2. QUANTITATIV E RELATION S 5 for al l x, an d [ T + (x)~T'(x)dxe. To pu t thi s i n quantitativ e for m w e nee d t o kno w ho w smal l w e ma y tak e e as a functio n o f th e degree s o f th e trigonometri c polynomial s T^(x). Sinc e th e characteristic functio n o f an interva l ca n b e expressed i n terms o f the saw-toot h function s(x) , w e begin b y considerin g one-side d approximation s t o s(x). We recall tha t Fejer' s kerne l AK{X) i s given b y the formula e Then Vaaler's polynomial i s (17) fc=i + 2 (K -hi) s i n 2 7 r ( ^ + 1 ) 2 ~ Ax-f-i(^)sin27rx . Since AK+I(X) i s a trigonometri c polynomia l o f degree if , th e su m i s of degre e at mos t K. Th e las t tw o term s ar e trigonometri c polynomial s o f degre e K + 1, but th e coefficient s o f e(±(K - f l)x) cancel , s o that th e combine d contributio n of the las t tw o terms i s a trigonometric polynomia l o f degree a t mos t K. B y a n elementary calculatio n i t ma y b e shown tha t 1 K (18) V K (x) = - J ^ / ( * / ( A ' + l))sin27rA a : k=i where f(u) —(1 u) cot nu jix, an d on e coul d tak e thi s t o b e th e definitio n of VK, but th e mos t interestin g propertie s o f VK{X) follow mor e readil y fro m th e definition (17) . The function VK(X) provide s a good approximatio n t o the saw-tooth functio n s(x). Lik e s(x), th e functio n VK(X) i s odd. Indeed , i t ma y b e show n tha t (19) V K (x) = s(x) + O (min(l , ^ 3 ) ) Here \\x\\ denotes th e natura l metri c o n T , namel y ||:r| | = min nG ^ \x n\. Whil e VK(X) i s presumably no t quit e th e bes t L 1 approximatio n t o s(x), i t lend s itsel f to constructin g th e bes t majorant , whic h i s given by th e Beurling polynomial (20) B K (x) = V K (x) + 2{K \l)AK+i(x). The relation s amon g thes e function s i s describe d i n th e followin g fundamenta l result, whic h w e refer t o a s
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