A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM FOR R k USING BINOMIAL DISTRIBUTIONS

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1 A PROOF OF THE WEIERSTRASS APPROXIMATION THEOREM FOR R k USING BINOMIAL DISTRIBUTIONS BEN KORDESH Abstract. We relate the probabilistic proof of the Weierstrass Approximatio Theorem [Lev84] to the covolutio proof [La83, Thm. XI-2.1]. The oly importat probability theory, the biomial distributio, relates to the Dirac sequece K [La83, p. 284] shifted by x. We use Chebyshev s Theorem to prove a aalogue to Lag s axiom DIR 3 [La83, p. 284]. We exted the etire proof of the Weierstrass Theorem to rectagles i R k. 1. Itroductio The Weierstrass Approximatio Theorem says a cotiuous fuctio o [0, 1] is the uiform limit of polyomials. Our proof fuses Lag s covolutios [La83, XI 1 2] ad Levasseur s probability [Lev84]. We rework Lag s Dirac sequeces ad the proof of Theorem 1.1 from [La83, XI- 1]. We replace the sequece of families of radom variables F,x, 0 x 1 ad = 0, 1, 2,... [Lev84] with probability distributios. We exted Dirac sequeces to higher dimesios, ad as a result exted the Weierstrass Theorem to higher dimesioal rectagles. Our motivatio came from hearig that compactly supported higher-dimesioal cotiuous fuctios ca be approximated by C fuctios through covolutio by pulse fuctios, which we have ot foud a good expositio for. The Stoe-Weierstrass Theorem [Rud76, Thm. 7.29] easily implies Theorem Stoe s proof is a outof-the-blue double compactess max/mi argumet which for k = 1 approximates polygoal fuctios with more jagged polygoal fuctios. Lebesgue s more elegat argumet for k = 1 showed that polygoal fuctios ca be writte i terms of the absolute value fuctio, which ca be approximated by polyomials. Thaks to Prof. Zabell for a NU Math course proof of the Weierstrass Theorem, which provided a blueprit for the proof of Theorem 2.1, ad which I foud easier to read tha [Lev84]. 2. Dirac Fuctioal Sequeces Let N be the set of all positive itegers. Fix k N. For t > 0 ad x R k, defie the ope ball B t (x) = {y R k : y x < t}. Defie Fuc(S) to be the set of all liear fuctios from a vector space S to R. Defie Adm(P ) ad C(P ) to be the sets of all admissible (bouded ad cotiuous except for a egligible subset of P ) [La83, p. 570] ad cotiuous (resp.) fuctios from a closed rectagle P R k to R. Let χ T : [0, 1] k R be the characteristic fuctio of T [0, 1] k. By a Dirac fuctioal sequece we mea a fuctio β : N [0, 1] k Fuc(Adm([0, 1] k )) where DIR 1: β,x (f) 0 if f 0, for all f Adm([0, 1] k ), N, ad x [0, 1] k. 1 as oted i 1

2 2 BEN KORDESH DIR 2: β,x (χ [0,1] k) = 1 for all N ad x [0, 1] k. DIR 3: For all ɛ, δ > 0, there exists N N such that if N ad x [0, 1] k, the β,x (χ [0,1]k B δ (x)) < ɛ. By DIR 1 ad liearity, if f g, the β,x (f) β,x (g), for all f, g Adm([0, 1] k ). Thus β(h) β( h ) ad β(h) β( h ) for all h Adm([0, 1] k ), ad thus (1) β(h) β( h ), for all h Adm([0, 1] k ). For all admissible [La83, p. 572, 574] S [0, 1] k, by DIR 1, DIR 2 ad liearity, (2) β,x (χ S ) 1, for all N ad x [0, 1] k, sice χ S χ [0,1] k. We ow rework [La83, Thm. XI-1.1] ad its proof. Theorem 2.1. Take a Dirac fuctioal sequece β ad a fuctio f C([0, 1] k ). For all N, defie f : [0, 1] k R by f (x) = β,x (f) for all x [0, 1] k. The the sequece {f } uiformly coverges to f. Proof. By compactess, f is uiformly cotiuous. Therefore for all ɛ > 0 there exists δ > 0 such that f(x) f(y) < ɛ if x y < δ, for all x, y [0, 1] k. By DIR 2 ad liearity, f (x) f(x) = β,x (f f(x)). Thus, by (1), (3) f (x) f(x) β,x ( f f(x) ). Sice f is cotiuous ad defied o a compact set, f has a maximum M. By the triagle iequality, f f(x) 2M. By uiform cotiuity, f(y) f(x) < ɛ for all y [0, 1] k B δ (x). Thus (4) f f(x) ɛχ [0,1]k B δ (x) + 2Mχ [0,1]k B δ (x). By (3), (4), DIR 1, liearity, (2) ad DIR 3, there exists N N such that, for all N ad x [0, 1] k, f (x) f(x) ɛβ,x (χ [0,1] k B δ (x)) + 2Mβ,x (χ [0,1] k B δ (x)) ɛ + 2Mɛ = (1 + 2M)ɛ. Sice we may take ay ɛ > 0 ad 2M + 1 is fixed, we are doe. [La83, Thm. XI-1.1] itself is false, because it assumes the fuctio f o R is piecewise cotiuous rather tha cotiuous. Let f be the characteristic fuctio of [0, 1]. Sice lim (f K )(0) = 1/2, ad ot 1 = f(0), f is a couterexample to Lag s theorem. Lag s error comes from claimig that f(x t) f(t) ɛ, for all t such that t < δ ad x S. We also eed to kow that x t S, for a compact S R. Lag makes the same error i his proof of [La93, Thm. VIII-3.1], where a compact A R replaces S. Lag s proof used a sequece of symmetric fuctios K : R R, for all N, kow as the Ladau kerels. Defie K,x : R R by K,x (t) = K (t x), for all x R, which is K shifted to the right by x. Take a cotiuous f : R R where f(x) = 0 for all x (0, 1). The, by the defiitio of covolutio ad symmetry, (f K )(x) = f(t)k (x t) dt = f(t)k,x (t) dt. There is a obvious resemblace betwee the right had side of the equatio ad the Dirac fuctioal sequece applicatio β,x (f) we used above.

3 THE WEIERSTRASS APPROXIMATION THEOREM 3 3. Fiite Probability Distributios o R k Expected Value, Variace ad Chebyshev s iequality. Fix k N. Assume a S φ(a) = 1, for a fiite set S ad φ: S [0, 1]. The we say (φ, S) is a fiite probability distributio. For ay rectagle P R k, let ProbDist(P ) be the set of all probability distributios (φ, S) for a fiite S P. For k N, we defie η : ProbDist([0, 1] k ) Fuc(Adm([0, 1] k )) by η(φ, S)(f) = a S φ(a)f(a). Usig stadard lambda calculus otatio, defie λy. y 2 : [0, 1] k R. Defie E: ProbDist([0, 1] k ) R k ad Var: ProbDist([0, 1] k ) R by (5) E(φ, S) = a S φ(a)a, Var(φ, S) = η(φ, S)(λa. a E(φ, S) 2 ), for all (φ, S) ProbDist([0, 1] k ). Note that E(φ, S) = η(φ, S)(λy.y) for k = 1. Usig familiar probability theory otatio, let µ = E(φ, S) ad σ 2 = Var(φ, S). Usig the dot product equatio y µ 2 = y 2 2y µ + µ 2, (6) µ = a S φ(a)a ad σ 2 = η(φ, S)(λy. y 2 ) µ 2 = µ 2 + a S φ(a) a 2. We ow prove a aalogue to Chebyshev s iequality. 2 Lemma 3.1. For all (φ, S) ProbDist([0, 1] k ), let γ = η(φ, S). Let µ = E(φ, S) ad σ 2 = Var(φ, S). The for all δ > 0, γ(χ [0,1]k B δ (µ)) σ2 δ 2. Proof. Let T = {x [0, 1] k : x µ δ}. The T = {x [0, 1] k : x µ 2 /δ 2 1}. Thus, for all x [0, 1] k, we kow χ T (x) x µ 2 /δ 2. Sice φ maps ito [0, 1], it is o-egative, ad thus γ(f) 0 for all f Adm([0, 1] k ) such that f 0. Thus γ(χ T ) γ(λx. x µ 2 /δ 2 ) = σ2 δ 2, by argumets used i 2. Sice [0, 1] k B δ (µ) = T, we are doe. Biomial Distributios. I this subsectio we fix x [0, 1]. For a positive Z ad k {0,..., }, we defie the biomial distributio [Lev84] of ad x: ( ) (7) b(, x; k) = x k (1 x) k. k By the biomial theorem, (8) b(, x; k) = 1, for all itegers 1. k=0 Let S,x = {k/ : 0 k }, for 1. We defie g,x : S,x [0, 1] by ( ) k (9) g,x = b(, x; k). 2 s_iequality has a proof of a result about radom variables, which our proof about probability distributios is a modificatio of.

