274 VERDI G. VERDIEV satisfying the condition (1.1) (v i+1? v i ) (v i? v i?1 ) < 0; i = 1; : : : ; N? 1; is called a sequence of extreme values (s. e

Size: px

Start display at page:

Download "274 VERDI G. VERDIEV satisfying the condition (1.1) (v i+1? v i ) (v i? v i?1 ) < 0; i = 1; : : : ; N? 1; is called a sequence of extreme values (s. e"

Sheena Watkins
5 years ago
Views:

1 ON ONE PROBLEM OF C. DAVIS Verdi G. Verdiev In this paper we present a rather general method of the proof of the existence of functions with prescribed extreme values. The method is based on the theory of extremal problems. 1. Introduction In 1956 C. Davis ([1], [2]) posed the following problem: For a real polynomial p() of degree N, denote the zeros of p 0 () (multiplicity counted) by i, i = 1; : : : ; N? 1. Assume that i are real, let 1 N?1. Now to what extent are the numbers p( i ) arbitrary? More precisely, give necessary sucient conditions on an (N? 1)-tuple of real numbers v 1, : : :, v N?1 in order that there exists a polynomial p() such that p 0 ( i ) = 0, p( i ) = v i, i = 1; : : : ; N? 1. C. Davis solved this problem in 1957 ([2]) for algebraic polynomials. Later many investigators (see [3]{[8]) gave other solutions of this problem solutions of an analogous setting for Chebyshev systems of polynomials splines. It should be pointed out that in [6], [7] more general results are obtained. These results are formulated below (see Theorem ). 2. Theorems on existence of an algebraic polynomial a spline with prescribed extreme values Let := [0; 1], m; N; n 2 N, T := ft i g n i=1, 0 t 1 < < t n 1. Denote by P N the family of all algebraic polynomials of order N by S n m the family of m-order polynomial splines of defect 1 on with nodes at the points t i 2 T: Denition. A sequence V = fv i g N i=0 ; v i 2 R; i = 0; 1; : : : ; N; 273

2 274 VERDI G. VERDIEV satisfying the condition (1.1) (v i+1? v i ) (v i? v i?1 ) < 0; i = 1; : : : ; N? 1; is called a sequence of extreme values (s. e. v.). Theorem 1.1. Let V = fv i g N i=0 be a s. e. v. Then there exists a unique algebraic polynomial bp() 2 P a unique system of points fb i g N i=0, 0 = b 0 < b 1 < < b N = 1; such that (1.2) (1.3) bp(b i ) = v i ; i = 0; 1; : : : ; N; bp 0 (b i ) = 0; j = 1; : : : ; N? 1: Theorem 1.2. Let m; n; N 2 N, m + n = N, let V = fv i g N i=0 be a s. e. v. Then there exists a unique spline bs() 2 Sm n a unique system of points fb i g N i=0 ; 0 = 0 b < 1 b < < N b = 1; such that (1.4) (1.5) (1.6) bs(b i ) = v i ; i = 0; 1; : : : ; N; bs 0 (b j ) = 0; j = 1; : : : ; N? 1; b i < t i < b i+m ; i = 1; : : : ; n: Denote by := 3. Setting of extremal problems n N?1 X o = ( 1 ; : : : ; N?1 ): i 0; i = 1 the (N? 1)-dimensional simplex. Put T := n = ( 1 ; : : : ; N?1 ) 2 : i=1 m+i X o j < t i < j ; t i 2 T :

3 ON ONE PROBLEM OF C. DAVIS 275 Let us introduce the following notation: 0 () := 0; i () := j ; 0 () := 0; i () := j ; i = 1; 2; : : : ; N: Given any s. e. v. V = fv i g N i=0 ; we construct a class of algebraic polynomials P a class of splines S as follows P := S := n? o p() 2 P N : p i () = v i ; 2 int ; v i 2 V ; n? s() 2 Sm n : s( i ) = vi ; 2 int T ; v i 2 V o : Let us consider the value p(2) (p() 2 P) as a function of a point = ( 1 ; : : : ; N?1 ) 2 int, that is, p(2) := p(2; ), the value s(2) (s() 2 S) as a function of a point = ( 1 ; : : : ; N?1 ) 2 int T, that is, s(2) := s(2; ). Now we can formulate the following extremal problems. Problem 3.1. p(2) := p(2; )! inf; p() 2 P; 2 int ; if v N?1 < v N ; p(2) := p(2; )! sup; p() 2 P; 2 int ; if v N?1 > v N : Problem 3.2. s(2) := s(2; )! inf; s() 2 S; 2 int T ; if v N?1 < v N ; s(2) := s(2; )! sup; s() 2 S; 2 int T ; if v N?1 > v N :

