Construction of generalized samplets in Banach spaces

Michael Multerer; Peter Balazs

REVIEW 3 major objections 4 minor 1 cited by

Construction of generalized samplets in Banach spaces

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper extends samplet bases with vanishing moments from point evaluations to frames of functionals in Banach spaces, and proves a localization bound for the resulting coefficients.

desk verdict The construction is a real generalization and the algebra is clean, but Theorem 11's localization estimate is not proven as stated; the proof needs extra module/restriction hypotheses and a step that equates the cutoff infimum with the local seminorm is unjustified. read the letter →

arxiv 2412.00954 v1 pith:HWWII4DA submitted 2024-12-01 math.FA cs.LGcs.NAmath.NA

Peter Balazs , Michael Multerer This is my paper · ORCID

classification math.FAcs.LGcs.NAmath.NA MSC 42C1546B1542C40

keywords sampletsBanachframesRieszbasesvanishingmomentsspectralclusteringmultiresolutionanalysislocalizationestimatecompactlysupporteddistributions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to remove the main restriction of samplet constructions, which until now worked only for point-evaluation measurements (Dirac deltas) on a data set. It claims that the same wavelet-style bases—localized functionals with vanishing moments—can be built from any finite collection of linear measurements on a Banach space, provided those measurements form an ℓ2-frame or a Riesz basis and the dual space consists of compactly supported distributions. Because the analysis operator is injective in these cases, the samplet basis is obtained by applying an isometry to the image of the analysis operator, and the multilevel hierarchy comes from spectral clustering of the functionals' supports. The paper's main theoretical output is a localization bound: the coefficient of a generalized samplet against a function v is controlled by how well v can be approximated, on the samplet's support, by the chosen primitives. If the claim holds, scattered-data compression, feature detection, and operator-adapted wavelet construction become instances of one Banach-space framework.

What carries the argument

The central object is the finite analysis operator $T_N^\star: B\to\mathbb{R}^N$ with entries $(T_N^\star v)_i=(f_i,v)_{B'\times B}$, whose image carries the Gram structure of the functionals. The construction is carried by Lemma 6: applying any isometry $U$ to the image of $T_N^\star$ turns the frame $\{f_i\}$ into a new sequence $\{\psi_i\}$ that is a Parseval frame—or, for Riesz bases and unitary $U$, an orthonormal basis—with respect to the inner product induced by the frame operator. Vanishing moments are enforced by choosing $U$ through a QR decomposition of the moment matrix $M=[(p_m,\phi_l)]$ relative to a chosen primitive space $\mathcal{P}$, so that $(\psi_i,p)=0$ for all $p\in\mathcal{P}$. The multilevel hierarchy is obtained by spectral bisection of a similarity graph built from the supports of the functionals, using the Fiedler vector of the graph Laplacian. The localization proof then cuts a test function $v$ with a smooth bump function $\chi$ supported near $\operatorname{supp}(\psi_i)$, which is legitimate because $B'$ embeds into the compactly supported distributions $\mathcal{E}'(\Omega)$.

What would settle it

Find a Banach space $B$ that has an $\ell^2$-frame and $B'\hookrightarrow\mathcal{E}'(\Omega)$ but is not closed under multiplication by $C_0^\infty$ cutoff functions, and exhibit $v\in B$ and a samplet $\psi_i$ for which $\inf_{p\in\mathcal{P}}\|v-p\|_{B,\operatorname{supp}(\psi_i)}$ is undefined while $(\psi_i,v)$ is finite; this would show the localization theorem needs an extra hypothesis beyond the stated assumptions.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that none of the essential samplet ingredients—localized basis functionals, vanishing moments, multilevel hierarchy, coefficient decay—depends on point evaluations. For a finite set $f_N:=\{f_1,\dots,f_N\}\subset B'$ that is an $\ell^2$-frame for its span (or a subset of a Riesz basis), with $B'\hookrightarrow \mathcal{E}'(\Omega)$, the authors construct a cluster tree by spectral bisection of a similarity graph defined through the supports of the $f_i$. They then define scaling functionals and samplets via a two-scale transform whose matrix is an isometry on the image of the analysis operator $T_N^\star$; choosing that isometry by a QR decomposition of the moment matrix makes the samplets annihilate a finite-dimensional space $\mathcal{P}$ of primitives. The resulting sequences are Parseval frames, and orthonormal bases when the underlying functionals form a Riesz basis, with respect to the Hilbert topology induced by the frame operator. Theorem 11 states the localization estimate $|(\psi_i,v)_{B'\times B}|\le \sqrt{B_N}\,\inf_{p\in\mathcal{P}}\|v-p\|_{B,\operatorname{supp}(\psi_i)}$; for polynomial primitives and Sobolev spaces it becomes the familiar diameter-dependent wavelet decay estimate.