4 4 BEN KORDESH The, by (8), we kow (g,x, S,x ) ProbDist([0, 1]). 3 We ow defie a fuctio β : N [0, 1] Fuc(Adm([0, 1])) by β,x = η(g,x, S,x ). By (5) ad (6), defie (10) µ,x = E(g,x, S,x ) = ag,x (a) a S,x ad (11) σ,x 2 = Var(g,x, S,x ) = We ow prove the followig two lemmas. Lemma 3.2. For all itegers 1, µ,x = x. a S,x a 2 g,x (a) µ 2,x. Proof. By (10) ad the defiitios of g,x ad S,x, 1 k µ,x = b(, x; k) = x b( 1, x; j) = x, k=1 by (8), (7), ad because kb(, x; k) = xb( 1, x; k 1) for all itegers k, such that 1 k, sice k ( ) ( k = 1 k 1) for such k,. Lemma 3.3. For all itegers 1, σ 2,x = j=0 x(1 x) 1 4. Proof. We kow x(1 x) 1/4 for all x [0, 1], sice (x 1/2) 2 0. Thus it suffices to prove a S,x a 2 g,x (a) = x ( 1)x2 +, sice by (11) ad Lemma 3.2, σ,x 2 = a S,x a 2 g,x (a) x 2. If = 1, the proof is straightforward, so we may assume > 1. As i the proof of Lemma 3.2, a 2 k 2 1 b(, x; k) (1 + j)b( 1, x; j) g,x (a) = 2 = x a S,x j=0 k=1 = x 1 + x j=1 jb( 1, x; j) = x + ( 1)x2 2 b( 2, x; l) Tesor Product of Probability Distributios. For k, l N, defie : ProbDist([0, 1] k ) ProbDist([0, 1] l ) ProbDist([0, 1] k+l ) by (φ, S) (ψ, T ) = (φ ψ, S T ) for all (φ, S) ProbDist([0, 1] k ) ad (ψ, T ) ProbDist([0, 1] l ), where (φ ψ)(x, y) = φ(x)ψ(y) for all x S ad y T. Lemma 3.4. Take k, l N, (φ, S) ProbDist([0, 1] k ), (ψ, T ) ProbDist([0, 1] l ). The E((φ, S) (ψ, T )) = (E(φ, S), E(ψ, T )). Proof. Sice (x, y) = (x, 0) + (0, y) for all x R k ad y R l, E((φ, S) (ψ, T )) = E(φ ψ, S T ) = φ(x)ψ(y)(x, y) ( = φ(x)ψ(y)x, l=0 ) φ(x)ψ(y)y. 3 (g,x, S,x) turs out to be defied by a radom variable, the average umber of heads i coi tosses with x the probability of a head, a probability theory cocept we will ot use.