4 276 VERDI G. VERDIEV 4. Solution of extremal problems Theorem 4.1. There exists b = (b 1 ; : : : ; b N?1 ) 2 int such that (4.1) inf p(2) = inf p(2; ) = p(2; b) if v N?1 < v N ; p()2p 2 (4.2) sup p()2p p(2) = sup p(2; ) = p(2; b) if v n?1 > v n : 2 Theorem 4.2. There exists b = (b 1 ; : : : ; b n?1 ) 2 int T such that (4.3) inf s(2) = inf s(2; ) = s(2; b) if v n?1 < v n ; s()2s 2T (4.4) sup s(2) = sup s(2; ) = s(2; b) if v n?1 > v n : s()2s 2T The idea of proofs of Theorems is based on the Weierstrass compactness principle. Since p(2) = p(2; )? p() 2 P s(2) = s(2; )? s() 2 S are continuous functions on T, respectively, +1; if vn?1 < v n lim p(2; ) = int 3! 2@?1; if v n?1 > v n ; +1; if vn?1 < v n lim s(2; ) = int T 3! 2@T?1; if v n?1 > v n ; then by the Weierstrass theorem there exist points b 2 int b 2 int T such that identities (4.3) (4.4) are valid.

5 ON ONE PROBLEM OF C. DAVIS The scheme of proofs of Theorems Beginning with the points b = (b 1 ; : : : ; b n?1 ) 2 int b = (b 1 ; : : : ; b n?1 ) 2 int T (which were found in Theorems ), we construct the points b = (b 0 ; b 1 ; : : : ; b n ) b = ( b 0 ; b 1 ; : : : ; b n ) so that b 0 := 0; b i = i ; 0 b := 0; i b := b i ; i = 1; : : : ; N: Then, we nd an interpolation polynomial bp () 2 R N an interpolation spline bs() 2 S n m satisfying the conditions: bp(b i ) = v i ; i = 0; 1; : : : ; N; bs(b i ) = v i ; i = 0; 1; : : : ; N: The polynomial bp() 2 P N the spline bs() 2 S n m are uniquely dened they satisfy the assumptions of Theorems Generalizations The represented method enables to solve various generalizations of the Davis problem in many directions. References [1] C. Davis, Problem 4714, Amer. Math. Monthly 63 (1956), no. 10, 729. [2] C. Davis, Extrema of a polynomial, Amer. Math. Monthly 64 (1957), no. 9, 679{680. [3] J. Mucielski S. Paszkowski, A generalization of Tschebyshe polynomials, Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys. 8 (1960), no. 7, 433{438. [4] W. J. Kammerer, Polynomial approximations of nitely oscillating functions, Math. Comp. 15 (1961), no. 74, 115{119. [5] J. Kudela, Construction des certains polyn^omes extremaux, Prace Matematyczne LXXVII (1963), no. 9, 31{36. [6] V. S. Videnskii, The existence uniqueness of a solution of an interpolation problem, Sb. Nauchn. Tr. Leningr. Mech. Inst. (1965), no. 50, 29{41. [7] S. H. Fitzgerald L. L. Shumaker, A dierential equation approach to interpolation at extremal points, J. Analyse Math. 22 (1969), 117{139. [8] M. B. Korobkova, A theorem on the existence of a spline-polynomial with a given sequence of extrema, Mat. Zametki 11 (1972), 251{258.

1 Solutions to selected problems

1 Solutions to selected problems 1. Let A B R n. Show that int A int B but in general bd A bd B. Solution. Let x int A. Then there is ɛ > 0 such that B ɛ (x) A B. This shows x int B. If A = [0, 1] and