Turán, Pál

English

Turán Pál

University of Szeged

.30 On the theory of the mechanical quadrature. By P . TURAN in Budapest. § 1. In what follows I communicate a few simple remarks on the theory •of mechanical quadrature to which also L. FEJSR1) devoted an important paper. These remarks though they are rather naturally connected to the clas-sical theory of mechanical quadrature of GAUSS—JACOBI, seem not to be •observed so far. These reveal an interesting property of those polynomials .TI*2I(X) (n, / fixed integers) which minimize the integral +1 • ( 1 - 1 ) I 2 l { n n ) = \ \ n n { x ) r d x .among the polynomials . ( 1 . 2 ) « B ( x ) = x» + a a x - 1 + . . . + a , . This polynomial n*,2!(x) minimizes obviously at the same time also the •expression +* L . • 0 , 3 ) . . H2l{Kn)={\\n„{x)\»dx]1. § 2 . The classical theorem of GAUSS—JACOBI deals with quadrature-formulae of the type +i •(2.1) \ f ( x ) d x = ¿ f ( x v ) Xv, v=l •where xu ..., x„ are different, arbitrarily prescribed numbers and the A's are •independent of / . Putting co(x) = f / ( x - x v ) and / v ( x ) = ^ (v—1,2,. . r i ) v=l <*> \ X V ) (X — Xv) n •we have f ( x ) = ]?f(xv)lv(x), for all polynomials f(x) of degree <Ln — 1, V=1 ' ) L. FEJ£R, Mechanische Quadraturen mit positiven Cotesschen Zahlen, Math. Zeit-schrift, 37 (1933), pp. 2 8 7 - 3 1 0 . P. Túrán : Theory of the mechanical quadrature. 31 and consequently +i + i <2.2) j f ( x ) d x = Z f ( x v ) § l v ( x ) d x . - i _ - l This is a quadrature-formula of type (2.1) with the „Cotes-numbers" l v = i I l v { x ) d x {y— 1, 2 , . . . , n), valid for all polynomials of degree Now the above-mentioned theorem of G a u s s - J a c o b i solves the question how to choose the "fundamental points" X 1 ) X 2 , . . . , X „ in order that the •quadrature-formula (2.2) be valid for "the greatest possible set" of poly-nomials. They proved that formula (2. 2) is valid for all polynomials of degree <L2n—1 if and only if xlt..., x„ are the zeros of the /zth Legendre-polynomial < 2 . 3 ) P„(X) = [ (X 2 —1)"]W. § 3. Now we consider mechanical quadratures of the type +i <3! 1) \ f ( x ) d x = Z f ( * v ) W + Z f ' ( x » ) W , J *=1 v=l - I . where the quantities A™ are independent of / . The existence of such a •quadrature-formula follows immediately from the formula of F E J É R 2 ) n n <3. 2) f ( x ) = Z f(xv) kd*) + Z / ' ( * , ) 1», 1 (X) v=l v={ •valid for all polynomials f(x) of degree <,2n — 1, where — again with the iiotation of § 2 — <3.3) lv,o(x) = (\-K¥4(x-^xv))n(x), lvA{x) = {x-xv)ll{x). V cü (x v) ) Integrating (3.2) over [—1, + 1 ] we obtain a formula of type (3. 1) with +1 +1 <3.4) W=\lVi0(x)dx, Á y=\lv>i(x)dx, 1 valid for all polynomials fix) of degree < 2 n — 1 . § 4. The n zeros of the Legendre-polynomial P„(x) of (2. 3) are real.-This fact implies that the Gauss—Jacobi formula is applied in praxis, e .g . in meteorology. Hence it is reasonable to modify a little Gauss—Jacobi 's problem asking .for a real system (x1; x2,..., x„) for which the quadrature-formula (3. 1)—(3.4) is true for a greater variety of polynomials than those of degree ¡i2n—\. It is easy to show that by no choice this formula 2) Implicitly in his paper: Lagrangesche Interpolation und die zugehörigen konju-gierten Punkte, Math. Annalen, 106 (1932), pp. 1—56. Our notation differs from his one; this •change is motivated by the considerations of § 8. '32 P. Túrán (3. 1)—(3. 4) can be made precise even to all polynomials of degree <L2n. The validity of formula (3. 1) for a class A of polynomials means namely that for any fx(x) and /2(x) of the class A, for which (4.1) Mxv)=Mxv), f[(xv)=f:Á(xv) (v=\,2,...,n) we have +i . . • (4 .2) 5(A(x)-Mx))dx= 0. - • But it follows from (4. 