Load-bearing premise

The whole construction assumes the dual space $B'$ embeds into compactly supported distributions on a domain and, in the localization proof, that elements of $B$ can be restricted to compact sets and multiplied by smooth cutoff functions; if a Banach space satisfying the stated hypotheses lacks that multiplicative and restriction structure, the main quantitative estimate does not follow.

Editorial extensions

If this is right

Generalized samplets can be built from local averages, derivative evaluations, boundary fluxes, or other linear measurements, so data compression and feature detection are no longer limited to point clouds.
In Sobolev spaces with polynomial primitives, the localization bound becomes the classical wavelet estimate $|(\psi_i,v)|\le C(\operatorname{diam}\operatorname{supp}\psi_i)^{m-k}\|v\|_{W^{m,p}}$, via the Bramble–Hilbert lemma.
When the functionals form a Riesz basis, the samplet transform is an orthonormal change of basis; when they only form a frame, it still produces a Parseval frame with respect to the frame-induced inner product.
The multilevel hierarchy is obtained from support similarities alone, so the construction applies to functionals supported on complicated shapes without a structured grid.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the hierarchy only needs a similarity matrix between functional supports, one could replace the geometric distance with learned or data-driven similarities, producing samplets adapted to a data manifold rather than to physical space.
The framework suggests a modular numerical pipeline—measurement functionals, support graph, spectral clustering, QR moment matrix—that could be implemented as a black-box multiresolution transform for unstructured linear measurements.
The localization theorem implicitly asks that $B$ be closed under multiplication by $C_0^\infty$ cutoff functions; spaces lacking this module property, beyond the stated dual-embedding assumption, would need an additional hypothesis for the bound to hold.
Choosing the primitives $\mathcal{P}$ to be operator-dependent spaces, such as ranges of powers of an elliptic operator, would make samplet coefficients decay according to the operator's smoothness, linking the construction directly to numerical homogenization and optimal recovery.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague — quick take before you spend time on this. The construction is a genuine generalization of samplets from point evaluations to general functionals that form ℓ2-frames or Riesz bases in Banach spaces. The algebra is sound: use the analysis operator's image, build a finite frame, do a QR on the moment matrix to get vanishing moments, then compose with an isometry. The three examples — RKHS point evaluation, Tausch–White wavelets, operator-adapted wavelets — are correctly identified as special cases. The spectral clustering step is a sensible replacement for the Euclidean clustering in [25]. If the paper only claimed the construction and the algebraic vanishing-moment property, I would be comfortable.

The problem is Theorem 11, the paper's quantitative payoff. The proof has two gaps, and the first compounds the second. It multiplies v ∈ B by a C∞₀ cutoff χ and restricts to supports. That requires B to be closed under pointwise multiplication by smooth cutoffs and to have a restriction/trace to compact sets. The stated assumption B′ ⊂ E′(Ω) does not give you that. The remark that E(Ω) is dense in B is not enough. So the argument already assumes a C∞-module structure that is not in the hypotheses.

Even granting that structure, the proof establishes |(ψᵢ,v)| ≤ √B_N inf_p inf_χ ‖χ(v−p)‖_B, with χ = 1 near supp ψᵢ. But the seminorm in the theorem is ‖v−p‖_{B,K} = inf_{ṽ : ṽ|_K = (v−p)|_K} ‖ṽ‖_B. Since χ(v−p) is one admissible candidate, the inner infimum is larger than or equal to the seminorm, not smaller. The last line of the proof asserts equality without an argument. So Theorem 11 is not proven as stated; the fix is not purely cosmetic. You need either a different definition of the local seminorm (e.g., infimum over functions equal on an open neighborhood of K) plus a trace/extension theorem, or additional hypotheses that make the cutoff infimum equal to the seminorm.

