Uniform Approximation and Bernstein Polynomials with Coefficients in the Unit Interval

Transcription

1 Unifor Approxiation and Bernstein Polynoials with Coefficients in the Unit Interval Weiang Qian and Marc D. Riedel Electrical and Coputer Engineering, University of Minnesota 200 Union St. S.E. Minneapolis, MN 55455, USA {qianx030, Ivo Rosenberg Matheatics and Statistics University of Montreal C.P. 6128, Succ. Centre-Ville Montreal, Quebec, Canada H3C 3J7 Abstract This paper presents two ain results. The first result pertains to unifor approxiation with Bernstein polynoials. We show that, given a power-for polynoial g, we can obtain a Bernstein polynoial of degree with coefficients that are as close as desired to the corresponding values of g evaluated at the points 0, 1,..., 1, provided that is sufficiently large. The second result pertains to a subset of Bernstein polynoials: those with coefficients that are all in the unit interval. We show that polynoials in this subset ap the open interval 0, 1 into the open interval 0, 1 and ap the points 0 and 1 into the closed interval [0, 1]. The otivation for this wor is our research on probabilistic coputation with digital circuits. Our design ethodology, called stochastic logic, is based on Bernstein polynoials with coefficients that correspond to probability values; accordingly, the coefficients ust be values in the unit interval. The atheatics presented here provide a necessary and sufficient test for deciding whether polynoial operations can be ipleented with stochastic logic. 1 Introduction The Weierstrass approxiation theore is a faous theore in atheatical analysis. It asserts that every continuous function defined on a closed interval can be uniforly approxiated as closely as desired by a polynoial function [1]. The Weierstrass Approxiation Theore: Let f be a continuous function defined on the closed interval [a, b]. For any ɛ > 0, there exists a polynoial function p such that for all x in [a, b], we have fx px < ɛ. The theore can be proved by a transforation with Bernstein polynoials [2]. By a linear substitution, the interval [a, b] can be transfored into the unit interval [0, 1]. Thus, the original stateent of the theore holds if and only if the theore holds for every continuous function f defined on the interval [0, 1]. This wor is supported by a grant fro the Seiconductor Research Corporation s Focus Center Research Progra on Functional Engineered Nano-Architectonics, contract No NT

2 A Bernstein polynoial of degree n is a polynoial expressed in the following for [3]: β,n b,n x, 1 where each β,n, 0, 1,..., n, is a real nuber and n b,n x x 1 x n. 2 The coefficients β,n are called Bernstein coefficients and the polynoials b 0,n x, b 1,n x,..., b n,n x are called Bernstein basis polynoials of degree n. Define the n-th Bernstein polynoial for f to be B n f; x f b,n x. n In 1912, Bernstein showed the following result [4, 5]: The Bernstein Theore: Let f be a continuous function defined on the closed interval [0, 1]. For any ɛ > 0, there exists a positive integer M such that for all x in [0, 1] and integer M, we have fx B f; x < ɛ. Note that the function B f; x is a polynoial on x. Thus, based on the Bernstein Theore, the Weierstrass Approxiation Theore holds. Given a power-for polynoial g of degree n, it is well nown that for any n, g can be uniquely converted into a Bernstein polynoial of degree [6]. Cobining this fact with the Bernstein Theore, we have the following corollary. Corollary 1 Let g be a polynoial of degree n. For any ɛ > 0, there exists a positive integer M n such that for all x in [0, 1] and integer M, we have β, g b, x < ɛ, where β 0,, β 1,,..., β, satisfy gx β, b, x. In the first part of the paper, we prove a stronger result than this: Theore 1 Let g be a polynoial of degree n 0. For any ɛ > 0, there exists a positive integer M n such that for all integers M and 0, 1,...,, we have β, g < ɛ, where β 0,, β 1,,..., β, satisfy gx β, b, x. 2

