Blatt, H.P.

Grothmann, R.

Kovacheva, R.K.

Ukrains’kyi Matematychnyi Zhurnal

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

K O R O T K I P O V I D O M L E N N QUDC 517.5H.-P. Blatt, R. Grothmann (Kathol. Univ. Eichsta¨tt-Ingolstadt, Germany),R. K. Kovacheva (Inst. Math. and Inform. Bulg. Acad. Sci., Sofia, Bulgaria)SIGN CHANGES IN RATIONAL L1w-APPROXIMATIONZNAKOZMINY U RACIONAL\NOMU L1w-NABLYÛENNILet f ∈ L1w[−1, 1], let rn,m(f) be a best rational L1w-approximation for f with respect to real rationalfunctions of degree at most n in the numerator and of degree at most m in the denominator, let m = m(n), andlet limn→∞(n −m(n)) = ∞. Then we show that the counting measures of certain subsets of sign changesof f − rn,m(f) converge weakly to the equilibrium measure on [−1, 1] as n → ∞. Moreover, we provediscrepancy estimates between these counting measures and the equilibrium measure.Nexaj f ∈ L1w[−1, 1] i rn,m(f) — najkrawe L1w-nablyΩennq dlq f vidnosno dijsnyx racional\nyxfunkcij stepenq ne bil\ße niΩ n u çysel\nyku ta stepenq ne bil\ße niΩ m u znamennyku, m = m(n)i limn→∞(n−m(n)) =∞. U c\omu vypadku prodemonstrovano, wo liçyl\ni miry pevnyx pidmnoΩynznakozmin f − rn,m(f) slabko zbihagt\sq do rivnovaΩno] miry na [−1, 1] pry n→∞. TakoΩ dovedenoocinky vidxylennq cyx liçyl\nyx mir vid rivnovaΩno] miry.1. Introduction. Let w be a weight function on I = [−1, 1] positive a.e. on [−1, 1] in thesense of Lebesgue. Let L1w[−1, 1] denote the class of all real-valued measurable functionsf on [−1, 1] such that f(t)w(t) is Lebesgue-integrable on [−1, 1] and let ‖f‖1,w be theweighted L1-norm in L1w[−1, 1], i.e.,‖f‖1,w :=1∫−1|f(t)|w(t)dt =∫|f |w.LetRn,m ={r =pq: deg p ≤ n, deg q ≤ m}be the class of all real-valued rational functions with numerator in Pn and denominator inPm. Here, Pk denotes the class of all algebraic polynomials of degree not exceeding k.For given f ∈ L1w[−1, 1], denote by rn,m(f) a best L1w-approximation of f with respectto Rn,m.We say that a function g ∈ L1w[−1, 1] does not change its sign at x0 ∈ [−1, 1] if f > 0(or f < 0) a.e. in some neighborhood U of x0 in [−1, 1]. All other points of [−1, 1] arecalled sign changes of f .We denote by µ the equilibrium measure of [−1, 1].In [1], Kro´o and Peherstorfer have proved the following result. Let n,m ∈ N0, 0 ≤≤ m < n+ 1, and f ∈ L1w[−1, 1] with w(x) ≡ 1. If f − rn,m(f) has no sign change on(α, β) ⊂ [−1, 1], thenµ[α, β] =arccosα− arccosβπ≤ 1n−m+ 2c© H.-P. BLATT, R. GROTHMANN, R. K. KOVACHEVA, 2006ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2 283284 H.-P. BLATT, R. GROTHMANN, R. K. KOVACHEVAand this upper bound is, in general, best possible. Our aim is to make more precisestatements about the distribution of sign changes of the error function, and to generalizeto general weights.To formulate our results, we need the notation of counting measures. Let A be a finitepoint set of k points. Then we define the discrete measure νA that associates the mass1kto every point of A. For specified subsets A of the set of sign changes of f − rn,m(f),the normalized counting measure νA will be compared with the equilibrium measure µ of[−1, 1]. In other words, we are interested in upper bounds for the discrepancyD[νA − µ] := sup−1≤α≤β≤1∣∣(νA − µ)([α, β])∣∣between νA and µ. We obtain distribution estimates for the sign changes of f − rn,m(f)on [−1, 1] that generalize our results for polynomial approximation in [2].2. Distribution of sign changes. In the following, c, ci, i = 1, 2, . . . , will denotepositive constants independent of n. Let w be positive a.e. and measurable on [−1, 1],∫w = 1,and, for 0 < ε ≤ 1, let us defineϕ(w, ε) := inf∫Aw : A ⊂ [−1, 1], µ(A) ≥ ε .By εn(w), we denote the unique solution of the equationϕ(w, ε) = e−nε.Then εn(w) → 0 as n→∞ andεn(w) ≥ c1 log nn, n = 1, 2, . . .(cf. [3], Lemma 2.3). Concerning the rate of εn(w) as n → ∞, we have εn(w) == O(lognn)for Jacobi weights and εn(w) = O(n−1/(1+α)) for weights of type w(x) == e−|x|αwith α > 0.The best approximation r = rn,m(f) =p0q0is well-known to be characterized by thefact that, for every p ∈ Pn and q ∈ Pm, we have∫pq0 − qp0q20sgn(f − r)w ≤∫Z(f−r)|pq0 − qp0|q20w (1)(cf. [1], Lemma 1), where we denoteZ(g) := {x : g(x) = 0}for functions g ∈ L1w[−1, 1] andsgn g(x) =1 if g(x) > 0,0 if g(x) = 0,−1 if g(x) < 0.ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2SIGN CHANGES IN RATIONAL L1w-APPROXIMATION 285It can easily be seen from (1) that the error function f−r changes its sign at least n+m+1times on [−1, 1] if µ(Z(f − r)) = 0 and r /∈ Rn−1,m−1.The function ϕ ∈ L1w[−1, 1] is called sign function if ϕ2 = 1 on [−1, 1]. ϕ is said tobe orthogonal to Pn (written ϕ ⊥ Pn) if∫ϕPw = 0 for all P ∈ Pn.Two sign functions ϕ and ψ are called equivalent if ϕ = ψ or ϕ = −ψ a.e. on [−1, 1].Let r = rn(f) =p0q0with no common factors, then the defect of r is defined asdn,m = dn,m(r) = min(n− grad p0, m− grad q0).Then (1) is equivalent to1∫−1pq20sgn(f − r)w ≤∫Z(f−r)|p|q20w for all p ∈ Pn+m−dn,m , (2)i.e., 0 is the best approximation of f − r in the subspace q−20 Pn+m−dn,m .According to Proposition 20 of Cheney and Wulbert [4], there exists a sign functionψ such that ψ(x) = sgn(f − r)(x) for all x ∈ [−1, 1] \ Z(f − r) and∫pq20ψw = 0 for all p ∈ Pn+m−dn,m .Then the sign changes of f − r are identical with the sign changes of ψ. Therefore, ψand, consequently, f − r has at least n+m− dm,n sign changes in (−1, 1).Concerning the distribution of these sign changes, we are able to prove estimates fora subset of n−m+ dn,m + 1 sign changes. To this end, we use the weaker condition∫p sgn(f − r)w ≤∫Z(f−r)|p|w for all p ∈ Pn−m+dn,m , (3)which holds because of (2).Theorem 1. Let f ∈ L1w[−1, 1] and let r = rn,m(f) be a best L1w-approximation tof on Rn,m, m < n + 1. Then there exist n −m + dn,m + 1 sign changes of f − r atpoints−1 < ξ(n,m)0 < . . . < ξ(n,m)n−m+dn,m < 1 (4)such that the normalized counting measure νn,m of the these points satisfiesD[νn,m − µ] ≤ c√εn−m+dn,m(w),where c is an absolute constant, not depending on n, m or f .Corollary 1. Let m = m(n) and letn−m(n) →∞ as n→∞,then the normalized counting measures νn,m of the point sets in (4) converge weakly to µfor n→∞.For special weights w, the numbers εn(w) are well-known and one gets explicit dis-crepancy estimates between νn,m and µ.ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2286 H.