I don't think this is fatal to the whole paper — the construction stands on its own, and the localization result is very likely true in the Sobolev/RKHS examples with a bit more care. But as written, the main quantitative claim is unsupported. The paper deserves a serious referee, but the referee should ask for a rewritten Theorem 11 with explicit hypotheses and a correct proof. I would not cite Theorem 11 in its current form. It is a worthwhile read for people working on multiresolution methods for unstructured data; just don't take the abstract as established.

Referee Report

3 major / 4 minor

Summary. The paper proposes a generalization of samplets—localized discrete signed measures with vanishing moments—to functionals in Banach spaces. The authors assume the functionals form an ℓ2-frame or a Riesz basis, construct a multiresolution hierarchy via spectral clustering of functional supports, and compute an orthogonal two-scale transform by QR-decomposing a moment matrix. The main quantitative result, Theorem 11, states that samplet coefficients satisfy |(ψ_i,v)| ≤ √B_N inf_{p∈P} ‖v−p‖_{B,supp(ψ_i)}. Three examples are presented: point-evaluation samplets in reproducing kernel Hilbert spaces, Tausch–White-type wavelets in H^1_0, and operator-adapted wavelets.

Significance. If the localization theorem were established in the claimed generality, the paper would unify several existing wavelet constructions (classical samplets, Tausch–White multiwavelets, operator-adapted wavelets) within a single Banach-space framework. The construction itself is elegant and internally consistent: the QR-based moment annihilation deterministically produces vanishing moments, the two-scale transform is isometric by construction, and the use of ℓ2-frames gives a clean abstract setting. However, Theorem 11 is the quantitative heart of the paper, and its proof has a load-bearing gap and rests on an ill-posed hypothesis. The paper does not, as written, prove the localization guarantee for the full class of Banach spaces claimed.

major comments (3)

[§4.3 (Theorem 11)] The proof of Theorem 11 relies on operations—restriction of elements of B to compact sets and pointwise multiplication by C∞_0 cutoffs—that are not defined or bounded under the stated hypotheses. The seminorm ‖·‖_{B,K} is defined via ṽ|_K = v, which presupposes a trace/restriction theory for B, and the proof replaces v by χ(v−p), which requires B to be a module over C∞_0. The hypothesis B' ⊂ E'(Ω) only constrains the dual space; it does not imply that elements of B are functions or that smooth cutoffs act continuously on B. Consequently, the statement of Theorem 11 is not well-posed and its proof is not valid in the claimed generality.
[§4.3, proof of Theorem 11] Even when the necessary module and restriction structure is available, the proof establishes only |(ψ_i,v)| ≤ √B_N inf_{p∈P} inf_{χ=1 near K} ‖χ(v−p)‖_B. For each fixed p, the set {χ(v−p)} is a subset of the feasible set in the definition of ‖v−p‖_{B,K}, so inf_χ ‖χ(v−p)‖_B is generally larger than ‖v−p‖_{B,K}. The final line of the proof asserts the reverse inequality without argument; closing this gap requires an explicit extension or trace theorem, or a redefinition of the seminorm as an infimum over admissible cutoffs. This is load-bearing because Corollary 13 and the claimed coefficient localization depend on Theorem 11.
[§4.1 (standing assumption)] The sentence 'This assumption is, for example, satisfied if E(Ω) is a dense subspace of B' is incorrect: density of C∞(Ω) in B yields only an embedding of B' into the space of all distributions D'(Ω), not into the space of compactly supported distributions E'(Ω). In fact, the global hypothesis B' ⊂ E'(Ω) is not satisfied by standard Sobolev spaces on bounded domains; for instance, the functional v ↦ ∫_Ω v dx is not compactly supported. This matters because the examples in §4.4 are of this type. The paper should either weaken the standing assumption to compact support of the finite functional set f_N or revise the examples and the abstract theorem accordingly.

minor comments (4)

