Applying the Stone-Weierstrass Theorem to the IT2FNN-2 Architecture. C This work deals with the many variations of the Stoneileierstrass Theorem for vector-valued functions and some of its applications. The book is largely self-contained. ✔ No Rental Trucks This is one of the foundational results on the topic of "why neural networks work"! : 46E25, 18B25, 54B40 0. A subalgebra A of C0(X, R) is said to vanish nowhere if not all of the elements of A simultaneously vanish at a point; that is, for every x in X, there is some  f  in A such that  f (x) ≠ 0. His collected works, comprising treatises, letters and papers written in German, Latin and French, were published in eight volumes between 1881 and 1891. It is possible to generalise the Weierstrass approximation theorem so that continuous functions on (0,1) can be approximated by polynomials uniformy on closed subintervals of (0,1). Using the Stone theorem. , Found inside – Page 595Hence is a compact convex set in Wd. □ 14.10 The Stone-Weierstrass approximation theorem The aim of this section is to show how the Krein-Milman theorem 2.22 opens a way to a proof of the algebra version of the Stone-Weierstrass ... , In this text, the whole structure of analysis is built up from the foundations. The Stone-Weierstrass theorem is an approximation theorem for continuous functions on closed intervals. {\displaystyle S} Section 11.7 The Stone-Weierstrass theorem. If a quaternion q is written in the form q = a + ib;+ jc + kd then the scalar part a is the real number (q − iqi − jqj − kqk)/4. Let A be a subset of C(M) s.t. Let Xbe any compact Hausdor space. Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. X WikiMatrix. when f ∈ C ( [ 0, 1] n), n > 1. ‖ f − p ‖ < ϵ / M. We also know that ∫ 1 0 p(x)f (x) dx = 0. Chebyshev polynomials. Thus, the Weierstrass theorem is equivalent to say that is dense in . , again with the topology of uniform convergence. The historical publication of Weierstrass (in German language) is freely available from the digital online archive of the Berlin Brandenburgische Akademie der Wissenschaften: Important historical works of Stone include: Stone–Weierstrass theorem, complex version, Stone–Weierstrass theorem, quaternion version, Stone–Weierstrass theorem, C*-algebra version, continuous functions on a compact Hausdorff space, "Density methods and results in approximation theory", "A survey on the Weierstrass approximation theorem", "On Hilbert extensions of Weierstrass' theorem with weights", "The Stone–Weierstrass theorem for quaternions", "A generalization of the Stone–Weierstrass theorem", Berlin Brandenburgische Akademie der Wissenschaften, https://en.wikipedia.org/w/index.php?title=Stone–Weierstrass_theorem&oldid=1050454082, Articles with unsourced statements from July 2018, Pages that use a deprecated format of the math tags, Creative Commons Attribution-ShareAlike License. If 91 is a subalgebra of. Papers and articles about periodic functions approximation. Let A be a subset of C(M) s.t. 309 - 311 View Record in Scopus Google Scholar ✔  Merced County The space X may be viewed as the space of pure states on Graduate students in mathematics, who want to travel light, will find this book invaluable; impatient young researchers in other fields will enjoy it as an instant reference to the highlights of modern analysis. How can polynomials approximate continuous functions? We finally show that the two classical approximation theorems at the beginning of this chapter are consequences of the Stone-Weierstrass theorem. This classic text is written for graduate courses in functional analysis. This text is used in modern investigations in analysis and applied mathematics. ( Stone-Weierstrass theorem (complex version) Theorem - Let X be a compact space and C ⁢ ( X ) the algebra of continuous functions X ℂ endowed with the sup norm ∥ ⋅ ∥ ∞ . ) ⊂ Application of the Stone Weierstrass Theorem. are less useful, and there is no Stone- Weierstrass-type theorem applicable to the reduced problem. We are now in a position to state and prove the Stone-Weierstrass the-orem. Contact US : Preparation theorem. C is continuous, then for each ² > 0, there is a polynomial P(x) such that jf(x)¡P(x)j < ² 8 x 2 [a;b]: Various structural generalizations of the Weierstrass approximation theorem have focused on generalized mappings, as in Stone =-= [22]-=-, and on alternate topologies, as in Krein [11]. THE GENERALIZED WEIERSTRASS APPROXIMATION THEOREM by M. H. Stone 1. We establish a new version of the theorem that applies to arbitrarily deep neural networks. Let