By Tatsuo Kawata

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Aimed toward the neighborhood of mathematicians engaged on usual and partial differential equations, distinction equations, and useful equations, this ebook includes chosen papers in accordance with the shows on the overseas convention on Differential & distinction Equations and purposes (ICDDEA) 2015, devoted to the reminiscence of Professor Georg promote.

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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)

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).