Simplification and Strengthening of Weyl s Definition of Asymptotic Equal Distribution of Two Families of Finite Sets

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1 Simplifictio d Stregtheig of Weyl s Defiitio of Asymptotic Equl Distributio of Two Fmilies of Fiite Sets Willim F. Trech Triity Uiversity, S Atoio, Texs, USA; wtrech@triity.edu Milig ddress: 95 Pie Le, Woodld Prk, CO USA Published i CUBO, A MATHEMATICAL JOURNAL, Vol. 6, No. 3 (47 54), October 2004 Abstrct Suppose tht < < b <, u u 2 u b, d v v 2 v b,. We simplify d streghthe Weyl s defiitio of symptotic equl distributio of U = {{u i} } d V = {{v i} } by showig tht the followig sttemets re equivlet: (i) (F(ui) F(vi)) = 0 for ll F C[, b]. (ii) ui vi = 0. (iii) F(ui) F(vi) = 0 for ll F C[, b]. Itroductio The followig defiitio is due to H. Weyl [, p. 62]. Defiitio. Suppose tht < < b <, {u i } [, b], d {v i} [, b],. The U = {{u i } } d V = {{v i } } re symptoticlly eqully distributed if (F(u i ) F(v i )) = 0, F C[, b].

2 2 Willim F. Trech We preset simple ecessry d sufficiet coditio for symptotic equl distributio d poit out tht stroger coclusio is implicit i Defiitio.. Without loss of geerlity, we my ssume tht u u 2 u b, v v 2 v b,. () Theorem.2 If () holds the the followig ssertios re equivlet: (F(u i ) F(v i )) = 0, F C[, b]; (2) u i v i = 0; (3) F(u i ) F(v i ) = 0, F C[, b]. (4) Obviously, (4) implies (2). The proof tht (3) implies (4) (Sectio 2) is strightforwrd. Our mi effort is devoted to showig tht (2) implies (3). Theorem.2 is specil cse of more geerl results i [4] cocerig symptotic reltioships betwee the eigevlues or sigulr vlues of two ifiite sequeces of mtrices {A } = d {B } = relted i some wy tht it is ot ecessry to specify here. However, [4] is quite techicl d of iterest mily to the lier lgebr commuity. We thik it is worthwhile to preset Theorem.2 i this expository rticle ddressed to lrger udiece. Give Theorem.2, we suggest replcig Defiitio. by the followig simpler defiitio while berig i mid tht (3) implies (4). Defiitio.3 U = {{u i } } d V = {{v i } } re symptoticlly eqully distributed if () holds d u i v i = 0. 2 Proof tht (3) implies (4) Suppose tht F C[, b] d ɛ > 0. By the Weierstrss pproximtio theorem, there is polyomil P such tht By the trigle iequlity, F(x) P(x) < ɛ/2, x b. F(u i ) F(v i ) F(u i ) P(u i ) + P(u i ) P(v i ) + P(v i ) F(v i ) < P(u i ) P(v i ) + ɛ. (5)

3 Asymptotic Equl Distributio 3 Let M = mx x b P (x). By the me vlue theorem, This d (5) imply tht From this d (3), P(u i ) P(v i ) M u i v i. F(u i ) F(v i ) < ɛ + M sup u i v i. F(u i ) F(v i ) ɛ. Sice ɛ is rbitrry, this implies (4). 3 Four Required Lemms We eed the followig lemms to show tht (2) implies (3). Lemm 3. (Helly s First Theorem) Let {φ m } m= be ifiite sequece of fuctios o [, b] d suppose tht there is fiite umber K such tht φ m (x) K, x b, d V b (φ m ) K, m. The there is subsequece of {φ m } m= tht coverges t every poit of [, b] to fuctio of bouded vritio o [, b]. Lemm 3.2 (Helly s Secod Theorem) Let {φ m } m= be ifiite sequece of fuctios o [, b] such tht V b (φ m ) K <, m, d The V b (φ) K d m φ m(x) = φ(x), x b. m F(x)dφ m (x) = F(x)dφ(x), F C[, b]. Lemm 3.3 Suppose tht φ() = φ(b) = 0, φ is of bouded vritio o [, b], d F(x)dφ(x) = 0, F C[, b]. The φ(x) = 0 t ll poits of cotiuity of φ. Thus, φ(x) 0 for t most coutbly my vlues of x. For proofs of Lemms , see [2, p. 222], [2, p. 233], d [3, p. ]. The followig lemm is lso kow [5, p. 08], but we iclude its short proof for coveiece.

