The weierstrass approximation theorem, of which one well known generalization is the stoneweierstrass theorem. Given o 0, there exists a polynomial po such that fx. It will be seen that the weierstrass approximation theorem is in fact a special case of the more general stoneweierstrass theorem, proved by stone in 1937, who realized that very few of the properties of the polynomials were essential to the theorem. Stone weierstrass theorem 16 acknowledgments 19 references 20 1. As an application, we now prove a theorem of weierstrass con cerning the approximation of continuous functions by polynomials. Weierstrass approximation theorem if the function f is continuous on a closed interval a,b, then f may be approximated uni. An elementary proof of the stone weierstrass theorem is given. An elementary proof of the stone weierstrass theorem bruno brosowski and frank deutsch1 abstract. Pdf a survey on the weierstrass approximation theorem. A second proof of weierstrass theorem here we consider the interval 0,1.
C0,1 can be arbitrarily approximated by polynomials in supremum norm. Weierstrass products given a sequence of complex numbers z j with no accumulation point in c, we will construct an entire function with zeros exactly the z j. The bolzanoweierstrass theorem, which ensures compactness of closed and bounded sets in r n. When weierstrass proved the approximation theorem, he was 70 years old. The wierstrass approximation theorem proof finally, choose n so that n m 2. It is interesting to learn that in the beginning, fejer was considered weak in mathematics at school and was required to have special tuition. Polynomials are far easier to work with than continuous functions and allow mathematicians and. We shall show that any function, continuous on the closed interval 0. Proof first show that by linear transformation the theorem for a,b will follow from the theorem for 0,1. Cauchy criterion, bolzanoweierstrass theorem we have seen one criterion, called monotone criterion, for proving that a sequence converges without knowing its limit. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially. Px for example, where px is a polynomial of degree d. First, bernstiens proof of the weierstrass approximation theorem, which is taken from 4, is examined.
An elementary proof of the stoneweierstrass theorem article pdf available in proceedings of the american mathematical society 811. Theorem 7 weierstrass division theorem let be regular in of degree and let. Psup weierstrass proof of the weierstrass approximation theorem anton r. Weierstrass approximation theorem for every continuous. I propose to follow george lorentzs approach at the start of his book bernstein polynomials 5, where the basic plan of the proof and the polynomials are retained, but. However, he greatly simpli ed his proof in 1948 into the one that is commonly used today.
We will prove the lemma by showing that if p, is an element of ue such that. We will prove the lemma by showing that if p, is an element of ue such that dp\ 1 and if there exists a borel. The weierstrass extreme value theorem, which states that a continuous function on a closed and bounded set obtains its extreme values. His proof heavily relied on intricate analysis concepts. The stoneweierstrass theorem 5 it is also useful to note that if for every basis element bcontaining fx, there is an open set wcontaining xin xsuch that fw. Weierstrass theorem an overview sciencedirect topics. The paper stating and proving what we in approximation theory call the weierstrass theorems, i. Full text of a higher order weierstrass approximation. Proof of weierstrass approximation theorem using band. Weierstrass approximation theorem if the function f is continuous on a closed interval a,b, then f may be approximated uniformly by polynomials. Lorentz chelsea publishing company, 1986, isbn 9780821875582.
Introduction a fundamental tool used in the analysis of the real line is the wellknown bolzanoweierstrass theorem1. Weierstrasss theorem if f is continuous on a, b then, given any. The proof of the weierstrass theorem by sergi bernstein is constructive. It has been accepted for inclusion in theses and dissertations by an authorized. In this note we give an elementary proof of the stone weierstrass theorem. Approximations of continuous functions by smooth functions. The writer, for instance, has noted a number of such appli. Pxj proof of the weierstrass approximation theorem and wikipedia gives a concise overview of the matter. The stoneweierstrass theorem generalizes the weierstrass theorem and was. Schep at age 70 weierstrass published the proof of his wellknown approximation theorem. In this paper we mainly consider two of weierstrass results. Introduction one useful theorem in analysis is the stone weierstrass theorem, which states that any continuous complex function over a compact interval can be approximated to an arbitrary degree of accuracy with a sequence of polynomials.
His proof was greatly simpli ed by sergei bernstein in 1912, using the nowfamous bernstein polynomials b knx n k. A simple proof of the weierstrass approximation theorem j. Weierstrass approximation theorem for continuous functions. The proof depends only on the definitions of compactness each open cover has a finite. In mathematical analysis, the weierstrass approximation theorem states that every continuous function defined on a closed interval a, b can be uniformly approximated as closely as desired by a polynomial function. Inequality in its most general form and will apply it in bernsteins proof of the weierstrass approximation theorem. The weierstrass preparation theorem climbing mount bourbaki.
Weierstrasss theorem with regard to polynomial approximation. A much more general version of the weierstrass approximation theorem is due to marshall stone, the stoneweierstrass theorem. Then m n 2 weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point p. In this paper, we will prove a famous theorem known as the weierstrass approximation theorem. The bernstein polynomials, which play a cen tral role in. Weierstrass and hadamard products january 12, 2015 1. Informally, the weierstrass approximation theorem wat asserts that any continuous function on a. Levasseur the american mathematical monthly, vol 91, no 4, april 1984, pp.
