SHs) or Legendre polynomials is involved in the expressions. Here, new relations between the SH coefficients of the stress and gravitational tensor elements are presented. Thereafter, integral equations are established from them to recover the elements of stress tensor from those of the gravitational tensor. The integrals have no closed-form kernels, but they are easy to invert and their spatial truncation errors are reducible. The integral equations are used to invert the real data of the gravity field and steady-state ocean circulation explorer mission (GOCE), in 2009 November, over the South American plate and its surroundings to recover the stress tensor at a depth of 35 km. The recovered stress fields are in good agreement with the tectonic and geological features of the area.

On the Link between Finite Difference and Derivative of Polynomials

On the Link between Finite Difference and Derivative of Polynomials

Limiting empirical distribution of zeros and critical points of random polynomials agree in general

In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences $\{a_n\}_{n\geq1}$ and $\{b_n\}_{n\geq1}$ of complex numbers whose limiting empirical measures are same. By choosing $\xi_n = a_n$ or $b_n$ with equal probability, define the sequence of polynomials by $P_n(z)=(z-\xi_1)\dots(z-\xi_n)$. We show that the limiting measure of zeros and critical points agree for this sequence of random polynomials under some assumption. We also prove a similar result for triangular array of numbers. A similar result for zeros of generalized derivative (can be thought as random rational function) is also proved. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables.

On the Derivative of a Polynomial with Prescribed Zeros

On the Derivative of a Polynomial with Prescribed Zeros

Some Zygmund type inequalities for the sth derivative of polynomials

For a polynomial P(z) of degree n having no zeros in |z| < 1, it was recently proved in [9] that $$\left| {{z^s}{P^{\left( s \right)}}\left( z \right) + \beta \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{{{2^s}}}P\left( z \right)} \right| \leqslant \frac{{n\left( {n - 1} \right)...\left( {n - s + 1} \right)}}{2}\left( {\left| {1 + \frac{\beta }{{{2^s}}}} \right| + \left| {\frac{\beta }{{{2^s}}}} \right|} \right)\mathop {\max }\limits_{\left| z \right| = 1} \left| {P\left( z \right)} \right|$$ for every β ∈ C with |β| ≤ 1, 1 ≤ s ≤ n and |z| = 1. In this paper, we obtain the Lp mean extension of the above and other related results for the sth derivative of polynomials.