6 CHAPTER 1 . UNIFOR M DISTRIBUTIO N VAALER'S LEMMA . If 0 x 1/ 2 then s(x) VK( X ) B K (x), while if 1/2 x 1 then VK(X) s{x) BK(X). IfT{x) is a trigonometric polynomial of degree K such that T(x) s(x) for all x then f T T(x)dx 2 (K+I) W ^ equality if and only ifT(x) = B^(x). We postpon e th e proo f o f thi s t o th e nex t section , an d conside r her e th e applications o f this t o unifor m distribution . Let 3 = [a , (3} b e a n ar c o f T, wit h a (3 a + l. Fro m Vaaler' s Lemm a w e know tha t —BK(—X) s(x) Bx(x) fo r al l x. Sinc e X3 {x) = (3 - a + s(x - (3) + s(a - x) except whe n x coincide s wit h on e o f the endpoint s o f [J , we pu t (21+) 5 + (x) =P-a + BK(x - p) + B K {a - x) and (21") SK(X) = f3 - a - B K {(3 -x)- B K (x - a). These ar e the Selberg polynomials. I t i s at onc e evident tha t S^(x) i s a trigono - metric polynomia l o f degree a t mos t K, tha t S^(x) x0(x) S~K( X ) r a ^ x - and tha t J T S^(x) dx = (3 a ± ^ p j Hence N n = l N K n = l -K On invertin g th e orde r o f summation w e see that thi s i s K = Y,S^{k)U N {-k). -K Since U N {0) = N an d S£(0 ) = (3 - a + j±^, i t follow s tha t D ( J V a , i 9 ) ^ T + £ S£(fc)£7 N (-k). 0|fc|K We now require an estimate for \S^(k)\. Sinc e S^(x) ha s been explicitly defined , one might argu e directly , bu t a serviceable estimat e i s obtained b y applying th e inequality |/(fc) | H/HL 1 to th e functio n f(x) = S^(x) - x 3 ix)- Henc e l^(fc)-x 3 (*)|||s^-x 3 ll^= l K + l
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