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Bernstein's Inequality (mathematical Analysis)
Bernstein's theorem is an inequality relating the maximum modulus of a complex polynomial function on the unit disk with the maximum modulus of its derivative on the unit disk. It was proven by Sergei Bernstein while he was working on approximation theory. Statement Let \max_ , f(z), denote the maximum modulus of an arbitrary function f(z) on , z, =1, and let f'(z) denote its derivative. Then for every polynomial P(z) of degree n we have : \max_ , P'(z), \le n \max_ , P(z), . The inequality is best possible with equality holding if and only if : P(z) = \alpha z^n,\ , \alpha, = \max_ , P(z), . Proof Let P(z) be a polynomial of degree n, and let Q(z) be another polynomial of the same degree with no zeros in , z, \ge 1. We show first that if , P(z), < , Q(z), on , then on . By Rouché's theorem, with has all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergei Bernstein
Sergei Natanovich Bernstein (russian: Серге́й Ната́нович Бернште́йн, sometimes Romanized as ; 5 March 1880 – 26 October 1968) was a Ukrainian and Russian mathematician of Jewish origin known for contributions to partial differential equations, differential geometry, probability theory, and approximation theory. Work Partial differential equations In his doctoral dissertation, submitted in 1904 to Sorbonne, Bernstein solved Hilbert's nineteenth problem on the analytic solution of elliptic differential equations. His later work was devoted to Dirichlet's boundary problem for non-linear equations of elliptic type, where, in particular, he introduced a priori estimates. Probability theory In 1917, Bernstein suggested the first axiomatic foundation of probability theory, based on the underlying algebraic structure. It was later superseded by the measure-theoretic approach of Kolmogorov. In the 1920s, he introduced a method for proving limit theorems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Theory
In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characterizing the approximation error, errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' will depend on the application. A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials. One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or Rational function, rational (ratio of polynomials) approximations. The objective is to make t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rouché's Theorem
Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions and holomorphic inside some region K with closed contour \partial K, if on \partial K, then and have the same number of zeros inside K, where each zero is counted as many times as its multiplicity. This theorem assumes that the contour \partial K is simple, that is, without self-intersections. Rouché's theorem is an easy consequence of a stronger symmetric Rouché's theorem described below. Usage The theorem is usually used to simplify the problem of locating zeros, as follows. Given an analytic function, we write it as the sum of two parts, one of which is simpler and grows faster than (thus dominates) the other part. We can then locate the zeros by looking at only the dominating part. For example, the polynomial z^5 + 3z^3 + 7 has exactly 5 zeros in the disk , z, b > 0). By the quadratic formula it has two zeros at -a \pm \sqrt. Rouché's theorem can be used to obtain mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Lucas Theorem
In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial ''P'' and the roots of its derivative ''P′''. The set of roots of a real or complex polynomial is a set of points in the complex plane. The theorem states that the roots of ''P′'' all lie within the convex hull of the roots of ''P'', that is the smallest convex polygon containing the roots of ''P''. When ''P'' has a single root then this convex hull is a single point and when the roots lie on a line then the convex hull is a segment of this line. The Gauss–Lucas theorem, named after Carl Friedrich Gauss and Félix Lucas, is similar in spirit to Rolle's theorem. Formal statement If ''P'' is a (nonconstant) polynomial with complex coefficients, all zeros of ''P′'' belong to the convex hull of the set of zeros of ''P''. Special cases It is easy to see that if ''P''(''x'') = ''ax''2 + ''bx'' + ''c'' is a second degree polynomial, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates—the magnitude or ''modulus'' of the product is the product of the two absolute values, or moduli, and the angle or ''argument'' of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes known as the Argand plane or Gauss plane. Notational conventions Complex numbers In complex analysis, the complex numbers are customarily represented by the symbol ''z'', which can be separated into its real (''x'') and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Lax
Peter David Lax (born Lax Péter Dávid; 1 May 1926) is a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics. Lax has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields. In a 1958 paper Lax stated a conjecture about matrix representations for third order hyperbolic polynomials which remained unproven for over four decades. Interest in the "Lax conjecture" grew as mathematicians working in several different areas recognized the importance of its implications in their field, until it was finally proven to be true in 2003. Life and education Lax was born in Budapest, Hungary to a Jewish family. Lax began displaying an interest in mathematics at age twelve, and soon his parents hired Rózsa Péter as a tutor for him. His parents Klara Kornfield and Henry Lax were both physici ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Brothers' Inequality
In mathematics, the Markov brothers' inequality is an inequality proved in the 1890s by brothers Andrey Markov and Vladimir Markov, two Russian mathematicians. This inequality bounds the maximum of the derivatives of a polynomial on an interval in terms of the maximum of the polynomial. For ''k'' = 1 it was proved by Andrey Markov, and for ''k'' = 2,3,... by his brother Vladimir Markov. Appeared in German with a foreword by Sergei Bernstein as The statement Let ''P'' be a polynomial of degree ≤ ''n''. Then for all nonnegative integers k : \max_ , P^(x), \leq \frac \max_ , P(x), . Equality is attained for Chebyshev polynomials of the first kind. Related inequalities * Bernstein's inequality (mathematical analysis) * Remez inequality Applications Markov's inequality is used to obtain lower bounds in computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Remez Inequality
In mathematics, the Remez inequality, discovered by the Soviet mathematician Evgeny Yakovlevich Remez , gives a bound on the sup norms of certain polynomials, the bound being attained by the Chebyshev polynomials. The inequality Let ''σ'' be an arbitrary fixed positive number. Define the class of polynomials π''n''(''σ'') to be those polynomials ''p'' of the ''n''th degree for which :, p(x), \le 1 on some set of measure ≥ 2 contained in the closed interval 1, 1+''σ'' Then the Remez inequality states that :\sup_ \left\, p\right\, _\infty = \left\, T_n\right\, _\infty where ''T''''n''(''x'') is the Chebyshev polynomial of degree ''n'', and the supremum norm is taken over the interval 1, 1+''σ'' Observe that ''T''''n'' is increasing on , +\infty/math>, hence : \, T_n\, _\infty = T_n(1+\sigma). The R.i., combined with an estimate on Chebyshev polynomials, implies the following corollary: If ''J'' ⊂ R is a finite interval, and ''E'' ⊂&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