3 Cobining Theore 1 with the fact that b, x 1, we can easily prove Corollary 1. In the second part of the paper, we consider a subset of Bernstein polynoials: those with coefficients that are all in the unit interval [0, 1]. Definition 1 Define U to be the set of Bernstein polynoials with coefficients that are all in the unit interval [0, 1]: { } U px n 1, 0 β 0,n, β 1,n,..., β n,n 1, such that px β,n b,n x. The question we as is: which polynoials can be converted into Bernstein polynoials in U? Definition 2 Define the set V to be the set of polynoials which are either identically equal to 0 or equal to 1, or ap the open interval 0, 1 into 0, 1 and the points 0 and 1 into the closed interval [0, 1], i.e., V {px px 0, or px 1, or 0 < px < 1, x 0, 1 and 0 p0, p1 1}. We prove that the two sets are equivalent: Theore 2 V U. In what follows, we will refer to a Bernstein polynoial of degree n converted fro a polynoial g as the Bernstein polynoial of degree n of g. When we say that a polynoial is of degree n, we ean that the power-for of the polynoial is of degree n. Exaple 1 Consider the polynoial gx 3x 8x 2 + 6x 3. It aps the open interval 0, 1 into 0, 1 with g0 0, g1 1. Thus, g is in the set V. Based on Theore 2, we have that g is in the set U. We verify this by considering Bernstein polynoials of increasing degree. The Bernstein polynoial of degree 3 of g is Note that here the coefficient β 2,3 2 3 < 0. The Bernstein polynoial of degree 4 of g is Note that here the coefficient β 3,4 1 4 < 0. gx b 1,3 x 2 3 b 2,3x + b 3,3 x. gx 3 4 b 1,4x b 2,4x 1 4 b 3,4x + b 4,4 x. The Bernstein polynoial of degree 5 of gx is Note that here all the coefficients are in [0, 1]. gx 3 5 b 1,5x b 2,5x + b 5,5 x. 3

4 Since the Bernstein polynoial of degree 5 of g satisfies Definition 1, we conclude that g is in the set U. Exaple 2 Consider the polynoial gx 1 4 x + x2. Since g0.5 0, thus g is not in the set V. Based on Theore 2, we have that g is not in the set U. We verify this. By contraposition, suppose that there exist n 1 and 0 β 0,n, β 1,n,..., β n,n 1 such that Since g0.5 0, therefore, gx β,n b,n x. β,n b,n Note that for all 0, 1,..., n, b,n 0.5 > 0. Thus, we have that for all 0, 1,..., n, β,n 0. Therefore, gx 0, which contradicts the original assuption about g. Thus, g is not in the set U. The reainder of the paper is organized as follows. In Section 2, we present soe atheatical preliinaries pertaining to Bernstein polynoials. In Section 3, we prove Theore 1. Based on this theore, in Section 4, we prove Theore 2. Finally, we conclude the paper with a discussion on applications of these theores to our research in probabilistic coputation with digital circuits. 2 Properties of Bernstein Polynoials We list soe pertinent properties of Bernstein polynoials. a The positivity property: For all 0, 1,..., n and all x in [0, 1], we have b The partition of unity property: b,n x 0. 3 The binoial expansion of the left-hand side of the equality x + 1 x n 1 shows that the su of all Bernstein basis polynoials of degree n is the constant 1, i.e., b,n x 1. 4 c Converting power-for coefficients to Bernstein coefficients: The set of Bernstein basis polynoials b 0,n x, b 1,n x,..., b n,n x fors a basis of the vector space of polynoials of real coefficients and degree no ore than n [6]. Each power basis function x j can be uniquely expressed as a linear cobination of the n + 1 Bernstein basis polynoials: x j σ j b,n x, 5 for j 0, 1,..., n. To deterine the eleents of the transforation atrix σ, we write x j x j x + 1 x nj 4