[§2.1, Definition 1] The upper frame bound is denoted by B, which conflicts with the Banach space B used throughout the paper; consider renaming one of them (e.g., use C for the frame bound) to avoid confusion.
[§4.3, proof of Theorem 11] The step |(u_i, T⋆_N χ(v−p))| ≤ ‖T⋆_N χ(v−p)‖ implicitly uses ‖u_i‖_{ℓ2} = 1, which holds because U is orthogonal; this should be stated explicitly for completeness.
[§4.1, similarity measure] The suggestion d(f_i,f_j) = dist(supp(f_i),supp(f_j)) is only a pseudo-metric when supports can overlap; this should be noted before invoking spectral clustering results that typically assume a metric or at least symmetric pairwise similarities.
[§4.4, Example 1] The statement that the resulting basis ~Ψ = U^⊺[κ̃_{x1},...,κ̃_{xN}]^⊺ is 'exactly' the dual embedded samplet basis of [7] would benefit from a precise specification of the matrix notation and the normalization conventions used in [7]; as written, the identification is not immediate.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: vanishing moments are imposed by the QR-based construction and the localization bound is a proved consequence; self-citations are background/example only.

full rationale

The paper neither fits parameters nor imports a load-bearing self-citation. The key construction in Sec. 4.2 builds samplets by a QR decomposition of the moment matrix: 'we employ the QR decomposition... By defining the first m_P functionals as scaling functionals and the remaining ones as samplets, we obtain samplets that are orthogonal to the primitives in P.' Thus the vanishing-moment property (ψ_i,p)=0 in (27) is enforced by construction, and Theorem 11 is a direct consequence of this property plus continuity of the analysis operator, not a fitted prediction. The cited prior work [25] and [7] (which share author M. Multerer) supplies background, terminology, the O(N) cost remark, and an identification of Example 1 with an earlier basis; none of these is load-bearing for the new abstract localization result, whose proof is attempted from stated hypotheses in the manuscript. A proof-gap concern exists in Theorem 11 (the argument passes from inf_p inf_χ ||χ(v-p)||_B to inf_p ||v-p||_{B,supp ψ_i}, which is not justified as written, and χ(v-p) may not lie in B under the stated hypotheses alone), but that is a correctness issue, not circularity: the theorem's conclusion is not an input to the construction. Accordingly no circular step is identified.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No new physical or mathematical entities are postulated; generalized samplets are linear combinations of existing functionals. The only free parameter-like quantity in the text is the bandwidth ℓ in the Gaussian similarity example, but it is a user-chosen example, not fitted and not used in any theorem. The key load-bearing axioms are the dual-embedding assumption, the frame/Riesz basis hypothesis, and the hidden cutoff-multiplication assumption in Theorem 11.

assumptions (5)

domain assumption The dual space B′ is imbedded into the space E′(Ω) of compactly supported distributions on a domain Ω ⊂ R^d.
Stated in Section 4.1, first paragraph. This is needed to define supports of functionals, perform spectral clustering based on support distance, and state the localization result. It holds for Sobolev spaces but not for all Banach spaces.
domain assumption The functionals f_N ⊂ B′ form an ℓ2-frame (or Riesz basis) with square-summable coefficients, making the analysis operator injective.
Definitions 1 and 2 in Section 2. The entire construction and stability analysis depend on this. The paper notes this forces B to be isomorphic to a Hilbert space (Eq. (10)).
ad hoc to paper Pointwise multiplication by C∞_0 cutoff functions is a bounded operation on B, and elements of B can be restricted to compact sets.
Used implicitly in the proof of Theorem 11: χ(v−p) ∈ B and the seminorm ‖·‖_{B,supp(ψ_i)} require this. Not stated as a hypothesis anywhere in the paper.
domain assumption Spectral clustering via the Fiedler vector produces a valid cluster tree with the hierarchical properties needed for the multilevel construction.
Algorithm 1 in Section 4.1 is adapted from [41,42]. The paper relies on consistency results from [42] for N → ∞ but does not prove them in this generalized functional setting.
standard math The QR decomposition of the moment matrix yields the claimed number of vanishing moments.
Section 4.2, Eqs. (23)-(24). This is a standard linear algebra fact applied to the rank-revealing QR, with the requirement |leaf| > m_P for leaf clusters to contain samplets.

