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Estimation in inverse problems and second-generation wavelets · 2006-ana-206

Functional Analysis · ICM 2006: Estimation in inverse problems and second-generation wavelets

Source assessment · not reviewedAs of 2026-08-18Source & evidence ↓
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Problem statement

Let (X,d,μ)(X,d,\mu) be a metric-measure space, let H=L2(X,μ)\mathbb H=L^2(X,\mu), and let K\mathbb K be a separable Hilbert space. Let K:H→KK:\mathbb H\to\mathbb K be a compact linear operator with singular-value decomposition

Kf=∑i≥0bi⟨f,ei⟩Hgi, Kf=\sum_{i\ge 0} b_i\langle f,e_i\rangle_{\mathbb H}g_i,

where (ei)i≥0(e_i)_{i\ge0} is an orthonormal basis of H\mathbb H, (gi)i≥0(g_i)_{i\ge0} is an orthonormal system in K\mathbb K, and there exist constants v>0v>0 and c1,c2>0c_1,c_2>0 such that

c1(i+1)−v≤bi≤c2(i+1)−vfor every i≥0. c_1(i+1)^{-v}\le b_i\le c_2(i+1)^{-v} \qquad\text{for every }i\ge0.

Determine whether, for every such operator KK and every such SVD basis (ei)(e_i), there exist index sets Λj\Lambda_j, centers xj,k∈Xx_{j,k}\in X, and a multiscale tight frame

{ψj,k:j≥0, k∈Λj}⊂H \{\psi_{j,k}:j\ge0,\ k\in\Lambda_j\}\subset\mathbb H

that simultaneously has the following two properties:

  1. There is a constant C>0C>0, independent of jj and kk, such that
∑i≥0∣⟨ei,ψj,k⟩Hbi∣2≤C22jvfor every j≥0 and k∈Λj. \sum_{i\ge0}\left|\frac{\langle e_i,\psi_{j,k}\rangle_{\mathbb H}}{b_i}\right|^2 \le C2^{2jv} \qquad\text{for every }j\ge0\text{ and }k\in\Lambda_j.
  1. For every N>0N>0, there is a constant CN>0C_N>0, independent of jj and kk, such that
∣ψj,k(x)∣≤CNμ(B(xj,k,2−j))(1+2jd(x,xj,k))−N |\psi_{j,k}(x)|\le \frac{C_N}{\sqrt{\mu(B(x_{j,k},2^{-j}))}} \bigl(1+2^j d(x,x_{j,k})\bigr)^{-N}

for μ\mu-almost every x∈Xx\in X, where B(x,r)={y∈X:d(x,y)<r}B(x,r)=\{y\in X:d(x,y)<r\}.

Here tightness means that

∥f∥H2=∑j≥0∑k∈Λj∣⟨f,ψj,k⟩H∣2for every f∈H. \|f\|_{\mathbb H}^2= \sum_{j\ge0}\sum_{k\in\Lambda_j} |\langle f,\psi_{j,k}\rangle_{\mathbb H}|^2 \qquad\text{for every }f\in\mathbb H.

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Progress & source assessment

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The exact problem is unresolved in the citable literature located through 4 August 2026. The literature constructs frames satisfying the two requirements for important structured SVD systems, and constructs localized spectral frames on broad classes of geometrically regular spaces, but no located result proves the assertion for every metric-measure space and every SVD basis. Nor was a published counterexample to that exact universal assertion found.

  • Dominique Picard and Gerard Kerkyacharian, "Estimation in inverse problems and second-generation wavelets," Proceedings of the International Congress of Mathematicians, Madrid 2006, Vol. III, European Mathematical Society, 2007, pp. 713-739, DOI: 10.4171/022-3/37. Section 2.2 explicitly says that, for an arbitrary linear operator and arbitrary SVD basis, it is not known whether the SVD-coherence condition can be combined with spatial localization. The paper verifies the coherence estimate for band-limited needlets when bi≍i−vb_i\asymp i^{-v} and reports localized constructions for structured bases, especially Fourier and Jacobi systems; it does not settle the arbitrary-basis case.

  • Gerard Kerkyacharian, Pencho Petrushev, Dominique Picard, and Thomas Willer, "Needlet algorithms for estimation in inverse problems," Electronic Journal of Statistics 1 (2007), 30-76, DOI: 10.1214/07-EJS014, arXiv: 0705.0274. The paper constructs SVD-band-limited needlets. Its support in indices i≍2ji\asymp 2^j, together with bi≍i−vb_i\asymp i^{-v}, yields the required coherence bound. Spatial localization is then established or assumed only in its wavelet and Jacobi-type scenarios, so the result does not cover arbitrary SVD bases.

  • Mauro Maggioni and Hrushikesh N. Mhaskar, "Diffusion polynomial frames on metric measure spaces," Applied and Computational Harmonic Analysis 24(3) (2008), 329-353, DOI: 10.1016/j.acha.2007.07.001. Theorem 2.1 gives a multiscale tight-frame decomposition from an arbitrary orthonormal system, and band limitation would supply the coherence estimate after indexing spectral parameters compatibly with the singular values. Its spatial kernel estimates, however, require additional summability and metric/spectral localization hypotheses. Those hypotheses are not automatic for an arbitrary basis, so this is partial progress only.

  • Gerard Kerkyacharian, George Kyriazis, Erwan Le Pennec, Pencho Petrushev, and Dominique Picard, "Inversion of noisy Radon transform by SVD based needlets," Applied and Computational Harmonic Analysis 28(1) (2010), 24-45, DOI: 10.1016/j.acha.2009.06.001, arXiv: 0809.3332. This constructs rapidly localized frames compatible with the polynomial SVD of the Radon transform. It adds a substantial operator-specific positive case, not a result for arbitrary compact operators.

  • Thierry Coulhon, Gerard Kerkyacharian, and Pencho Petrushev, "Heat kernel generated frames in the setting of Dirichlet spaces," Journal of Fourier Analysis and Applications 18 (2012), 995-1066, DOI: 10.1007/s00041-012-9232-7, arXiv: 1206.0463. The paper constructs band-limited, rapidly localized frames on doubling Dirichlet spaces when a compatible self-adjoint operator has Gaussian and Holder-continuous heat kernels. These additional geometric and spectral assumptions give a broad sufficient framework, but they neither follow from the hypotheses of the present problem nor apply to every ordering of every SVD basis.

Thus the established mechanism is clear under compatible spectral localization, cubature, heat-kernel, or polynomial structure. What remains unresolved is whether no such extra relation between d,μd,\mu and (ei)(e_i) is needed, or alternatively whether a citable counterexample disproves that universal claim.

References
  1. [1]ICM Conjectures · 2006-ana-206 link
  2. [2]ICM 2006: Estimation in inverse problems and second-generation wavelets link

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