5 and perfor a binoial expansion on the right hand side. This gives j x j n,n x, jb for j 0, 1,..., n. Therefore, we have j σ j { σj n 1, j j for j 0, for j >. 6 Suppose that a power-for polynoial of degree no ore than n is gx a,n x 7 and the Bernstein polynoial of degree n of g is gx β,n b,n x. 8 Substituting Equations 5 and 6 into Equation 7 and coparing the Bernstein coefficients, we have n 1 β,n a j,n σ j a j,n. 9 j j Equation 9 provide a eans for obtaining Bernstein coefficients fro power-for coefficients. d Degree elevation: Based on Equation 2, we have that for all 0, 1,...,, 1 +1 b,+1 x x 1 x 1 b, x, b +1,+1 x x 1 x +1 + x +1 1 x or b, x +1 b,+1 x + +1 b +1,+1 x b,+1x b +1,+1x. 10 Given a power-for polynoial g of degree n, for any n, g can be uniquely converted into a Bernstein polynoial of degree. Suppose that the Bernstein polynoials of degree and +1 degree + 1 of g are β, b, x and β,+1 b,+1 x, respectively. We have β, b, x 5 +1 β,+1 b,+1 x. 11

6 Substituting Equation 10 into the left-hand side of Equation 11 and coparing the Bernstein coefficients, we have β 0,, for 0 β,+1 +1 β 1, β,, for 1 12 β,, for + 1. Equation 12 provides a eans for obtaining the coefficients of the Bernstein polynoial of degree + 1 of g fro the coefficients of the Bernstein polynoial of degree of g. We will call this procedure degree elevation. For convenience, given a Bernstein polynoial gx n β,nb,n x, we can also express it as gx c,n x 1 x n, 13 where c,n n β,n, 14 for 0, 1,..., n. Substituting Equation 14 into Equation 12, we have 3 A Proof of Theore 1 c 0,, for 0 c,+1 c 1, + c,, for 1 c,, for + 1. Suppose that the polynoial g is of degree n. Applying Equation 15 recursively, we can express c, as a linear cobination of c 0,n, c 1,n,..., c n,n. Lea 1 Let g be a polynoial of degree n. For any n, suppose that the Bernstein polynoial of degree of g is gx c, x 1 x. Let c, 0 for all < 0 and all >. Then for all 0, 1,...,, we have c, n n Proof: We prove the lea by induction on n. i 15 c +n+i,n. 16 Base case: For n 0, the right-hand side of Equation 16 reduces to 0 0 c,n c,, so the equation holds. Inductive step: Suppose that Equation 16 holds for soe n and all 0, 1,...,. Consider + 1. Since we assue that c 1, c +1, 0, Equation 15 can be written as c,+1 c 1, + c,, 17 6

7 for all 0,..., + 1. With our convention that c i,n 0 for all i < 0 and i > n, it is easily seen that c 1, 0 n n i c 1+n+i,n, c +1, 0 n n i c +1+n+i,n. Together with the induction hypothesis, we conclude that for all 1, 0,...,, + 1 c, n n Based on Equations 17 and 18, for all 0, 1,..., + 1, we have c,+1 n n i c 1+n+i,n + i c +n+i,n. 18 n n In the first su, we change the suation index to j i 1. We obtain c,+1 n1 j1 n 0 Applying the basic forula c,+1 c +n1,n + n c +n+j,n + j + 1 c +n1,n + n1 r q n1 n r 1 + q n j + 1 n j [ n j + 1 r 1 q + j c +n+j,n n j, we obtain c +n+j,n + c,n c +n+j,n. ] c +n+j,n + +1n + 1 n Thus Equation 16 holds for + 1. By induction, it holds for all. Rear: Equation 16 can be forulated as c, in{,n} iax{0,+n} i n c,n. n c 1+n+i,n. n c i,n, 19 i for all n and 0, 1,...,. Indeed, in Equation 16, first use the basic forula and then change the suation index to j + n + i to obtain c, n n c +n+i,n n i j+n Note that c j,n 0 iplies 0 j n. This yields Equation 19. n c j,n. j r r q r q 7

8 Lea 2 Let n be a positive integer. For all integer, and i such that we have Proof: For siplicity, we define δ n i > n, 0, ax{0, + n} i in{, n}, 20 i 1 ni n2. 21 n i i 1 ni n!!! i! n + i!! ni1 j j j n i. Now 1 i n + i n + 1 i1 i1 i j 1 ni1 j We obtain an upper bound for n i i1 n i 1 i. i j by replacing j in Equation 22 with its least value, 0: 1 ni1 1 i i i ni. i We need the following siple inequality: for real nubers 0 x y 1 and a non-negative integer l, l1 y l x l y x y j x l1j y xl. 23 Fro Equation 20, we obtain 0 i in{, n} and so we can use Equation 23 for 0 x i y 1, l n i 0. We obtain i δ 1 ni n i i i ni ni i i i n i Since 0 1, 0 1, and 0 i n, we obtain i ni i in i i. ni i 22 i in i i 8 in i > n2.

