Exact Inequalities for the Norms of Factors of Polynomials

1994 ◽  
Vol 46 (4) ◽  
pp. 687-698 ◽  
Author(s):  
Peter B. Borwein
Keyword(s):  
Chebyshev Polynomial ◽  
Algebraic Polynomial ◽  
Supremum Norm

AbstractThis paper addresses a number of questions concerning the size of factors of polynomials. Let p be a monic algebraic polynomial of degree n and suppose q1q2 … qi is a monic factor of p of degree m. Then we can, in many cases, exactly determine Here ‖ . ‖ is the supremum norm either on [—1, 1] or on {|z| ≤ 1}. We do this by showing that, in the interval case, for each m and n, the n-th Chebyshev polynomial is extremal. This extends work of Gel'fond, Mahler, Granville, Boyd and others. A number of variants of this problem are also considered.

scholarly journals Concentration of the error between a function and its polynomial of best uniform approximation

1998 ◽  
Vol 41 (3) ◽  
pp. 447-463 ◽  
Author(s):  
Maurice Hasson
Keyword(s):  
Uniform Approximation ◽  
Algebraic Polynomial ◽  
General Class ◽  
Supremum Norm ◽  
Best Uniform Approximation ◽  
Class A ◽  
Image Position ◽  
Class Of Functions

Let f be a continuous real valued function defined on [−1, 1] and let En(f) denote the degree of best uniform approximation to f by algebraic polynomial of degree at most n. The supremum norm on [a, b] is denoted by ∥.∥[a, b] and the polynomial of degree n of best uniform approximation is denoted by Pn. We find a class of functions f such that there exists a fixed a ∈(−1, 1) with the following propertyfor some positive constants C and N independent of n. Moreover the sequence is optimal in the sense that if is replaced by then the above inequality need not hold no matter how small C > 0 is chosen.We also find another, more general class a functions f for whichinfinitely often.

Markov's and Bernstein's Inequalities on Disjoint Intervals

1981 ◽  
Vol 33 (1) ◽  
pp. 201-209 ◽  
Author(s):  
Peter B. Borwein
Keyword(s):  
Algebraic Polynomial ◽  
Supremum Norm ◽  
Algebraic Polynomials ◽  
Image Position

In 1889, A. A. Markov proved the following inequality:INEQUALITY 1. (Markov [4]). If pn is any algebraic polynomial of degree at most n thenwhere ‖ ‖A denotes the supremum norm on A.In 1912, S. N. Bernstein establishedINEQUALITY 2. (Bernstein [2]). If pn is any algebraic polynomial of degree at most n thenfor x ∈ (a, b).In this paper we extend these inequalities to sets of the form [a, b] ∪ [c, d]. Let Πn denote the set of algebraic polynomials with real coefficients of degree at most n.THEOREM 1. Let a < b ≦ c < d and let pn ∈ Πn. Thenfor x ∈ (a, b).

scholarly journals Solvability of an infinite system of integral equations on the real half-axis

2020 ◽  
Vol 10 (1) ◽  
pp. 202-216
Author(s):  
Józef Banaś ◽  
Weronika Woś
Keyword(s):  
Integral Equations ◽  
Measure Of Noncompactness ◽  
Infinite System ◽  
Main Tool ◽  
Supremum Norm ◽  
Nonlinear Integral Equations ◽  
Measures Of Noncompactness ◽  
System Of Integral Equations ◽  
The Real ◽  
Half Axis

Abstract The aim of the paper is to investigate the solvability of an infinite system of nonlinear integral equations on the real half-axis. The considerations will be located in the space of function sequences which are bounded at every point of the half-axis. The main tool used in the investigations is the technique associated with measures of noncompactness in the space of functions defined, continuous and bounded on the real half-axis with values in the space l∞ consisting of real bounded sequences endowed with the standard supremum norm. The essential role in our considerations is played by the fact that we will use a measure of noncompactness constructed on the basis of a measure of noncompactness in the mentioned sequence space l∞. An example illustrating our result will be included.