C0(X, R) be the space of real-valued continuous functions on X which vanish at infinity; that is, a continuous function  f  is in C0(X, R) if, for every ε > 0, there exists a compact set K ⊂ X such that  | f |  < ε on X \ K. Again, C0(X, R) is a Banach algebra with the supremum norm. Theorem 14.2 (Stone-Weierstrass Theorem (1937)). Soc. Math. Stone-Weierstrass Approximation Theorem: Let M be a compact metric space. Weierstrass and Approximation Theory 3 It is in this context that we should consider Weierstrass' contributions to approxi-mation theory. But against this, its algebraic nature is lost, as compact convergence can seldom be constructed from CY alone. That is, A is dense in C(M). The Stone–Weierstrass theorem can be used to prove the following two statements which go beyond Weierstrass's result. If 91 is a subalgebra of C(T) which contains constants and separates points, then the elements of C(T) can be uniformly approximated by the elements of 91. If Aseperates points in Xand contains the constant functions, then A= C(X) in the uniform metric ˆ(f;g) =kf gkwhere kfk= max x2C jf(x)j. ✔  Stanislaus County Proposition 2. not lead to a practical method of approximation. The theorem was rst proved by Stone in 1937 in [2]. This presentat. Still less can continuous convergence, a sibling of compact conver­ gence, be created solely from CY. Note: 3 lectures. We do some foundational works before proving Theorem 1. THE STONE-WEIERSTRASS THEOREM AND ITS APPLICATIONS TO L2 SPACES PHILIP GADDY Abstract. Let Kbe a compact Hausdorff space and Aan algebra of C(K,R) . However, the discussion of Stone [3] re-mains the most direct and, when all details are considered, the short-est proof of the Stone-Weierstrass approximation theorem. In this article, we discuss the basic theme of approximating functions by polynomial functions. Applications of the Weierstrass approximation theorem abound in mathematics, and we discuss a few of them, including one addressing Gaus-sian quadrature. 1. consists of all those functions that can be obtained from the elements of ⊂ Thus there exists a polynomial p p such that ∥f −p∥ < ϵ/M. ... The text contains carefully worked out examples which contribute motivating and helping to understand the theory. There is also an excellent selection of exercises within the text and problem sections at the end of each chapter. C Our containers make any commercial or household project cost effective. Throughout the course of this paper, we will rst prove the Stone-Weierstrass Theroem, after providing some initial de nitions. 3.3. World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. Weierstrass approximation theorem. {\displaystyle {\mathfrak {A}}} Class. This book presents a comprehensive account of the theory of spaces of continuous functions under uniform, fine and graph topologies. , Roughly one third of this book isconcerned with developing some of the principal applications of function theory in several complex variables to Banach algebras. {\displaystyle S} ) ( A presentation of the Weierstrass approximation theorem and the Stone-Weierstrass theorem and a comparison of these two theorems are the objects of this thesis. The Stone-Weierstrass theorem says given a compact Hausdorff space X X, one can uniformly approximate continuous functions f: X → ℝ f: X \to \mathbb{R} by elements of any subalgebra that has enough elements to distinguish points. Then we may state : The space of complex-valued continuous functions on a compact Hausdorff space In 1937, M. Stone5 generalized Weierstrass approximation theorem to compact Hausdor spaces: Theorem 2.9 (Stone-Weierstrass Theorem for compact Hausdor space, Version 1). the Stone-Weierstrass theorem. , and because for real subsets, f The Stone-Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a, b], an arbitrary compact Hausdorff space X is considered, and instead of the space of polynomials, more general subalgebras of C(X) are involved. A celebrated theorem of Weierstrass states that any continuous real-valued function f defined on the closed interval [0, 1] ⊂ ℝ is the limit of a uniformly convergent sequence of polynomials. Polynomial approximation of differentiable functions has been studied in numerical analysis of differential and partial differential equations. This approach not only sheds an interesting light on that theorem, but will help the reader understand the nature of the lemma. He wants to find subalgebras of C(X, R) which are dense. Professor M. H. Stone would not begin to work on "The generalized Weierstrass approximation