9 Therefore, δ Siilarly, we obtain a lower bound for i 1 ni n i n i the first product and with n i in the second product, obtaining n i1 i 1 ni1 1 i i1 j i j i i n + i ni i i n i > n2. 24 by replacing the index j in Equation 22 with i in 1 i i n + i n + i ni1 ni. 1 i n Thus, proceeding as above, we have i δ 1 ni n i i ni i i n + i ni i n + i [ i ] i i [ ni ni ] n + i ni i i +. i n + i i Due to Equation 20, we have and so we obtain 0 i i 1, 0 n + i n + i 1, δ i i i + i ni n + i ni. 25 n + i Applying Equation 23 twice to the right-hand side of Equation 25, we obtain δ i i + n i i n + i n + i i2 i Fro Equation 20, we have n i2 + n + i. Therefore, δ i 1 ni 0 i 1, 0 n + i 1. n i i2 + n i 2 ni + nn i n2. 26 Equations 24 and 26 together yield Equation 21. Now we give a proof of Theore 1. 9

10 Theore 1 Let g be a polynoial of degree n 0. For any ɛ > 0, there exists a positive integer M n such that for all integer M and 0, 1,...,, we have β, g < ɛ, where β 0,, β 1,,..., β, satisfy that gx β, b, x. Proof: For n 0, g is a constant polynoial. Suppose that gx y, where y is a constant value. We select M 1. Then, for all integers M and all integers 0, 1,...,, we have β, y g. Thus, the theore holds. { } n 2 For n > 0, we select M such that M > ax c i,n, 2n, where the real nubers ɛ c 0,n, c 1,n,..., c n,n satisfy gx c i,n x i 1 x ni. 27 Now consider any M. Since 2n ax { n 2 ɛ } c i,n, 2n < M, we have n > n. Consider the following three cases for. 1. The case where n n. Here ax{0, + n} 0 and in{, n} n. Thus, the suation indices in Equation 19 range fro 0 to n. Therefore, β, c n, i c i,n. 28 Substituting x with in Equation 27, we have g i c i,n 1 ni. 29 By Lea 2, since 0 < n < and 0, Equation 21 holds for all 0 ax{0, + n} i in{, n} n. Thus, by Equations 21, 28, 29 and the well-nown inequality x i x i, we have [ β n i i, g 1 ] ni c i,n n i i 1 ni c i,n n2 c i,n. 10

11 Since n2 ɛ c i,n < M, we have n 2 c i,n < ɛ. 30 Therefore, for all n n, we have β, g < ɛ. 2. The case where 0 < n. Since > 2n, we have + n < n < 0. Thus, ax{0, + n} 0 and in{, n}. Thus, the suation indices in Equation 19 range fro 0 to. Therefore, β, c n, i c i,n. 31 When + 1 i n, we have that i and so i 1 ni i1 1 ni < n n2. 32 By Lea 2, since 0 < n < and 0, Equation 21 holds for all 0 ax{0, + n} i in{, n}. Thus, by Equations 21, 29, 30, 31, 32 and the inequality x i x i, we have β n i i, g c i,n 1 ni c i,n [ n i i 1 ] ni i c i,n 1 ni c i,n i+1 n i i 1 ni c i i,n + 1 ni c i,n n2 c i,n < ɛ. i+1 3. The case where n <. Since > 2n, we have n < n <. Thus, ax{0, + n} + n and in{, n} n. Now, the suation indices in Equation 19 range fro + n to n. Therefore, β, c n, i c i,n. 33 i+n When 0 i + n 1, we have that n i. Thus, i 1 ni 1 i 1 ni1 < n n

