Math 2.0

Integral transforms, operational calculus Definition

Ill-posedness of real Laplace inversion

Builds on Well-posed problemLaplace transformLebesgue integralLerch's uniqueness theorem

Definition

Let L2(0,∞)L^2(0, \infty) be the space of square-integrable functions on (0,∞)(0, \infty) with the norm ∥f∥2=(∫0∞∣f(t)∣2 dt)1/2\norm{f}_2 = \left( \int_0^\infty \abs{f(t)}^2 \, dt \right)^{1/2}. For f∈L2(0,∞)f \in L^2(0, \infty), its Laplace transform ℒf(s)=∫0∞f(t)e−st dt\mathcal{L} f(s) = \int_0^\infty f(t) e^{-s t} \, dt is defined for real s>0s > 0, and ℒ\mathcal{L} is a bounded linear operator from L2(0,∞)L^2(0, \infty) to L2(0,∞)L^2(0, \infty). The real Laplace inversion problem is: given F∈L2(0,∞)F \in L^2(0, \infty), find f∈L2(0,∞)f \in L^2(0, \infty) with ℒf=F\mathcal{L} f = F on (0,∞)(0, \infty). Ill-posedness of this problem means that it is not well posed in the sense of Hadamard. Here the solution is unique when it exists, but the inverse map ℒf↦f\mathcal{L} f \mapsto f is not continuous. That is, there are fnf_n with ∥fn∥2\norm{f_n}_2 bounded away from 00 and ∥ℒfn∥2→0\norm{\mathcal{L} f_n}_2 \to 0.

Remarks

Uniqueness holds by Lerch's uniqueness theorem. Instability, with the check: let fn(t)=e−tsin⁡(nt)f_n(t) = e^{-t} \sin(n t). Then ∥fn∥22=∫0∞e−2tsin⁡2(nt) dt=n24(1+n2)≥18\norm{f_n}_2^2 = \int_0^\infty e^{-2t} \sin^2(n t) \, dt = \frac{n^2}{4(1 + n^2)} \ge \frac{1}{8} for n≥1n \ge 1. Its transform is ℒfn(s)=n(s+1)2+n2\mathcal{L} f_n(s) = \frac{n}{(s + 1)^2 + n^2}, and ∥ℒfn∥22≤∫−∞∞n2(u2+n2)2 du=π2n→0\norm{\mathcal{L} f_n}_2^2 \le \int_{-\infty}^{\infty} \frac{n^2}{(u^2 + n^2)^2} \, du = \frac{\pi}{2 n} \to 0. So no estimate ∥f∥2≤C∥ℒf∥2\norm{f}_2 \le C \norm{\mathcal{L} f}_2 can hold. Existence also fails for most data: the range of ℒ\mathcal{L} consists of restrictions of holomorphic functions, so a generic F∈L2(0,∞)F \in L^2(0, \infty) is not a transform at all. Contrast: inversion from values of FF on a vertical line in the complex plane, by the Bromwich integral, is a Fourier inversion and is stable in L2L^2.

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Continuous function, concept, no evidence levelContinuous functionconceptFunction, concept, no evidence levelFunctionconceptLebesgue integral, concept, no evidence levelLebesgue integralconceptNormed vector space, concept, no evidence levelNormed vector spaceconceptLaplace transform, definition, no evidence levelLaplace transformdefinitionWell-posed problem, definition, no evidence levelWell-posed problemdefinitionLerch's uniqueness theorem, statement, no evidence levelLerch's uniqueness th…statementIll-posedness of real Laplace inversion, definition, no evidence levelIll-posedness of real…definition
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  1. Continuous function concept
  2. Function concept
  3. Lebesgue integral concept
  4. Normed vector space concept
  5. Laplace transform definition
  6. Well-posed problem definition
  7. Lerch's uniqueness theorem statement
  8. Ill-posedness of real Laplace inversion definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Well-posed_problem
About this page
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MSC44A10, 47A52, 65R30
DefinesIll-posedness of real Laplace inversion
UsesWell-posed problem, Laplace transform, Lebesgue integral, Lerch's uniqueness theorem

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