By Errett Bishop
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Extra info for Constructive Measure Theory
If f ∈ Lp (0, ω) and Sf ∈ Lq (0, ω), then Sf, U f = f, S ∗ U f . 4) Proof. 3) that S ∗ U f ∈ Lq (0, ω). 4) are well deﬁned. 4) is obtained directly from the following property of an involution: f1 , f2 = U f2 , U f1 , f1 ∈ Lq (0, ω), f2 ∈ Lp (0, ω). 5) 2. 1 from Chapter 1 can be modiﬁed for the case of the equations of the ﬁrst kind as follows. 1. 3. 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).
2. Solutions of equations of the ﬁrst kind 35 1. We introduce the function r(x, t) = N2 (ω − t)N1 (x) − N1 (x − t)N2 (x). 3) 0 where f (x) is an arbitrary function in Lq (−ω, ω). (l) We denote by Wp the set of functions ϕ(x) such that ϕ(l) (x) ∈ Lp (0, ω). (2) We deﬁne an operator T on Wp by ω ω ϕ (t)r(x, t) dt + ϕ(ω)N2 (x) − Tϕ = 0 ϕ (x − t + ω)N2 (t) dt x ω ω − ϕ (x − t + s)r(t, s) ds dt. 6) we obtain into Lp (0, ω). 3), B(x, λ) = T eiλx . 5) imply that ST eiλx = eixλ , T SB(x, λ) = B(x, λ). 6) we prove the following theorem.
10) we have ω ∗ AS−SA M (x)+N (t) δ ω−t dt. 7) holds for f (x) = δ(x) and f (x) = δ(ω − x). 3. 7) is valid for f ∈ D. This proves the theorem. 12) 0 which maps the functions from D into the functions which are continuous on the segment [0, ω]. Let us prove that for any pair of the functions f (x) and g(x) from D the following equality holds: Sf, g = f, S ∗ g . 13) Indeed, we put f (x) = α δ(x) + β δ(ω − x) + f1 (x), g(x) = γ δ(x) + ν δ(ω − x) + g1 (x), where f1 (x), g1 (x) ∈ L(0, ω). 12) we obtain ω Sf = αk(x) + βk(x − ω) + k(x − t)f1 (t) dt, 0 ω ∗ S g = γk(−x) + νk(ω − x) + k(t − x)g1 (t) dt.