theorem" and published the paper in 1948. ) First, if you wish to talk about polynomials, you just need the Weierstrass Approximation theorem. WEIERSTRASS' PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM ANTON R. SCHEP At age 70 Weierstrass published the proof of his well-known Approximation Theorem. JOURNAL OP APPROXIMATION THEORY 34, 1-11 (1982) Upper Semicontinuous Functions and the Stone Approximation Theorem GERALD BEER Department of Mathematics, California State University, Los Angeles, California 90032 Communicated by Garrett Birkhoff Received October 19, 1978 INTRODUCTION In convex function theory it has long been recognized as useful to identify a convex function with its epigraph . A third proof of the Weierstrass Theorem making use of integrals is also presented. See [3]. 2021 © Casey Portable Storage. Re The Stone-Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions both regressive and progressive: instead of the real interval [a, b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C(X) is . It says that every continuous function on the interval [a, b] [a,b] [a, b] can be approximated as accurately desired by a polynomial function. The second half of this book, and consequently the second semester, covers differentiation and integration, as well as the connection between these concepts, as displayed in the general theorem of Stokes. We can provide inside storage at our facility or you can keep it on site at your home or business. We'll pick up your loaded container and bring it to one of our local storage facilities. for every continous function its fourier series converges. 9, p. 25 (Stone-Weierstrass theorem: a broad generalization of the classical Weierstrass theorem on the approximation of functions) found : Eisenreich. The Weierstrass Uniform Approximation Theorem W. L. Green Research Horizons - October 11, 2006 In 1885 Karl Weierstrass published the first proof of what is now known as the Weierstrass Uniform Approximation Theorem. This is a text for students who have had a three-course calculus sequence and who are ready to explore the logical structure of analysis as the backbone of calculus. A version of the Stone–Weierstrass theorem is also true when X is only locally compact. This book is a reissue of classic textbook of mathematical methods. The Weierstrass theorem states that for any continuous function f of one variable there is a sequence of polynomials that uniformly converge to f. To my surprise, I couldn't find any reference to similar results (either positive or negative) for the multivariate case, i.e. A more serious application of the lemma will be made later in a paper on the Bernstein approximation problem. This book provides an introduction to those parts of analysis that are most useful in applications for graduate students. We now consider a subset of the IT2FNN-2 on Figure 2. One of the most elegant and elementary proofs of this classic result is that which uses the Bernstein polynomials of f. one for each integer n ≥ 1. These include: The Weierstrass approximation theorem, of which one well known generalization is the Stone-Weierstrass theorem The Bolzano-Weierstrass theorem, which ensures compactness of closed and bounded sets in Rn The Weierstrass extreme value theorem, which states that a continuous function on. Proc. WEIERSTRASS APPROXIMATION THEOREM 5 Theorem 5. A theorem obtained and originally formulated by K. Weierstrass in 1860 as a preparation lemma, used in the proofs of the existence and analytic nature of the implicit function of a complex variable defined by an equation $ f( z, w) = 0 $ whose left-hand side is a holomorphic function of two complex variables. Marshall Stone's generalisation to compact Hausdorff spaces is natural and important in mathematics. Subj. Stone-Weierstrass Theorem 16 Acknowledgments 19 References 20 1. In this paper, Stone's theorem is used to prove a more general completeness theorem, which includes as special cases Plancherel's theorem, the . Stone's original proof of the theorem used the idea of lattices in C(X, R). In 1937, Stone generalized Weierstrass approximation theorem to compact Haus-dor spaces: Theorem 2.7 (Stone-Weierstrass Theorem for compact Hausdor space, Version 1). Journal of Approximation Theory 136 (2005) 45-59 . No, for several reasons. I discuss the Weierstrass polynomial approximation theorem and provide a simple proof! Let Q n(x) = C n(1 x2)n be functions restricted to [ 1;1], with C n chosen to make Q n satisfy (a). C əm] (mathematics) If S is a collection of continuous real-valued