12 By Lea 2, since 0 < n < and 0, Equation 21 holds for all + n ax{0, + n} i in{, n} n. Thus, by Equations 21, 29, 30, 33, 34 and the inequality x i x i, we have β n i i, g c i,n 1 ni c i,n i+n [ n ] ni i i i+n n i i+n n2 c i,n < ɛ. i 1 c i,n i 1 ni c i,n + In conclusion, if M, then for all 0, 1,...,, we have β, g < ɛ. 4 A Proof of Theore 2 +n1 +n1 1 ni c i,n i 1 ni c i,n We deonstrate that the sets U and V defined in the introduction see Definitions 1 and 2 are one and the sae. We deonstrate that U V and V U separately. First, we prove the forer the easier one. Then we use Theore 1 to prove the latter. Theore 3 U V. Proof: Let n 1 and β,n 0, for all 0 n. Then the polynoial px Let n 1 and β,n 1, for all 0 n. Then, by Equation 4, the polynoial px Thus 0 U and 1 U. Fro the definition of V, 0 V and 1 V. Now consider any polynoial p U such that p 0 and p 1. 0 β 0,n, β 1,n,..., β n,n 1 such that px β,n b,n x. β,n b,n x 0. β,n b,n x 1. There exist n 1 and Fro Equations 3, 4 and the fact that 0 β 0,n, β 1,n,..., β n,n 1, for all x in [0, 1], we have 0 px β,n b,n x b,n x 1. We further clai that for all x in 0, 1, we ust have 0 < px < 1. By contraposition, we assue that there exists a 0 < x 0 < 1, such that px 0 0 or px 0 1. Since for 0 < x 0 < 1, we have 0 px 0 1, thus px 0 0 or 1. 12

13 We first consider the case that px 0 0. Since 0 < x 0 < 1, it is not hard to see that for all 0, 1,..., n, b,n x 0 > 0. Thus, px 0 0 iplies that for all 0, 1,..., n, β,n 0. In this case, for any real nuber x, px n β,nb,n x 0, which contradicts the assuption that px 0. Siilarly, in the case that px 0 1, we can show that px 1, which contradicts the assuption that px 1. In both cases, we get a contradiction; this proves the clai that for all x in 0, 1, 0 < px < 1. Therefore, for any polynoial p U such that p 0 and p 1, we have p V. Since we showed at the outset that 0 U, 1 U, 0 V and 1 V, thus, for any polynoial p U, we have p V. Therefore, U V. Next we prove the clai that V U. We will first show that each of four possible different categories of polynoials in the set V are in the set U. The different categories are tacled in Theores 4 and 5 and Corollaries 2 and 3. Theore 4 Let g be a polynoial of degree n apping the open interval 0, 1 into 0, 1 with 0 g0, g1 < 1. Then g U. Proof: Since g is continuous on the closed interval [0, 1], it attains its axiu value M g on [0, 1]. Since gx < 1, for all x [0, 1], we have M g < 1. Let ɛ 1 1 M g > 0. By Theore 1, there exists a positive integer M 1 n such that for all integers M 1 and 0, 1,...,, we have β, g < ɛ 1, where β 0,, β 1,,..., β, satisfy that gx β, b, x. Thus, for all M 1 and all 0, 1,...,, β, < g + ɛ 1 M g + 1 M g Denote by r the ultiplicity of 0 as a root of gx where r 0 if g0 0 and by s the ultiplicity of 1 as a root of gx where s 0 if g1 0. We can factorize gx as gx x r 1 x s hx, 36 where hx is a polynoial, satisfying that h0 0 and h1 0. We show that h0 > 0. By the way of contraposition, suppose that h0 0. Since h0 0, we have h0 < 0. By the continuity of the polynoial hx, there exists soe 0 < x < 1, such that hx < 0. Thus, gx x r 1 x s hx < 0. However, gx > 0, for all x 0, 1. Therefore, h0 > 0. Siilarly, we have h1 > 0. gx Since gx > 0 for all x in 0, 1, we have hx x r > 0 for all x in 0, 1. In view of 1 x s the fact that h0 > 0 and h1 > 0, we have hx > 0, for all x in [0, 1]. Since hx is continuous on the closed interval [0, 1], it attains its iniu value h on [0, 1]. Clearly, h > 0. Let ɛ 2 h > 0. By Theore 1, there exists a positive integer M 2 n r s, such that for all integers d M 2 and 0, 1,..., d, we have γ,d h < ɛ 2, where γ 0,d, γ 1,d,..., γ d,d satisfy d that d hx γ,d b,d x