-P. BLATT, R. GROTHMANN, R. K. KOVACHEVAAs a special example, we consider generalized Jacobi weights where, as in the poly-nomial case, the estimate of Theorem 1 can be sharpened.Theorem 2. Let −1 = t1 < t2 < . . . < tk = 1 be fixed points, a1, . . . , ak > −1 befixed numbers and w a weight function that satisfiesw(x) ≥ c1k∏i=1|x− ti|aiwith c1 > 0. Then the normalized counting measure νn,m of (4) satisfyD[νn,m − µ] ≤ c (log(n−m+ dn,m))2n−m+ dn,mfor all n ≥ m+ 2, where c is an absolute constant.3. Proofs. In the following, we use a lemma of Kro´o and Peherstorfer [5].Lemma 1 [5]. Let ϕ be a sign function, ϕ ⊥ Pn, and assume that ϕ has exactly n+1sign changes at the point y1, . . . , yn+1 withy0 = −1 < y1 < y2 < . . . < yn+1 < 1 = yn+2.If the sign function ψ is not equivalent to ϕ and ψ ⊥ Pn, then ψ has a sign change ineach interval (yi, yi+1), 0 ≤ i ≤ n+ 1.Denote by Un,w(x) the Chebychev polynomial of second kind with respect to w; i.e.,Un,w(x) is a monic polynomial in Pn and‖Un,w‖1,w = min{‖P‖1,w : P monic in Pn},Un,w is unique and characterized by exactly n sign changes (or zeros) at points−1 < y(n)1 < y(n)2 < . . . < y(n)n < 1.We denote by νn,w the normalized zero-counting measure of Un,w. Then for all n == 1, 2, . . . ,D[νn,w − µ] ≤ c√εn(w) (5)with some absolute constant c > 0 (Theorem 2 in [2]).Proof of Theorem 1. Letϕ(x) :=sgn Un−m+dn,m+1,w(x) for x with Un−m+1,w(x) = 0,1 elsewhere.Then ϕ satisfies the conditions of Lemma 1.Because of (3), 0 is a best L1w-approximation to f − r from the space Pn−m+dn,m .Thus, there exists a sign function ψ ⊥ Pn−m+dn,m such thatψ(x) = sgn (f − r)(x) for all x ∈ [−1, 1] \ Z(f − r)(cf. [4], Proposition 20).Then either ψ = ±ϕ a.e. on [−1, 1] or, due to Lemma 1, ψ has a sign change in eachinterval (y(n−m+dn,m+1)i , y(n−m+dn,m+1)i+1), 0 ≤ i ≤ n−m+ dn,m + 1,ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2SIGN CHANGES IN RATIONAL L1w-APPROXIMATION 287where−1 < y(n−m+dn,m+1)1 < . . . < y(n−m+dn,m+1)n−m+dn,m+1 < 1are the zeros of Un−m+dn,m+1,w(x) andy(n−m+dn,m+1)0 = −1, y(n−m+dn,m+1)n−m+dn,m+2 = 1.Finally, we get from (5) the desired discrepancy result.Corollary 1 is an immediate consequence of Theorem 1. Theorem 2 follows from aresult in [2], stating that, for generalized Jacobi weights,D[νn,w − µ] ≤ c (logn)2nfor all n = 2, 3, . . . , where c is again an absolute constant.1. Kro´o A., Peherstorfer Fr. Interpolatory properties of best rationalL1-approximations // Constr. Approxim.– 1988. – 4. – P. 97 – 106.2. Blatt H.-P., Grothmann R., Kovacheva R. K. On sign changes in weighted polynomial L1-approximation// Acta math. hung. – 2002.3. Blatt H.-P. A discrepancy lemma for oscillating polynomials and sign changes of the error function of bestapproximations // Ann. Numer. Math. – 1997. – 4. – P. 55 – 66.4. Cheney E. W., Wulbert D. E. Existence and unicity of best approximation // Math. scand. – 1969. – 24. –P. 113 – 140.5. Kro´o A., Peherstorfer Fr. Interpolatory properties of best L1-approximation // Math. Z. – 1987. – 96. –P. 249 – 255.Received 16.05.2005ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2

Sign changes in rational L¹w-approximation

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Sign changes in rational L¹w-approximation

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)