Algebraic Polynomial Based Topological Study of Graphite Carbon Nitride (g-) Molecular Structure.

2021 ◽  
pp. 1-22
Author(s):  
Abdul Rauf ◽  
Muhammad Ishtiaq ◽  
Mehwish Hussain Muhammad ◽  
Muhammad Kamran Siddiqui ◽  
Qammar Rubbab
Keyword(s):  
Molecular Structure ◽  
Algebraic Polynomial ◽  
Carbon Nitride

scholarly journals Approximation by Boolean sums of linear operators: Telyakovskiǐ-type estimates

1990 ◽  
Vol 42 (2) ◽  
pp. 253-266 ◽  
Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska
Keyword(s):  
Upper Bound ◽  
Algebraic Polynomial ◽  
Present Note ◽  
Type Inequality ◽  
Linear Operators ◽  
Polynomial Operators ◽  
Boolean Sums ◽  
General Conditions

In the present note we study the question: “Under which general conditions do certain Boolean sums of linear operators satisfy Telyakovskiǐ-type estimates?” It is shown, in particular, that any sequence of linear algebraic polynomial operators satisfying a Timan-type inequality can be modified appropriately so as to obtain the corresponding upper bound of the Telyakovskiǐ-type. Several examples are included.

Model reduction by Chebyshev polynomial techniques

1979 ◽  
Vol 24 (5) ◽  
pp. 741-747 ◽  
Author(s):  
Y. Bistritz ◽  
G. Langholz
Keyword(s):  
Model Reduction ◽  
Chebyshev Polynomial

Three-Dimensional Vibration Analysis of Twisted Cylinder With Sectorial Cross Section

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
D. Zhou ◽  
S. H. Lo
Keyword(s):  
Cross Section ◽  
Chebyshev Polynomial ◽  
Numerical Solutions ◽  
Twist Angle ◽  
Ritz Method ◽  
Three Dimensional ◽  
Cylindrical Domain ◽  
Linear Elasticity Theory ◽  
Boundary Functions ◽  
Polynomial Series

The three-dimensional (3D) free vibration of twisted cylinders with sectorial cross section or a radial crack through the height of the cylinder is studied by means of the Chebyshev–Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. A simple coordinate transformation is applied to map the twisted cylindrical domain into a normal cylindrical domain. The product of a triplicate Chebyshev polynomial series along with properly defined boundary functions is selected as the admissible functions. An eigenvalue matrix equation can be conveniently derived through a minimization process by the Rayleigh–Ritz method. The boundary functions are devised in such a way that the geometric boundary conditions of the cylinder are automatically satisfied. The excellent property of Chebyshev polynomial series ensures robustness and rapid convergence of the numerical computations. The present study provides a full vibration spectrum for thick twisted cylinders with sectorial cross section, which could not be determined by 1D or 2D models. Highly accurate results presented for the first time are systematically produced, which can serve as a benchmark to calibrate other numerical solutions for twisted cylinders with sectorial cross section. The effects of height-to-radius ratio and twist angle on frequency parameters of cylinders with different subtended angles in the sectorial cross section are discussed in detail.

Monotone methods and attractivity results for Volterra integro-partial differential equations

1981 ◽  
Vol 89 (1-2) ◽  
pp. 135-142 ◽  
Author(s):  
Andrea Schiaffino ◽  
Alberto Tesei
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Past History ◽  
Main Tool ◽  
Parabolic Type ◽  
Supremum Norm ◽  
Partial Differential ◽  
Diffusion Effects ◽  
Monotone Methods ◽  
Globally Attractive

SynopsisA Volterra integro-partial differential equation of parabolic type, which describes the time evolution of a population in a bounded habitat, subject both to past history and space diffusion effects, is investigated; general homogeneous boundary conditions are admissible. Under suitable conditions, the unique nontrivial nonnegative equilibrium is shown to be globally attractive in the supremum norm. Monotone methods are the main tool of the proof.

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