functions on a compact space E, which contains the constant functions, and if for any pair of distinct points x and y in E there is a function ƒ in S such that ƒ(x) is not equal to ƒ(y), then for any continuous real . S If f : [a;b] ! C Then for any The Stone-Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a, b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C(X), [clarification needed] the set of . C 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. f However, the discussion of Stone [3] re-mains the most direct and, when all details are considered, the short-est proof of the Stone-Weierstrass approximation theorem. Casey Portable Storage three areas in the Central Valley with warehouses located in Stockton, Modesto and Atwater, CA.  Not only do we provide do-it-yourself solutions, we also offer full service moving and storage services. Physical Description 2, iii, 36 leaves n See also Rudin (1973, §5.7). taking the real parts of the generated complex unital (selfadjoint) algebra agrees with the generated real unital algebra generated. Some years ago the writer discovered a gen-eralization of the Weierstrass approximation theorem suggested by an inquiry into certain algebraic properties of the continuous real functions on a topological space [1]. Cambridge Philos. ( , with the weak-* topology. [3][4][5][6][7] According to the mathematician Yamilet Quintana, Weierstrass "suspected that any analytic functions could be represented by power series".[7][6]. Let \(P([a,b])\) be the set of polynomial functions defined in Example 5.13 . The book has a modern approach and includes topics such as: •The p-norms on vector space and their equivalence •The Weierstrass and Stone-Weierstrass approximation theorems •The differential as a linear functional; Jacobians, Hessians ... The Weierstrass approximation theorem says that an arbitrary continuous function on a finite closed interval can be approximated uniformly by polynomials to . The Stone-Weierstrass Theorem states the following. Although there are standard reference texts like Rudin's . It turns out that the crucial property that a subalgebra must satisfy is that it separates points: a set A of functions defined on X is said to separate points if, for every two different points x and y in X there exists a function p in A with p(x) ≠ p(y). → Applications of the Weierstrass approximation theorem abound in mathematics { to Gaussian quadrature for instance. A subset L of C(X, R) is called a lattice if for any two elements  f, g ∈ L, the functions max{ f, g}, min{ f, g} also belong to L. The lattice version of the Stone–Weierstrass theorem states: The above versions of Stone–Weierstrass can be proven from this version once one realizes that the lattice property can also be formulated using the absolute value | f | which in turn can be approximated by polynomials in  f . Stone-Weierstrass theorem. This book is meant as a text for a first-year graduate course in analysis. əm] (mathematics) If S is a collection of continuous real-valued functions on a compact space E, which contains the constant functions, and if for any pair of distinct points x and y in E there is a function ƒ in S such that ƒ(x) is not equal to ƒ(y), then for any continuous real . Found inside – Page 187We will present a series of preliminary lemmas and definitions before finally proving the Stone - Weierstrass Theorem. We start with a classical result which we proved carefully in (Peterson (99) 2020). 9.1 Weierstrass Approximation ... Or, we'll take care of driving your Casey container to your new home or business. Our purpose is to establish a good compromise between (A) and (B). However, the discussion of Stone [3] re-mains the most direct and, when all details are considered, the short-est proof of the Stone-Weierstrass approximation theorem. The Stone-Weierstrass theorem may be stated as follows. processing and scattered data approximation. Weierstrass's Classical Theorem The starting point of all our discussions is: Weierstrass's Theorem (1885). f the Stone-Weierstrass theorem looks after all cases of interest. This page was last edited on 17 October 2021, at 22:41. This classic text offers a clear exposition of modern probability theory. The body of results has grown to an extent which seems to justify this book. The intention here is to make these results as accessible as possible. The book addresses essentially two questions. {\displaystyle {\mathfrak {A}}} Then is dense in . M.H. Using exponential functions, polynomials, rational By using the Stone-Weierstrass theorem