4 4 Willim F. Trech Lemm 3.4 Suppose tht x x 2 x d y y 2 y. Let {l, l 2,...l } be permuttio of {, 2,..., } d defie The Q(l, l 2,..., l ) = (x i y li ) 2. Q(l, l 2,..., l ) Q(, 2,..., ). (6) Proof The proof is by iductio. Let P be the stted propositio. P is trivil. Suppose tht > d P is true. If l =, P implies P. If l = s <, choose r so tht l r =, d defie l i if i r d i, l i = s if i = r, if i =. The Q(l, l 2,..., l ) Q(l, l 2,..., l ) = (x y s ) 2 + (x r y ) 2 Sice l =, P implies tht Q(l, l 2,..., l ) Q(, 2,..., ). (x y ) 2 (x r y s ) 2 Therefore (7) implies (6), which completes the iductio. 4 Proof tht (2) implies (3) We will show tht if (2) holds the From Schwrz s iequlity, = 2(x x r )(y y s ) 0. (7) (u i v i ) 2 = 0. (8) u i v i ( ) /2 (u i v i ) 2, so (8) implies (3). The proof of (8) is by cotrdictio. If (8) is flse, there is ɛ 0 > 0 d icresig sequece {l k } k= of positive itegers such tht l k l k (u ilk v ilk ) 2 ɛ 0, k. (9)

5 Asymptotic Equl Distributio 5 However, we will show tht if (2) holds, the y icresig sequece {l k } k= of positive itegers hs subsequece { k } k= such tht k k k (u ik v ik ) 2 = 0, (0) cotrdictig (9). If S is set, let crd S be the crdility of S. For x b, let ν (x;u) = crd { i u i < x } d ν (x;v) = crd { i v i < x }. () Defie d If F C[, b], the ρ (x;u) = ρ (x;v) = { ν (x;u)/, x < b,, x = b, { ν (x;v)/, x < b,, x = b. (2) (3) d F(u i ) = F(v i ) = F(x)dρ (x;u) (4) F(x)dρ (x;v) (5) [2, p. 23]. The sequeces {ρ ( ;U)} = d {ρ ( ;V)} = both stisfy the hypotheses of Lemm 3.. Therefore, there is subsequece {m k } k= of {l k} k= such tht γ(x;u) := ρ m k (x;u) (6) k exists for x b, d there is subsequece { k } k= of {m k} k= such tht exists for x b. Clerly, (6) implies tht γ(x;v) := k ρ k (x;v) (7) γ(x;u) = k ρ k (x;u), x b. (8) From () (3), γ( ; U) d γ( ; V) re odecresig, γ(;u) = γ(;v) = 0, d γ(b;u) = γ(b;v) =. (9) Therefore, (7), (8), d Lemm 3.2 imply tht k F(x)dρ k (x;u) = F(x)dγ(x;U), F C[, b], (20)

6 6 Willim F. Trech d k F(x)dρ k (x;v) = Now (2), (4), (5) (20), d (2) imply tht F(x)dγ(x;V), F C[, b]. (2) F(x)dγ(x;U) = F(x)dγ(x;V), F C[, b]. This, (9), d Lemm 3.3 with φ = γ( ;U) γ( ;V) imply tht γ(x;u) = γ(x;v) except for t most coutbly my vlues of x i [, b]. If ɛ > 0, choose 0,,..., m so tht d Let Defie d The d = 0 < < < m = b, j j < ɛ, j m, (22) γ( j ;U) = γ( j ;V), j m. (23) I j = [ j, j ), j m, I m = [ m, m ]. ν k ( ;U), j =, U jk = ν k ( j ;U) ν k ( j ;U), 2 j m, k ν k ( m ;U), j = m, ν k ( ;V), j =, V jk = ν k ( j ;V) ν k ( j ;V), 2 j m, k ν k ( m ;V), j = m. U jk = crd { i uik I j }, Vjk = crd { i vik I j }, from (2), (3), (7), (8), d (23). Sice U jk V jk = 0, j m, (24) k k d mi(u jk, V jk ) = U jk + V jk U jk V jk, 2 m m U jk = V jk = k, j= j=

7 Asymptotic Equl Distributio 7 it follows tht where From (24), m mi(u jk, V jk ) = k r k, (25) j= r k = 2 m U jk V jk. j= r k = 0. (26) k k From (22) d (25), there is permuttio τ k of {,..., k } such tht for k r k vlues of i; hece k Now Lemm 3.4 implies tht Hece, from (26), (u ik v τk(i), k ) 2 < ɛ (u ik v τk(i), k ) 2 < k ɛ + r k (b ) 2. k (u ik v ik ) 2 < k ɛ + r k (b ) 2. sup k k k (u ik v ik ) 2 ɛ. Sice ɛ is rbitrry, this implies (0), which completes the proof. 5 Ackowledgmet I thk Professor Polo Tilli for suggestio tht ebled me to complete the proof i Sectio 4. Refereces [] U. Greder, G. Szegö, Toeplitz Forms d Their Applictios, Uiv. of Clifori Press, Berkeley d Los Ageles, 958. [2] I. P. Ntso, Theory of Fuctios of Rel Vrible, Frederick Ugr Publishig Co., New York, 955. [3] F. Riesz d B. Sz. Ngy, Fuctiol Alysis, Frederick Ugr Publishig Co., New York, 955.

8 8 Willim F. Trech [4] W. F. Trech, Absolute equl distributio of fmilies of fiite sets, Lier Algebr Appl. 367 (2003), [5] J. H. Wilkiso, The Algebric Eigevlue Problem, Clredo Press, Oxford, 965.