14 Thus, for all d M 2 and all 0, 1,..., d, γ,d > h ɛ 2 h h 0. d Cobining Equations 36 and37, we have gx x r 1 x s hx x r 1 x s d d+r+s γ d,d d+r+s +r d + r + s + r β,d+r+s b,d+r+s x, d γ,d b,d x x r 1 x s x +r 1 x d+s d+r r d d γ,d x 1 x d γ d r,d r b,d+r+s x d+r+s where β,d+r+s are the coefficients of the Bernstein polynoial of degree d + r + s of g and 0, for 0 < r and d + r < d + r + s β,d+r+s γ r,d r d > 0, for r d + r. d+r+s Thus, when d + r + s M 2 + r + s, we have for all 0, 1,...,, β, According to Equations 35 and 38, if we select an 0 ax{m 1, M 2 + r + s}, then gx can be expressed as a Bernstein polynoial of degree 0 : 0 gx β,0 b,0 x, with 0 β,0 1, for all 0, 1,..., 0. Therefore, g U. Theore 5 Let g be a polynoial of degree n apping the open interval 0, 1 into 0, 1 with g0 0 and g1 1. Then g U. Proof: Denote by r the ultiplicity of 0 as a root of gx. We can factorize gx as gx x r hx, 39 where hx is a polynoial satisfying h0 0. By a siilar reasoning as in the proof of Theore 4, we obtain h0 > 0. Since for all x in 0, 1], hx gx x r > 0, we have for all x in [0, 1], hx > 0. Since hx is continuous on the closed interval [0, 1], it attains its iniu value h on [0, 1]. Clearly, h > 0. Let ɛ 1 h > 0. By Theore 1, there exists a positive integer M 1 n r such that for all integers d M 1 and 0, 1,..., d, we have γ,d h < ɛ 1, where γ 0,d, γ 1,d,..., γ d,d satisfy d hx d γ,d b,d x

15 Thus, for all d M 1 and all 0, 1,..., d, γ,d > h ɛ 1 h h 0. d Cobining Equations 39 and 40, we have gx x r hx x r d γ d,d d+r +r d d d γ,d b,d x x r γ,d x 1 x d d+r x +r 1 x d γ d r,d d+r r b,d+r x β,d+r b,d+r x, d + r + r where β,d+r are the coefficients of the Bernstein polynoial of degree d + r of g and 0, for 0 < r β,d+r γ r,d r d > 0, for r d + r. d+r r d+r Thus, when d + r M 1 + r, we have for all 0, 1,...,, Let β, g x 1 gx. 42 Then g aps the open interval 0, 1 into 0, 1 with g 0 1, g 1 0. ultiplicity of 1 as a root of g x. Thus, we can factorize g x as Denote by s the g x 1 x s h x, 43 where h x is a polynoial satisfying that h 1 0. As in the proof of Theore 4, we obtain h 1 > 0. Since for all x in [0, 1, h x g x 1 x s > 0, we have for all x [0, 1], h x > 0. Since h x is continuous on the closed interval [0, 1], it attains its iniu value h on [0, 1]. Clearly, h > 0. Let ɛ 2 h > 0. By Theore 1, there exists a positive integer M 2 n s such that for all integers q M 2 and 0, 1,..., q, we have γ,q h < ɛ 2, where γ0,q q, γ 1,q,..., γ q,q satisfy h x Thus, for all q M 2 and all 0, 1,..., q, γ,q > h q q γ,q b,qx. 44 Cobining Equations 42, 43 and 44, we have ɛ 2 h h 0. gx 1 g x 1 1 x s h x 1 1 x s 1 1 x s q γ,q q x 1 x q 1 15 q γ,q b,qx q γ q,q q+s q + s x 1 x q+s.