5 THE WEIERSTRASS APPROXIMATION THEOREM 5 By a versio of Fubii s Theorem for summatios, φ(x)ψ(y)x = φ(x)ψ(y)x = y T x S y T = y T(ψ(y)E(φ, S)) = E(φ, S). ( ψ(y) ) φ(x)x x S Thus E((φ, S) (ψ, T )) = (E(φ, S), E(ψ, T )), by a similar result for (ψ, T ). Lemma 3.5. Take k, l N, (φ, S) ProbDist([0, 1] k ), (ψ, T ) ProbDist([0, 1] l ). The Var((φ, S) (ψ, T )) = Var(φ, S) + Var(ψ, T ). Proof. By Lemma 3.4, Fubii s Theorem, the distributive law ad sice (φ, S) ad (ψ, T ) are probability distributios, Var((φ, S) (ψ, T )) = φ(x)ψ(y) (x, y) (E(φ, S), E(ψ, T )) 2 = = y T ψ(y) x S φ(x)ψ(y)( x E(φ, S) 2 + y E(ψ, T ) 2 ) φ(x) x E(φ, S) 2 + x S φ(x) y T ψ(y) y E(ψ, T ) 2 =Var(φ, S) + Var(ψ, T ) Obviously is associative. Take k N. Take x = (x 1,..., x k ) [0, 1] k. Defie (G,x, S,x ) = (g,x1, S,x1 )... (g,xk, S,xk ) ProbDist([0, 1] k ). Thus S,x = {i/ : i {0,..., }} k ad (12) G,x (i 1 /,..., i k /) = By Lemmas 3.4, 3.5, 3.2 ad 3.3, k b(, x s ; i s ). s=1 (13) E(G,x, S,x ) = x ad Var(G,x, S,x ) = k s=1 x s (1 x s ) 4. A Special Case of the Stoe-Weierstrass Theorem Fix k N. For all x [0, 1] k, N ad f Adm([0, 1] k ), we defie Defiitio 4.1. β,x (f) = η(g,x, S,x )(f) = a S,x f(a)g,x (a). Thus, for all f Adm([0, 1] k ), by (12), (14) β,x (f) = (i 1/,...,i k /) S,x f(i 1 /,..., i k /) k b(, x s ; i s ). s=1 k 4. We will prove β is a Dirac fuctioal sequece. By (12), G,x is always o-egative if N ad x [0, 1] k, ad thus β satisfies DIR 1. For all N ad x [0, 1] k, by (14), (8), the distributive law ad sice χ [0,1] k(s,x ) = {1}, β,x (χ [0,1] k) = (i 1/,...,i k /) S,x s=1 k b(, x s ; i s ) = 1.

6 6 BEN KORDESH Thus β satisfies DIR 2. We will ow prove β satisfies DIR 3. Lemma 4.2. For all ɛ, δ > 0, there exists N N such that if N ad x [0, 1], the β,x (χ [0,1] Bδ (x)) < ɛ. Proof. By (13) ad Lemma 3.1, for all N ad x [0, 1], β,x (χ [0,1] Bδ (x)) k 4δ 2. So β satisfies DIR 1, DIR 2 ad DIR 3, ad thus β is a Dirac fuctioal sequece. For all f Adm([0, 1] k ) ad N, defie f : [0, 1] k R by f (x) = β,x (f). By (14) ad (7), f is a polyomial fuctio from [0, 1] k to R. Sice β is a Dirac fuctioal sequece, by Theorem 2.1, we have Theorem 4.3. For all k N, f C([0, 1] k ) ad N, defie f : [0, 1] k R by f (x) = β,x (f) for all x [0, 1] k. The the sequece of polyomial fuctios {f } uiformly coverges to f. Refereces [La83] S. Lag, Udergraduate Aalysis, Udergraduate Texts i Math., Spriger, New York, [La93], Real ad Fuctioal Aalysis, Graduate Texts i Math., vol. 142, Spriger, New York, [Lev84] K. Levasseur, A Probabilistic Proof of the Weierstrass Approximatio Theorem, Amer. Math. Mothly 91 (1984), [Rud76] W. Rudi, Priciples of Mathematical Aalysis, McGraw Hill Ic., New York, 1976.