Real and complex analysis with applications to other
sciences, CIMPA School, University of Buea, Cameroon,
May 2011
Best analytic and meromorphic approximation in Hardy classes;
application to inverse potential problems, Juliette Leblond1
We will discuss some approximation issues by analytic (holomorphic) functions in smooth domains of the complex plane, from incomplete and corrupted
boundary values. Motivations are provided from a number of inverse problems
concerning Laplace-type partial differential equations (elliptic PDE) and recovery of harmonic functions, which arise from physical applications. For such
issues, a well-posed and robust formulation is provided by means of bounded
extremal problems in Hardy spaces (normed spaces of holomorphic functions).
There, constructive resolution schemes are available, which involve Toeplitz
operators and provide strong links with Carleman extrapolation formulas.
Such a best approximation approach may also be used in slightly more general
framework, where Hardy classes of solutions may still be defined, and that
may be discussed during the tutorial:
• 2D pseudo-analytic functions, related to the (complex conjugated) Beltrami equation, or to variable conductivities situations in elliptic PDE,
• harmonic functions in 3D spherical domains or shells, where analytic
functions coincides with gradients of harmonic ones.
This original and efficient approach to inverse problems for PDE consist in
approximating the available boundary data in classes of solutions (harmonic
or analytic functions), rather than in the classical discretization of the involved
(Laplace or elliptic) operator.
1
Overview, revisited (3h)
1.1
First hour
• Some inverse problems for Laplace equations.
For example, inverse Cauchy type boundary value problem for Laplace
equation in a domain Ω ⊂ R2 with smooth (C 1 ) boundary. Let I ⊂ ∂Ω:
(IP) Given pointwise values on I of a solution ud to ∆u = 0 in Ω and of
its normal derivative ∂n ud , recover ud and ∂n ud on the complementary
part ∂Ω \ I of the boundary.
1
INRIA Sophia-Antipolis Méditerranée, Team APICS, BP 93, 06902 Sophia-Antipolis
Cedex, France. [email protected]
Also, inverse problems for Laplace equation in R3 , and for more general
elliptic partial differential equations (with a variable isotropic conductivity coefficient) in R2 ; associated direct Dirichlet problems.
Applications to physical issues for electric, magnetic, gravitational potentials (medical imaging, plasma fusion, geophysics), through Maxwell
and Newton equations.
In these realistic situations, the given pointwise values of ud , ∂n ud above
may be subject to measurements errors with respect to which a robust
behaviour of the resolution scheme is required.
• Harmonic / analytic functions.
Links between harmonic functions (solutions to Laplace equations) and
analytic (holomorphic) functions, in (simply connected smooth) plane
domains, e.g. in the unit disk D ⊂ C ' R2 (or conformally equivalent to
D).
Cauchy-Riemann equations (harmonic conjugation, compatibility condition). Function g holomorphic (or analytic) in D: g = u + iv, for
compatible (conjugated) harmonic functions u, v.
¯ ∆'
Other relations between the operators div ' ∇·, grad ' ∇, ∂, ∂,
2
¯ through the isometry C ' R [3, 7, 8, 11, 13, 14].
∂ ∂,
Formulation of (IP) in terms of recovery
R issues for analytic functions, for
complex valued function fd = ud + i ∂n ud built from data on I:
Given pointwise values on I ⊂ ∂Ω of a function fd analytic in Ω, recover
fd on the complementary part ∂Ω \ I of the boundary.
1.2
Second hour
• Hardy spaces2 .
Basic properties of Hardy spaces H 2 = H 2 (D), in Hilbertian case (functions analytic in D, square-summable w.r.t. Lebesgue measure on circles
Tr ⊂ D, r < 1). Boundary values belong to L2 (T) (T = ∂D unit circle),
representation on Fourier bases. Uniqueness properties, behaviour of zeroes, factorization, maximum principle. Cauchy-Riemann equations up
to T.
Best approximation (or extremal) problems on T [8, 11, 14] (whole boundary, analytic projection is orthogonal L2 (T) → H 2 .
Toeplitz (and Hankel) operators, Nehari (and Adamjan-Arov-Krein) theorems [16].
2
Link with Sandra’s part of the course.
1.3
Third hour
• Bounded extremal problems in Hardy spaces.
Density properties of traces of H 2 functions (on strict subsets I ⊂ T)
ill-posedness of (exact) extrapolation (Hadamard).
Well-posedness of best constrained approximation of boundary data (after preliminary robust interpolation step of pointwise values) by traces
of H 2 functions on I:
(BEP) Given a function f ∈ L2 (I), we look for a best approximation
g∗ ∈ H 2 to f in L2 (I), subject to some norm constraint on T \ I.
Existence, uniqueness of optimal solution in g∗ ∈ H 2 , and constructive
issues [1, 5].
