By A. I. Kostrikin, I. R. Shafarevich

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17) i s c o m p l e t e d as before, u s i n g d u a l i t y for p = 1, a n d u s i n g t h e R i e s z - T h o r i n i n t e r p o l a t i o n t h e o r e m for o t h e r values o f p. • W e n o t e t h e significant role p l a y e d i n t h e proofs o f b o t h o f t h e above t h e o r e m s b y t h e estimate o n t h e D i r i c h l e t kernel D m and its derivative. In this s i m p l e case, t h i s c o u l d be done b y direct c a l c u l a t i o n s . I n t h e case o f o r t h o g o n a l p o l y n o m i a l s o n t h e w h o l e line, t h e c o r r e s p o n d i n g estimates o n t h e C h r i s t o f f e l D a r b o u x k e r n e l a n d its d e r i v a t i v e w i l l be considerably m o r e difficult.

11) and the inequality constraint for convexity: Q (x) > 0 for 0 < x < a. 11). This works if aV" < V\. com by KAINAN UNIVERSITY on 02/15/15. For personal use only. (Vi - a V ) ( s - a ) . 2 0 4 2 If V i < aV2 then necessarily, > 0. We define Q" to be a constant in a neighborhood of 0 and a linear function i n the remainder of the interval up to a. The function Q is then defined by taking antiderivatives. We distinguish between two cases. If V i < a V < 2Vi then let, i n this proof only, 2 aV A:=Vi- 2 k:=V -aV 0 + 1 —V .

2. For integer n > 1, An(') = perrt i[(1)(P)]-2 f P2(t)w(t)dt. 4) The minimum is attained if and only if P(t) = a En k:01 (1,(pk)pk(t) for some constant a. ) (X) = pgri 1 [13/(4-2 f P2(t)w(t)dt, with the minimum attained if and only if P(t) = akKn(x,t) for some constant a. com by KAINAN UNIVERSITY on 01/30/15. For personal use only. 8) = An(4)-1 f P2(t)w(t)dt. Hence, 2(t)w(t)dt. 4). 4). ■ Next, we describe an extremal property of the largest zero Xn. 3. For integer n > 1, f xP2(x)w(x)dx PEll. 9) - where the maximum is attained if and only if P(x) = apn(x)(x — Xn)-1for some constant a.