1) that the polynomial f1(x)—f2(x) is divisible by w2(x); hence if A is the class of polynomials of degree < 2 n , we have f1(x)—f2(x) — C(o2(x), i . e . from (4.2) we would have for all c +i c j co2(x)dx — 0. But this is impossible for a proper polynomial co(x) with real zeros only. § 5 . Now we pass a step further. We consider quadrature-formulae of type +i (5 .1 ) [ f ( x ) d x = - Z f ( x v ) W + é f ( x J W ' + Z r ( x » ) W , J V— 1 V=l v—l -1 ' where the Á^'s are independent of / . It is again easy to show the existence of such a quadrature-formula (5. 1), valid for all f(x) polynomials of degree < L 3 n — 1 . Following namely the reasoning FEJÉR used to determine lvo(x) and lv\(x) in (3.2), we obtain8) the representation n n ti (5. 2) f{x) = Z f(xv) lv, o (x • + Z /' (xv) lv, i (x) + Z f" (xv) lv, 2 (x) V—l v—l , v=l valid for all / ( x ) of degree <3n — \. Here we have with the notation of § 2 lv,o(x) = ¡1 -3i:(xv)(x-xv) + ^-[5(K(xv))2-^'(xv)] (x—xv)2l ll(x), (5.3) ' - ' L, 1 (x) = (x — xv) [ 1 — 3 /; (xv) (x — x„)] 11 (x), lV: 2 (x) = y (x—xvf 11 (x), i .e . we obtain (5. 1) with +i. +i +i (5 .4) 4°» = \lr,o(x) dx, \hA{x) dx, ^=\lVi2(x) dx. -i -i -i § 6. Now we raise again the question to determine those systems (XJ , . . . , x„) of n different points for which the quadrature-formula (5. 1)—(5. 4) is valid for all polynomials of degree <L4n — 1. If B denotes this class of polynomials 3) The same formula was established also by Mr. I. RAISZ in an unpublished paper. P. Túrán: Theory of the mechanical quadrature. 47 and f^x), / 2 (x) denote any two members of the class B with (6 .1 ) A(xv) =f2(xv), M =/2'(xv), (xv) = /2" (x„) (v — 1, 2,.. , n), then we must have • 4 - 1 . (6.2) . \V,(x)-Mx))dx--=0. . . . . . . Fixing fx{x) in B and choosing ( 6 . 3 ) . : / Ï W = / I ( x ) + V ( x ) / 2 ( x ) where h(x) is an arbitrary polynomial of degree i i /z — 1, / 2 (x) belongs ob-viously to the class B and satisfies (6. 1); hence by (6.2) we must have ' , . • 41 (6. 4) j OJ'! (x) // (x) dx - 0 for all polynomials h(x) of degree ^L/z — 1. Since any. two polynomials with the property (6. 1) fulfill the relation (6.3), the condition (6. 4) is necessary and sufficient to the validity of the quadratureTformula (5. 1)—(5.4). § 7 . Now we have to determine, whether or not there is an a>(x) with the "higher orthogonality-property." (6 .4) . We suppose that such an <u(x) exists. We show first that all the zeros of co(x) lie in the interior of the interval [—1, + 1 ] and are simple. To prove this by a classical argument we remark that from (6. 4) obviously +i - . (7 .1.) ' j w 3 ( x ) x » i / x = 0 ( p — 0, 1 , . . .', n— 1). ' — i If <o(x) would have in the interior of [—I , + 1 ] only k<n s ign-changing places, say — 1 < ^ < f 2 < . . . < ^ < + 1 , then we would have '!: k >•-0 _ !• 1 in [—! , + 1 ] and hence • +i • ' i • ' • ' • Jfc k n ' ' «*(*) r i ( x - î v ) d x = Z cv co*(x) xvdx = 0 V=1 •• v=0 J - 1 - 1 owing to ( 7 . 1 ) ; an obvious contradiction. Hence k=n and the assertion concerning the zeros, of (o(x) is proved. This implies of course that the coefficients of w(x )a re all real too. Since any polynomiarreM(x) = x , 1- l- . . . + may be written in the form jr„(x)i=<y(x) + /z(x) with an h{x) of degree < n — 1 we have • -M +i +i +i J = { | ^ ( x ) | 4 û f x — J | w ( x ) | 4 i / x . = i [ w ( x ) + 7 z ( x ) ] 2 [ c l > ( x ) + h ( x ) ] 2 d x — j | a ( x ) | 4 i /x . - i - i • - i - l . where h(x) denotes that polynomial, whose coefficients are conjugate-complex ' 3 4 P. Túrán to those of h(x). Hence . • • +1 i J = 2 f (x) (h (.x) + h(x)) dx + • f [ | h (x) | 2 + w (x ) (h(x) + h(x) )]2 dx + ( 7 , 2 ) . - . • + ] . . . -j- 2 J |7J (X) | 2 CO2 (x) dx. . But the first integral in (7. 