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3). 6). Proof. 3) remain valid for S acting in Lp (0, ω). 3) holds, it follows that for a certain C we have Lm+1 Here f p p ≤ Cm Lm p ≤ C m+1 m! 8) is the norm in the space Lp (0, ω). 8) the series ∞ Bγ (x, λ) = (iλ)m Lm+1 M! m=0 converges for |λ| < C −1 . Consequently, SBγ (x, λ) = eiλx , |λ| < C −1 . 11) x where ω aγ (λ) = iλ ω Bγ (t, λ) dt, bγ (λ) = 1 + iλ 0 Bγ (t, λ)N (t) dt. 10) we derive ω ei(x−t)λ uγ (t, λ) dt. 13) 34 Chapter 2. Equations of the First Kind with a Difference Kernel Next we write the functions aγ (λ) and a(λ) in the following form: aγ (λ) = iλ Bγ (x, λ), S ∗ U N2 , a(λ) = iλ SBγ (x, λ), U N2 .

2. 3). 6) belongs to D and SB(x, λ) = eixλ . 14) Proof. 1) we introduce the norm ω f D = |α| + |β| + |f1 (t)| dt. 2) that for some c Lm+1 D ≤ cm Lm Hence, the series ≤ cm+1 m!. D ∞ B(x, λ) = (iλ)m Lm+1 m! m=0 converges for |λ| < c−1 . We also see that B(x, λ) ∈ D and SB(x, λ) = eiλx . 6). 17) x where f (x) ∈ D, and g(x, t) is a continuous function of x and t (0 ≤ x, t ≤ ω). 17) holds for f (x) = δ(x) and f (x) = δ(ω − x). 17) holds also for f (x) ∈ L(0, ω). 16) by the method of successive approximations.

Proof. 3) remains valid for N1 , N2 ∈ D. 4) we deduce that T eiλx = B(x, λ). Hence, ST eiλx = eiλx , that is, ST ϕ = ϕ, ϕ ∈ C (2) . 20). Suppose that this equation has more than one solution. Then there is a non-trivial solution of Sf = 0. 21) that S ∗ U f = 0, where U f = f (ω − x). 2, we have eixλ , U f = SB(x, λ), U f . 10), we obtain eixλ , U f = 0, that is, f = 0. This proves the theorem.

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