16 Further using 4, we obtain q+s gx b,q+s x q γ,q q q+s q+s b,q+s x β,q+s b,q+s x, where the β,q+s s are the coefficients of the Bernstein polynoial of degree q + s of g: 1 γ,q q β,q+s q+s < 1, for 0 q 1, for q < q + s. Thus, when q + s M 2 + s, we have for all 0, 1,...,, β, According to Equations 41 and 45, if we select an 0 ax{m 1 + r, M 2 + s}, then gx can be expressed as a Bernstein polynoial of degree 0 : 0 gx β,0 b,0 x, with 0 β,0 1, for all 0, 1,..., 0, Therefore, g U. Lea 3 If a polynoial p is in the set U, then the polynoial 1 p is also in the set U. Proof: Since p is in the set U, there exist n 1 and 0 β 0,n, β 1,n,..., β n,n 1 such that By Equation 4, we have 1 px b,n x px β,n b,n x. β,n b,n x 1 β,n b,n x γ,n b,n x, where γ,n 1β,n satisfying 0 γ,n 1, for all 0, 1,..., n. Therefore, 1p is in the set U. Corollary 2 Let g be a polynoial of degree n apping the open interval 0, 1 into 0, 1 with 0 < g0, g1 1. Then g U. Proof: Let polynoial h 1 g. Then h aps 0, 1 into 0, 1 with 0 h0, h1 < 1. By Theore 4, h U. By Lea 3, g 1 h is also in the set U. Corollary 3 Let g be a polynoial of degree n apping the open interval 0, 1 into 0, 1 with g0 1 and g1 0. Then g U. 16

17 Proof: Let the polynoial h 1 g. Then h aps 0, 1 into 0, 1 with h0 0, h1 1. By Theore 5, h U. By Lea 3, g 1 h is also in the set U. Cobining Theore 4, Theore 5, Corollary 2 and Corollary 3, we show that V U. Theore 6 V U. Proof: Based on the definition of V, for any polynoial p V, we have one of following five cases. 1. The case where p 0 or p 1. In the proof of Theore 3, we have shown that 0 U and 1 U. Thus p U. 2. The case where p aps the open interval 0, 1 into 0, 1 with 0 p0, p1 < 1. By Theore 4, p U. 3. The case where p aps the open interval 0, 1 into 0, 1 with 0 < p0, p1 1. By Corollary 2, p U. 4. The case where p aps the open interval 0, 1 into 0, 1 with p0 0 and p1 1. By Theore 5, p U. 5. The case where p aps the open interval 0, 1 into 0, 1 with p0 1 and p1 0. By Corollary 3, p U. In suary, for any polynoial p V, we have p U. Thus, V U. Based on Theores 3 and 6, we have proved Theore 2: V U. 5 Discussion We are interested in Bernstein polynoials with coefficients in the unit interval because this concept has applications in the area of digital circuit design. Specifically, the concept is a atheatical prerequisite for a design ethodology that we have been advocating called stochastic logic [7 9]. We provide a brief overview of this application and point the reader to further sources. Stochastic logic ipleents Boolean function with inputs that are rando Boolean variables. A Boolean function f on n variables x 1, x 2,..., x n is a apping f : {0, 1} n {0, 1}. With stochastic logic, the variables x 1, x 2,..., x n are a set of independent rando Boolean variables, i.e., for 1 i n, x i has a certain probability p i 0 p i 1 of being one and a probability 1 p i of being zero. With rando Boolean variables as inputs, the output is also a rando Boolean variable: the function f has a certain probability p o of being one and a probability 1 p o of being zero. If ipleented by digital circuitry, stochastic logic can be viewed as coputation that transfors input probabilities into output probabilities [8]. Given an arbitrary Boolean function f and 17