2) is 0 owing to (6. 4) and hence we have ¿ / ¡ > 0 ; equality only in the case h(x)^Q. Hence if a polynomial <o(x) with property (6 .4) exists, then it minimizes the integral f4(n„) of (1. 1). But the existence and uniqueness of a solution of this extremal-problem was proved by JACKSON 4 ) -Hence we proved the following' T h e o r e m I. Among the quadrature-formulae (5. 1) valid for all poly-nomials f(x) of degree ¿3n— 1 there is exactly one choice of ( x 1 ; . . . , x„) for which the formula is valid for all polynomials of degree <14/2—1. This ( x , , . . x „ ) - s y s t e m consists of the n real distinct zeros in the interior of [ — i , 1] of that polynomial n* 4(x) — xn +.... which minimizes the integral I4{nn) of (1 .1 ) in the class (1.2). § 8 . The generalisation of the quadrature-formula (5.1) is immediate. Given any system of n distinct points ( x l r . . . , x„) and n integers mx,m2,...,mn, HERMITE5) proved the existence and uniqueness of polynomials lvo(x', l v \ ( x ) , . . . , lv>mv.x{x) (v = \ , 2 n ) , . each of degree ¿ (m{- \ -m 2 + . . . + m„—1), such that W:(x,u) — 0 for all v,f i , k ,h (1 ¿ v ^ n , 1 0 ^ k < m v — 1, 1), except for v — /i and k — h \ in the latter case Then we have the representation (8. 1) / ( x ) == Z [/(*«) U (x) + / ' (x.) lv,(x) + . . . + / ' " » " 0 (x.) /„, „,„_, (x)], valid for all polynomials of degree f j (mx + . . . -f mn— 1), for the difference of the expressions on both sides of (8. 1) is divisible by (x—Xj)'"1 (x —x2)m2. .(x—x„)""\ i . e . identical zero. Hence integrating (8.1) over [—1, +.1] we obtain the quadrature-formula of L. TSCHAKALOFF") 4) D. JACKSON, On functions of closest approximation, Transactions American Math. Society, 22 (1921), pp. 1 1 7 - 1 2 8 . 6) CH: HERMITE, Sur la formule d'interpolation de Lagrange, Journal für die reine und angewandte Math., 84 (1878), pp.70—79. . • L. TSCHAKALOFF, Über eine allgemeine Quadraturformel, ' Comptes Rendus de l'Acad. Bulgare des Sciences, 1 (1948), pp:9—12. The point of his paper is a method for computation of the m ' s . • ; Theory of the mechanical quadrature . 35 +1 f(x)dx= Z [/(**) £)+f'(xv)ti) + .. v = l ' • - 1 . . . . valid for all polynomials ' of degree . 1), which was the start ing point of these investigations. . § 9 . Specializing .m1 = m2 — ..-. .— m„ = k we obtain the quadrature-formula " <9. I) f f ( x ) dx = ¿ 1 f(xv) +/' ixv) W + . . . +/*"» ixv) iVX J v = l . • - 1 • . valid for all polynomials fix) of degree <kn — 1. In this case the functions iv j ix) of § 8 can explicitly be represented following FEJ£R'S procedure2) and s o the quantities Asking by which choice, of the xv's the formula (9; 1) will be exact for all polynomials f i x ) of degree <;(A:-)- 1) n — 1 we obtain similarly that there is no real system . ., x„) if -A: is even, and for odd A: if and only if Xt,..., x„ are the zeros of the minimizing polynomial ?r* i+1(x) of § 1. § 10. Is. this result compatible with the theorem of Gauss—Jacobi, •explained in § 2 which corresponds, to the special .case k— 1 ? It is a well-iknown property of the nth Legendre-polynomial (2. 3) that, when properly normalized, it minimizes the integral : /2(TO of (1. 1). .Hence our results con-stitute a generalization of GAUSS—JACOBI'S theorem. § 1 1 . All these considerations can be applied to the theory of mechanical •quadrature of MEHLER—CHRISTOFFEL—CHEBYSHEV—STIELTJES, where a weight-function is permitted, i. e.. of quadrature-formulae of the type +i • ••(11.1) \hx)p{x)dx='jtf{xv)№' J • . v = l -1 where the Xf's are independent of f i x ) but dependent .in. general on p(x). T h e theorems so obtained will not be formulated explicitly except in the case . ( U . 