By Arne Schneck
The aim of this ebook is twofold. Our place to begin is the layout of layered media with a prescribed mirrored image coefficient. within the first a part of this booklet we exhibit that the gap of bodily realizable mirrored image coefficients is very constrained via a couple of homes. within the moment half we contemplate a limited approximation challenge in Hardy areas. this is seen as an optimization challenge for the frequency reaction of a causal LTI method with constrained achieve.
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Additional info for Bounds for optimization of the reflection coefficient by constrained optimization in Hardy spaces
31). 2. CONTINUITY OF THE DIRECT PROBLEM notation. Multiplying by ϕ ∈ C ∞(0, d), d 0= u′′ (x)ϕ(x) + k 2n2 (x)u(x)ϕ(x) dx 0 ′ d ′ = u (d)ϕ(d) − u (0)ϕ(0) + 0 −u′ (x)ϕ′(x) + k 2 n2 (x)u(x)ϕ(x) dx = ik (u(d)ϕ(d) + u(0)ϕ(0)) d + 0 k 2n2 (x)u(x)ϕ(x) − u′ (x)ϕ′(x) dx − 2ikϕ(0). 33) where d Bn [u, ϕ] = ik (u(d)ϕ(d) + u(0)ϕ(0)) + 0 k 2n2 (x)u(x)ϕ(x) − u′ (x)ϕ′(x) dx. The careful reader should not confuse the Sobolev space H 1 (0, d) with the Hardy spaces H p (D) and H p (C+ ). 8. 33). 31). Proof. 33). Then u(x) = u(x) − u(0) − 1 (u(d) − u(0))x is a weak solution of d u′′(x) + k 2 n2(x)u(x) = f (x), u(0) = u(d) = 0, with f (x) = −k 2n2 (x) u(0) + d1 (u(d) − u(0))x .
E(x, t) = Ein (x)e−iωtey with Ein (x) = eikx. The layered medium gives rise to a reflected field E(x, t) = Eref (x)e−iωtey with Eref (x) = R(k)e−ikx. The number R(k) is called reflection coefficient (from the left). The reflected field is also a plane wave, but it travels in the opposite direction. Then the mapping L : Ein → Eref is an LTI system which maps eik· to R(k)eik· . Because the system should map a real incident field to a real reflected field, its impulse response should only take values in R.
Together with (c) we get for x ≥ 0 x 0 m2 (t, k)(1 − n2(t))D˙ k (x − t) dt x t ≤ 0 ≤ |k| x + x2 2 exp |k| 0 |1 − n2 (s)| ds |1 − n2(t)| (1 + |k|(x − t)) dt x 2 sup |1 − n (t)| exp |k| t∈[0,x] 0 |1 − n2 (t)| dt . 10). The estimates for m1 can be obtained in the same way. 3. 7) for m1 and m2 , one sees easily that it converges uniformly in k on compact subsets of C. Since the partial sums themselves are analytic in C, m1 (x, ·) and m2 (x, ·) are analytic in C. 4. One easily sees that since n is real-valued, the symmetry relations (or reality conditions) mj (x, −k) = mj (x, k) and uj (x, −k) = uj (x, k), j = 1, 2, hold.