18 a set of input probabilities p 1, p 2,..., p n that correspond to the probabilities of the input rando Boolean variables being one, the output probability p o is a function on p 1, p 2,..., p n. In fact, we have shown that the general for of the function is a ultivariate polynoial on variables p 1,..., p n with integer coefficients and with the degree of each variable no ore than one [9]. Exaple 3 Consider stochastic logic based on the Boolean function fx 1, x 2, x 3 x 1 x 2 x 1 x 3, where eans logical AND conjunction, eans logical OR disjunction, and eans logical negation. The Boolean function f evaluates to one if and only if the 3-tuple x 1, x 2, x 3 taes values fro the set {0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1}. The probability of the output being one is p o Prf 1 Pr x 1, x 2, x 3 : x 1, x 2, x 3 {0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1} Prx 1 0, x 2 0, x Prx 1 0, x 2 1, x Prx 1 1, x 2 1, x Prx 1 1, x 2 1, x 3 1. If x 1, x 2, and x 3 are independent rando Boolean variables with probability p 1, p 2, and p 3 of being one, respectively, then we obtain p o 1 p 1 1 p 2 p p 1 p 2 p 3 + p 1 p 2 1 p 3 + p 1 p 2 p 3 1 p 1 p 3 + p 1 p 2 p 1 p 2 + p 3 p 1 p 3, 46 which confirs that the function coputed by stochastic logic is a ultivariate polynoial on arguents p 1, p 2, and p 3 with integer coefficients and with the degree of each variable no ore than 1. In design probles, we encounter univariate polynoials that have real coefficients and degree greater than 1. Soeties it is possible to ipleent these by setting soe of the probabilities p i to be a coon variable x and the others to be constants. For exaple, if we set p 1 p 3 x and p in Equation 46, then we obtain the polynoial gx 1.75x x 2. With different underlying Boolean functions and different assignents of probability values, we can ipleent any different univariate polynoials. An interesting and yet practical question is: which univariate polynoials can be ipleented by stochastic logic? Define the set W to be the set of univariate polynoials that can be ipleented. We are interested in characterizing the set W. In [9] we showed that U W, i.e., if a polynoial can be expressed as a Bernstein polynoial with all coefficients in the unit interval, then the polynoial can be ipleented by stochastic logic. In this paper, we proved that V U. Thus, we have V W. Further, in [9] we showed that W V, i.e., if a polynoial can be ipleented by stochastic logic, then it is either identically equal to 0 or equal to 1, or it aps the open interval 0, 1 into the open interval 0, 1 and aps the points 0 and 1 into the closed interval [0, 1]. Therefore, we conclude that W V, i.e., a polynoial can be ipleented by stochastic logic if and only if it is either identically equal to 0 or equal to 1, or it aps the open interval 0, 1 into the open interval 0, 1 and aps the points 0 and 1 into the closed interval [0, 1]. This necessary and sufficient conditions allows us to answer the question of whether any given polynoial can be ipleented by stochastic logic. Based on the atheatics, we have proposed a constructive design ethod [9]. An overview of the ethod and its applications in circuit design will appear in a forthcoing Research Highlights article in Counications of the ACM [10]. 18

19 References [1] P. J. Davis, Interpolation and Approxiation. Blaisdell Publishing Copany, [2] G. Lorentz, Bernstein Polynoials. University of Toronto Press, [3] J. Berchtold and A. Bowyer, Robust arithetic for ultivariate Bernstein-for polynoials, Coputer-Aided Design, vol. 32, no. 11, pp , [4] S. N. Bernstein, Déonstration du théorèe de Weierstrass fondée sur le calcul des probabilités, Counications of the Kharov Matheatical Society, vol. 13, pp. 1 2, [5] K. M. Levasseur, A probabilistic proof of the Weierstrass approxiation theore, The Aerican Matheatical Monthly, vol. 91, no. 4, pp , [6] R. Faroui and V. Rajan, On the nuerical condition of polynoials in Bernstein for, Coputer Aided Geoetric Design, vol. 4, no. 3, pp , [7] B. Gaines, Stochastic coputing systes, in Advances in Inforation Systes Science. Plenu, 1969, vol. 2, ch. 2, pp [8] B. Brown and H. Card, Stochastic neural coputation I: Coputational eleents, IEEE Transactions on Coputers, vol. 50, no. 9, pp , [9] W. Qian and M. D. Riedel, The synthesis of robust polynoial arithetic with stochastic logic, in Design Autoation Conference, 2008, pp [10], The synthesis of stochastic logic, Counications of the ACM to appear,