2 ) ' = • In this case — as S . BERNSTEIN discovered7) — the Chebyshev-polynomial T„ix) with . ' •(11.3) . T„ (cos ^ - ^ L - c o s minimizes rail functionals 7) S. BERNSTEIN, Sur les polynomes orthogonaux relatifs à un segment fini, Journal .de Math., (9) 9 (1930), pp. 1 2 7 - 1 7 7 et (9) 10 (1931), pp. 2 1 9 - 2 8 6 . 36 P : Túrán +1 0 1 - 4 ) J l { n n ) = ^ ^ M t d x (Ar^.l, fixed) . . -1 in the class (1.2) . . Hence we obtained the following new property of the zeros of T„(x): T h e o r e m II. Given an arbitrary odd integer k we can determine the numbers W so that quadrature-formula +i ' • • ¡f(x)dx = Z J v=l (11.5) - . / | c o s 2n 7t\Xv - f / lcos 2 n n\v + , , Ai-Dl 2v— 1 | -(¡t-i) 4 - . . . + / „ cos 2n 7t\Zv is valid for all polynomials f(x) of degree ^(k-{-\)n— 1. § 12. These results make desirable to find an explicit expression of the. extremal-polynomials n*ik(x) (for all / : ;>1) which minimize (1. 3) with A: instead of 21 in the class (1.2) or develop a similar asymptotical theory of them which exists8) in the case k=2. As to the explicit representation the following cases are only known to me. k= \.. The minimizing polynomial is the polynomial U„(x) with q , ( c o s f r ) = - L : ^ ( n + i ) » • •• .. 2" s i n ^ (Result of KORKINE and ZO'LOTAREFF.) k=2. The minimizing polynomial is the nth Legendre-polynomial. k= + oo. The minimizing polynomial is the polynomial Tn(x) of (11. 3). (Classical result of CHEBYSHEV.) . As to the asymptotical theory of these polynomials little is known. Among the four main questions of the theory, namely a) asymptotic behaviour on the segment f—1, + 1 ] , b) asymptotic, behaviour outside the segment [—1, +1] -c) asymptotic determination of the individual zeros, d) uniform distribution of the zeros, only the last one is in a somewhat satisfactory shape. As we have shown9), the zeros of n*,K(x) are "uniformly-distributed on the unit-circle" in the sense that writing them in the form (k and n fixed) xv = xVin = cos&V/n — cos&v (v—\,2 ,...,n) we have for ail 0 <a < ¡3¡gn • 8) See e. g. G. SZEGÖ, Orthogonal polynomials (New York, 1939). °) P. ERDÖS and P. TUUÀN, On (he uniformly dense distribution of certain sequen-ces of points, Annals of Math., 41 (1940), pp. 162 -173 . • P. Túrán: Theory of the mechanical quadrature . 37 •Z l - ^ - n < c(k)]jn logn . As to the question b) we obtained certain results, but — as Prof. G . SZEGŐ mentioned in a conversation — h e has found a sharper asymptotical formula f o r < t ( z ) outside the interval [—1, -f-1]. Essentially the same is quite recently announced by GERONIMUS 1 0 ) . § 13. Finally we return to the quadrature-formula (5.1). FEJÉR1) has shown the importance of the fact that the Cotes-numbers AV in (2. 1) are non-negative in some cases; The same advántáges can be derived for the quadrature-formula (5. 1) if the numbers AF, A™, AF are. non-negative. In what follows I shall show that if the are the zeros of'7i*4(x), then <13. 1) ; • /<;> > 0 (v - = 1,2 n). The corresponding questions for X® and A™ remain open. To prove the assertion (13. 1) we remark that from (5.3) and (5.4) . +i ' +i But [1— lv{x)]!(x—x„) is obviously a polynomial of degree ti — 2, i. e., from the. orthogonality-property (6.4), the last integral is Ö. Hence +i • = I (x-x„yn(x)dx> 0. Q. e. d. (Received August 8, 1949.) 10)'JA. L. GERONIMUS, On asymptot ic properties of polynomials deviating least from zero in the space LI, Doklady Akad. Nauk SSSR, 62. (1948), pp. 9—12. I know this paper only from its review in the Math. Reviews. 

On the theory of the mechanical quadrature

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On the theory of the mechanical quadrature

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