together with Lemmas 6 and 7, we establish that the proposed IT2FNN-0 possesses the universal approximation capability. Re Perhaps surprisingly, even a very badly behaved continuous function is a uniform limit of polynomials. The Stone-Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a, b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on are shown to suffice, as is detailed below. A presentation of the Weierstrass approximation theorem and the Stone-Weierstrass theorem and a comparison of these two theorems are the objects of this thesis. ✔  San Joaquin County. Suppose A is an algebra of continuous real-valued functions on X that separates points in X and contains the constant functions. References B. BROSOWSKI AND F. DEUTSCH, An elementary proof of the Stone-Weierstrass theorem, Proc. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. Then A is dense in C(X). Theorem 1 (Weierstrass). To Weierstrass is due also the corresponding theorem on approximation hy trigonometric sums : If f(x) is a given periodic function of period 2. tt, continuous for all real values of x, and e is a given posi­ A more serious application of the lemma will be made later in a paper on the Bernstein approximation problem. the nature of the lemma. A is an Algebra, (1) A separates points of M, (2) A contains the constant functions (3) Then = C(M). In this note we will present a self-contained version, which is essentially his proof. X Introduction The Stone-Weierstrass Approximation Theorem. Amer. {\displaystyle \operatorname {Re} f_{n}\to \operatorname {Re} f} Our containers allow you to do your move at your own pace making do-it-yourself moving easy and stress free. A1 Weierstrass and StoneWeierstrass Approximation Theorems Let us recall two of from INFORMATIC 101 at Universidad Nacional de Asuncion " "Eminently suitable for self-study, this book may also be used as a supplementary text for courses in general (or point-set) topology so that students will acquire a lot of concrete examples of spaces and maps."--BOOK JACKET. C Found inside – Page 342.4 A Stone–Weierstrass Type Theorem The problem, whether interesting special families of functions are dense in the ... functions on a given space, has drawn attention ever since Weierstrass gave his well known approximation theorem. Let be a subalgebra of C ⁢ ( X ) for which the following conditions hold: Using a progressive but flexible format, this book contains a series of independent chapters that show how the principles and theory of real analysis can be applied in a variety of settings—in subjects ranging from Fourier series and ... Let AˆC(X;R) be a subalgebra which vanishes at no point and separates points. WEIERSTRASS' APPROXIMATION THEOREM AND FEJER´ 'S THEOREM Unless we say otherwise, all our functions are allowed to be complex-valued. ✔ We Do The Driving It is well known (e.g. ✔ All The Space You Need A short elementary proof of the Bishop-Stone-Weierstrass theorem Math. (b) follows because Q n is positive, and (d) follows by de nition. And we cannot really get any "nicer" functions than polynomials. The Stone-Weierstrass Theorem 3 Note. 1. Proof of Theorem 5.1. A rich collection of homework problems is included at the end of most chapters. The book is suitable as a text for a one-semester graduate course. Stone-Weierstrass theorem: lt;p|>In |mathematical analysis|, the |Weierstrass approximation theorem| states that every |cont. Stone-Weierstrass Theorem Yongheng Zhang Theorem 1 (Stone-Weierstrass). A That is, given / G C(T) and e > 0 there exists g G 91 such that sup,eT|/(/) — g(t)\ < e. For any >0, there is a polynomial, p, such that jf(t) p(t)j< for all t 2[0;1], that is jjp f jj 1< Proof We rst derive some equalities. , Weierstrass theorem. The Stone-Weierstrass theorem is a substantial generalization which you would consult when you want to consider things that. Various applications of these theorems are given.Some attention is devoted to related theorems, e.g. the Stone Theorem for Boolean algebras and the Riesz Representation Theorem.The book is functional analytic in character. The rst, Weierstrass [1872], is Weierstrass' example of a continuous nowhere di erentiable function. Likewise being the scalar part of −qi, −qj and −qk : b,c and d are respectively the real numbers (−qi − iq + jqk − kqj)/4, The theorem has many other applications to analysis, including: Slightly more general is the following theorem, where we consider the algebra Let γ>0 and let Iγ = [−1/2 + γ,1/2 − γ]. ||g|| for all  f, g). For a bounded uniformly continuous function f: R !R de ne for h>0 S hf(x) = 1 h p ˇ Z 1 1 f(u)e (u . Edited by fellow German mathematician Karl Weierstrass (1815-97), Volume 5 appeared in 1890. As far as I read I need to use the Stone-Weierstarss Theorem to prove this. Although it is exemplified by . Bernstein's Theorem states that B n (f) → f . WEIERSTRASS' PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM ANTON R. SCHEP At age 70 Weierstrass published the proof of his well-known Approximation Theorem. X We need only prove (c). This 1937—1948 theorem is probably the final conceptual brick to the edifice of which Weierstrass laid the first stone in 1885. The original Universal Approximation Theorem is a classical theorem (from 1999-ish) that states that shallow neural networks can approximate any function. We will denote the interval [0;1] by I. This follows since we have assumed ∫ 1 0 xnf (x) dx = 0 . This book is about the subject of higher smoothness in separable real Banach spaces. This classic book is a text for a standard introductory course in real analysis, covering sequences and series, limits and continuity, differentiation, elementary transcendental functions, integration, infinite series and products, and ... Idea. { Stone-Weierstrass Theorem, Version 1. Let AˆC(X;R) be a subalgebra which vanishes at no point and separates points. If f ∈C[0,1] and ε>0 then there exists a polynomial P such that "f −P"sup <ε.If f is The complex unital *-algebra generated by This self-contained book brings together the important results of a rapidly growing area. The Stone-Weierstrass theorem may be stated as follows. , then the real parts of those functions uniformly approximate the real part of that function, This theorem implies the real version, because if a net of complex-valued functions uniformly approximates a given function, Let Abe an algebra as a subset of C(X) where X is a compact space. 0 1997 Elsevier Science B.V. 1991 Math. Theorem: Let [a,b] be a compact interval contained in the real line. In this paper we mainly consider two of Weierstrass' results. Let Xbe any compact Hausdor space. The theorem generalizes as follows: This version clearly implies the previous version in the case when X is compact, since in that case C0(X, R) = C(X, R). {\displaystyle S\subset C(X,\mathbb {R} )\subset C(X,\mathbb {C} ),} Nachbin's theorem gives an analog for Stone–Weierstrass theorem for algebras of complex valued smooth functions on a smooth manifold (Nachbin 1949). See the last paragraph in this section. Let X be a compact Hausdorff space. Stone [1]) that the Stone-Weierstrass approximation theorem can be used to prove the completeness of various systems of orthogonal polynomials, e.g. In this paper we will look at three proofs of the Weierstrass Approximation Theorem. A text for a first graduate course in real analysis for students in pure and applied mathematics, statistics, education, engineering, and economics. Stone-Weierstrass Approximation Theorem: Let M be a compact metric space. Found inside – Page 161[T] e] Complex Version of the Weierstrass Approximation Theorem. ... and hence the StoneWeierstrass theorem implies the Weierstrass approximation theorem. b) Observe that the real polynomials in a form a subalgebra of C[-1, ... f C Nachbin's theorem is as follows (Llavona 1986): In 1885 it was also published an English version of the paper whose title was On the possibility of giving an analytic representation to an arbitrary function of real variable. The Wierstrass Approximation Theorem Theorem Let f be a continuous real-valued function de ned on [0;1]. Stone−Weierstrass Theorem [4]. Found inside – Page 3999.3.5 The Stone - Weierstrass Theorem * We consider next a famous generalization of the Weierstrass approximation theorem . A straightforward generalization would simply be that any continuous real - valued function f : M + R for M a ... 7. Just give us a ring at (209) 531-9010 for more info. R The following is one of the statements that we want to prove in this chapter. For a bounded uniformly continuous function f: R !R de ne for h>0 S hf(x) = 1 h p ˇ Z 1 1 f(u)e (u . → Found inside – Page 321Section 44 Theorem 44.6 was proved in 1885 by Weierstrass (Uber die analytische Darstellbarkeit sogenannter willkiirlicher F unctionen reeller Argumente). ... See also Stone (The Generalized Weierstrass Approximation Theorem). by throwing in the constant function 1 and adding them, multiplying them, conjugating them, or multiplying them with complex scalars, and repeating finitely many times.
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