In 1885, weierstrass being 70 years of age proved a rather astounding theorem that on the interval 0. Let us try to prove this theorem with some standard results and objects from calculus. Then m n 2 weierstrasss theorem leaving no stoneunturned b sury b sury was associated with resonance during 1999 2005. The weierstrass approximation theorem scholar commons. This statement is the weierstrass approximation theorem.
In terms of the weierstrass approximation theorem, it is possible to approximate a function as a polynomial by combining more and more bfs. The weierstrass approximation theorem larita barnwell hipp university of south carolina follow this and additional works at. Pdf the celebrated and famous weierstrass approximation theorem characterizes the set of continuous functions on a compact interval via uniform. Proof of weierstrass approximation theorem using bandlimited. The approximation theorem, for him, was an extension of this result to continuous functions. This is the conclusion of the famous weierstrass approximation theorem, named for karl weierstrass. Keywords weierstrass approximation, fejers theorem, fejer kernel, polynomial approximation, hausdorff theorem, moments, bernstein polynomial. The nonexistence of a continuous linear projection 9.
The same is not quite true of his proof of theorem 2. The stone weierstrass theorem 823 is related to the hahnbanach theorem and is discussed in the same loomis reference. The first step is to show that polynomial approximations exist to arbitrary accuracy. Part of themathematics commons this open access thesis is brought to you by scholar commons. The weierstrass theorem justifies our preoccupation with polynomial curves because it asserts that to within any arbitrary tolerance all continuous curves are essentially polynomials. Marshall stone, with the stoneweierstrass theorem, generalized the result to. The celebrated and famous weierstrass approximation theorem characterizes the set of continuous functions on a compact interval via uniform approximation by algebraic polynomials.
Two papers of runge published almost at the same time, gave another proof of the theorem, but unfortunately, the theorem was not titled weierstrassrunge. Throughout this article, f will be a continuous realvalued function on an interval a, b, and p will be a real polynomial that approximates f on a, b. It states that such a function is, up to multiplication by a function not zero at p, a polynomial in one fixed variable z, which is monic, and whose coefficients of lower degree terms are analytic functions in the remaining variables and zero at p. You can see what the problem is, at least roughlylagranges polynomial doesnt behave locallyi. Approximation by polynomials and trigonometric polynomials 8. The lebesgue integral, chebyshevs inequality, and the. In this article, he celebrates a wonderful theorem which he introduces as. Weierstrasss classical theorem the starting point of all our discussions is. Bernstein proves the weierstrass approximation theorem. However, this approach has undesirable limitations in practical operations. The weierstrass approximation theorem there is a lovely proof of the weierstrass approximation theorem by s. Then there is a unique expression, where and has degree. Weierstrass and approximation theory department of mathematics.
Applications of the weierstrass approximation theorem abound in mathematics to gaussian quadrature for instance. C0,1 can be arbitrarily approximated by polynomials in. He extends ffrom a,b so that it is continuous and bounded on all of. The stoneweierstrass theorem generalizes the weierstrass approximation theorem in two directions. Then there is a sequence of polynomials pnx that converges uniformly to fx on a. A proof of the weierstrass approximation theorem is obtained using the fourier transform and bandlimited functions.
Weierstrass s classical theorem the starting point of all our discussions is. Twenty years later, another proof was given by the 19year old fejer this is what is charming about mathematics. Weierstrass approximation theorem 3 proof of theorem 1. Speci cally, brand 3, gaughan 4, and watson 9 prove the result via the bisection method outlined above. Weierstrass theorem before weierstrass 3 is not valid, e. In this note we will present a selfcontained version, which is essentially his proof. The approximating polynomials in the weierstrass theorem need not be interpolating. Pxj weierstrass approximation theorem wat asserts that any continuous function on a. Once the theorem is proved for this case, the general theorem will follow by a change of variables. The proof of this theorem can be found in most elementary texts on real analysis. The rst, weierstrass 1872, is weierstrass example of a continuous nowhere di erentiable function.
We present a short proof of the bolzanoweierstrass theorem on the real line which avoids monotonic subsequences, cantors intersection theorem, and the heineborel theorem. We will prove it in class and the proof can be found in rudins book. At the same time many interesting applications to classical problems of analysis have been observed by those working with the generalized approximation theorem. Pdf an elementary proof of the stoneweierstrass theorem. By weierstrass approximation theorem, every continuous realvalued function on closed interval can be uniformly approximated by a sequence of polynomials. Weierstrass approximation theorem the basic question well look at today is how to approximate a complicated function fx with a simpler function px fx.
A simple proof of the weierstrass approximation theorem. The weierstrass approximation theorem clemson cecas. The stoneweierstrass theorem is an approximation theorem for continuous functions on closed intervals. A survey on the weierstrass approximation theorem emis. In particular, there exists r0 such that fx0 for jxjr. We shall prove this result using the weierstrass division theorem, for which we shall give an algebraic proof. At age 70 weierstrass published the proof of his wellknown approximation. His proof heavily relied on intricate analysis concepts and involved building up a sequence of polynomials from a convolution with a gaussian heat kernel. It is one of the most important results in analysis. The stoneweierstrass theorem 823 is related to the hahnbanach theorem and is discussed in the same loomis reference.
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