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Be a system of functions in (a, b) such that 0 a f 0",(x) "(x) dw(x) _ f for m 4 n (m, n = 0, 1, ... ), (3'4) m=n} where the integral is taken in the Stieltjes sense (Stieltjes-Riemann or Stie)tjesa A,,,>0 for Lebesgue). The system {(5n} is then called orthogonal over (a, b) with respect to dm(x). If A,=A,= ... = 1, the system is orthonormal. The Fourier coefficients of any function f with respect to (0n) are 1 en = f (x) n(x) do)(x), n Jab (3'5) and the series c000 + c1g1 + ... is the Fourier series off.

We get G5[AB]<1. Now let a = {an} and b = {bn} be any two sequences such that S,[a] and 6,,[b] are positive and finite, and let us set A n = an/C,,,,[a], Bn = bn/CB,,[b] for all n. Then C,[A] = 6,[B] = 1, so that S[AB] < 1. nnbn f <',[a] G,[b]. 4) These inequalities are called Holder's inequalities. They are trivially true if 6,[a] = 0 or CS,[b] = 0. 4), summation being replaced by integration. 5) reduce to the familiar Schwarz inequalities. 2) if and only if a' = b'. 3) shows that the sign of equality holds there if and only if f A.

For A(x + 21r) - 0(x) = F(x + 2n) - F(x) - 2rrco = 0. 2) may be called a mass distribu- tion (of positive and negative masses, in general) on the circumference of the unit circle. If (a,,8) is an arc on this circumference and 0 <,8 - a < 27r, then F(f) - F(a) is, by definition, the mass situated on the semi-open are a < x

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