# Fourier Analysis in Probability Theory by Tatsuo Kawata PDF

By Tatsuo Kawata

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A special case is the following discrete case: If p and q are nonnegative numbers such that (l//>) + (l/#) = 1, then for any {ak}y {bk}f k = 1, 2,. . such that Σ I ** \v < °°> k=l Σ I ** \q < °°> fc=l then Σ I akbk I < ° ° Ar=l and oo / o o Σ\αφ,\ < Σ Κ Ν \l/p/ oo \l/q ( ΣI**Ie) · The equality holds if and only if \ ak\v = c \bk\q independent of k. MINKOWSKI INEQUALITY. / ( ω ) + #(ω) e Z> and Γr , l 1/p (6) for some constant c If p > 0 and / ( ω ) e Z>, #(ω) e Z>, then Γr ηΐ/ρ [J0Ι/(ω) + *(ω)Ι"4κ] =s[jJ/Hlp4"J r r l1^ for p^l, (7) +[Jflk(ft')lp^J 26 I.

Finally we mention a theorem due to Polya and Szegö ([1], p. 147). 4. (i) there are positive constants A and B such that \f(z)\ (ii) 0, \f{ir)\<\, (iii) Let f(z) be analytic in Re z ^ 0. Suppose that \f(-ir)\^l; limsup[log|/(r)|/r] ^ 0 . r-H-oo Then in the entire half-plane Re z ^ 0, |/(#) 1 ^ 1 . 16. Inner Product Space Consider a nonempty space X of abstract elements (or objects). An operation called addition is defined for any pair of elements x, y> of X and is denoted by x + y.

4. Suppose that a periodic function f(x) is of bounded variation. Let the total variation of f(x) over [—π, π] be T(—ττ, π). Then (9) \cn\£(4n)-*T(-n,n), I an | < (2η)^Τ(-π, π), \bn\ ^ (2η)"1Γ(-π, π). (10) Proof. From (5), | cn | < (4π)-! Γ |/(x + (π/η)) —/(*) | <£r. J —π For a periodic function, this integral is equal to (4π)" 1 Γ J —n \f[x + (htjn)] —f{x + [ ( Α - 1 ) π / η ] } | dx for any integer k. Adding these integrals for k = —n-\- 1,. , 0, 1 , . . , w and dividing the sum by 2«, we obtain | cn | =S (8«π) Γ <{Τ(—π,π)βηπ) g | / ( * + (*π/»)) - / [ * + ((Ä Γ έ& = Τ(—π, A similar argument applies to obtain (10).