• Back to solutions to inverse problems (IP).
- Preliminary robust interpolation step of the
R given pointwise boundary
data on I: from ud , ∂n ud , build fd ' ud + i ∂n ud on I.
- Compute the solution g∗ to (BEP) associated with f = fd , and appropriated constraint. Resolution algorithm from semi-explicit Carleman
type integral formula:
(i) get Fourier representation3 of χI f ,
(ii) computate iteratively the resolvent of Toeplitz operator with symbol
χ∂Ω\I .
- Then, a robust approximate solution to (IP) is given by the real and
imaginary parts of g∗ . Behaviour of error (criterion value) w.r.t. constraint level.
2
Tutorial (2h)
2.1
First hour
Applications, links with (generalized) analytic functions:
• Harmonic identification setup (frequency domain identification for linear
stationary causal controlled systems, transfer functions, Laplace4 and
z-transform) [12].
• Plasma shaping, elliptic diffusion PDE div (σ grad u) = 0, pseudo-analytic
functions.
3
4
Link with Aline & Abderrazek part of the course.
Link with David & Cyrille part of the course.
• Analytic functions in domains of R3 : gradients of harmonic functions.
• Inverse problems of singularities recovery (sources, defaults): approximation in meromorphic or rational classes of functions, with constrained
poles (quantity, location).
2.2
Second hour
• Banach spaces5 H p (D) for 1 ≤ p < ∞, H ∞ (D). Hardy spaces H 2 (D),
for domains D ' D ⊂ C conformally (simply connected plane domains,
e.g. D half-plane; also, H 2 (A), for annulus A ⊂ D (annular or multiply
connected plane domains).
• Hardy classes of pseudo-analytic functions Hν2 (A), [9, 10].
• Hardy classes H 2 (B), ball B ⊂ R3 , [2, 15].
References
[1] D. Alpay, L. Baratchart, J. Leblond, Some extremal problems linked with
identification from partial frequency data, Proc. 10th conference in Analysis and optimization of systems, Sophia Antipolis, 1992, LNCIS, 185,
Springer Verlag, 563-573, 1993.
[2] B. Atfeh, L. Baratchart, J. Leblond, J.R. Partington. Bounded extremal
and Cauchy-Laplace problems on the sphere and shell, J. of Fourier Analysis and Applications, 16(2), 177-203, 2010.
[3] L. Ahlfors, Lectures on quasiconformal mappings, Wadsworth and
Brooks/Cole Advanced Books and Software, Monterey, CA, 1987.
[4] K. Astala, L. Päivärinta, A boundary integral equation for Calderón’s
inverse conductivity problem, Proc. 7th Int. Conf. on Harmonic Analysis
and PDEs., Madrid (Spain), 2004, Collect. Math. , Extra. vol., 127–139,
2006.
[5] L. Baratchart, J. Leblond. Hardy approximation to Lp functions on subsets of the circle with 1 ≤ p < ∞, Constructive Approximation, 14, 41-56,
1998.
[6] L. Baratchart, J. Leblond, J.R. Partington. Hardy approximation to L∞
functions on subsets of the circle, Constructive Approximation, 12, 423436, 1996.
5
Link with Sandra & Brett parts of the course, with other normed spaces of analytic
functions.
[7] S. Chaabane, I. Fellah, M. Jaoua, J. Leblond, Logarithmic stability estimates for a Robin coefficient in 2D Laplace inverse problems, Inverse
problems, 20(1), 49-57, 2004.
[8] P. L. Duren, Theory of H p spaces, Pure and Applied Mathematics, 38,
Academic Press, New York-London, 1970.
[9] Y. Fischer, J. Leblond, Solutions to conjugate Beltrami equations and
approximation in generalized Hardy spaces, Adv. Pure Applied Math., 2,
47-63, 2010.
[10] Y. Fischer, J. Leblond, J.R. Partington, E. Sincich, Bounded extremal problems in Hardy spaces for the conjugate Beltrami equation
in simply connected domains, Appl. Comp. Harmonic Anal., in press,
10.1016/j.acha.2011.01.003.
[11] J. Garnett, Bounded analytic functions, Pure and Applied Math. 96,
Academic Press, 1981.
[12] J. Leblond, Approximation et interpolation par des classes de Hardy sur
des sous-ensembles du cercle : problèmes extrḿaux bornś et problèmes de
Loewner, HdR, Univ. Nice, Sophia-Antipolis, 1998.
[13] Ch. Pommerenke, Boundary behaviour of conformal maps, Springer Verlag, 1992.
[14] W. Rudin, Real and complex analysis, Tata McGraw-Hill, 2006.
[15] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean
Spaces, Princeton University Press, 1971.
[16] N. Young, An introduction to Hilbert space, Cambridge University Press,
1988.