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Pith. "Pith review of Construction of generalized samplets in Banach spaces." pith.science (2026). https://pith.science/paper/HWWII4DA

@misc{pith2026241200954,
  author       = {Pith},
  title        = {Pith review of: Construction of generalized samplets in Banach spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HWWII4DA}},
  note         = {Machine review of arXiv:2412.00954}
}

read the original abstract

Recently, samplets have been introduced as localized discrete signed measures which are tailored to an underlying data set. Samplets exhibit vanishing moments, i.e., their measure integrals vanish for all polynomials up to a certain degree, which allows for feature detection and data compression. In the present article, we extend the different construction steps of samplets to functionals in Banach spaces more general than point evaluations. To obtain stable representations, we assume that these functionals form frames with square-summable coefficients or even Riesz bases with square-summable coefficients. In either case, the corresponding analysis operator is injective and we obtain samplet bases with the desired properties by means of constructing an isometry of the analysis operator's image. Making the assumption that the dual of the Banach space under consideration is imbedded into the space of compactly supported distributions, the multilevel hierarchy for the generalized samplet construction is obtained by spectral clustering of a similarity graph for the functionals' supports. Based on this multilevel hierarchy, generalized samplets exhibit vanishing moments with respect to a given set of primitives within the Banach space. We derive an abstract localization result for the generalized samplet coefficients with respect to the samplets' support sizes and the approximability of the Banach space elements by the chosen primitives. Finally, we present three examples showcasing the generalized samplet framework.

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Works this paper leans on

43 extracted references · 43 canonical work pages · cited by 1 Pith paper

[25]

Harbrecht and M

H. Harbrecht and M. Multerer. Samplets: Construction a nd scattered data compres- sion. J. Comput. Phys. , 471:111616, 2022

work page 2022
[1]

D. Alm, H. Harbrecht, and U. Kr¨ amer. The H2-wavelet method. J. Comput. Appl. Math., 267:131–159, 2014

work page 2014
[2]

B. Alpert. A class of bases in L2 for the sparse representation of integral operators. SIAM J. Math. Anal. , 24(1):246–262, 1993

work page 1993
[3]

P. Balazs. Matrix-representation of operators using fr ames. Sampl. Theory Signal Image Process., 7(1):39–54, January 2008

work page 2008
[4]

Balazs and K

P. Balazs and K. Gr¨ ochenig. A guide to localized frames a nd applications to Galerkin- like representations of operators. In I. Pesenson, H. Mhask ar, A. Mayeli, Q. T. L. Gia, and D.-X. Zhou, editors, Frames and Other Bases in Abstract and Function Spaces , Applied and Numerical Harmonic Analysis series (ANHA). Bir khauser/Springer, 2017

work page 2017
[5]

Balazs and H

P. Balazs and H. Harbrecht. Frames for the solution of ope rator equations in Hilbert spaces with ﬁxed dual pairing. Numerical Functional Analysis and Optimization , 40(1):65–84, 2019

work page 2019
[6]

Balazs, N

P. Balazs, N. Holighaus, T. Necciari, and D. Stoeva. Fram e theory for signal process- ing in psychoacoustics. In R. Balan, J. J. Benedetto, W. Czaj a, and K. Okoudjou, editors, Excursions in Harmonic Analysis Vol. 5, , pages 225–268. Springer, 2017

work page 2017
[7]

Baroli, H

D. Baroli, H. Harbrecht, and M. Multerer. Samplet basis p ursuit: Multiresolution scattered data approximation with sparsity constraints. IEEE Trans. Sig. Proc. , 72:1813–1823, 2024

work page 2024

Show all 43 references

[8]

Bramble and S

J. Bramble and S. Hilbert. Estimation of linear function als on Sobolev spaces with application to fourier transforms and spline interpolatio n. SIAM J. Numer. Anal. , 7(1):112–124, 1970

1970
[9]

P. G. Casazza, D. Han, and D. Larson. Frames for banach spa ces. Cont. Math. , 247:149–182, 1999

1999
[10]

Christensen

O. Christensen. An Introduction to Frames and Riesz Bases . Birkh¨ auser, Boston, 2003

2003
[11]

Christensen and D

O. Christensen and D. T. Stoeva. p-frames in separable Banach spaces. Adv. Comput. Math., 18(2-4):117–126, 2003

2003
[12]

C. Chui. An Introduction to Wavelets . Academic Press, San Diego, 1992

1992
[13]

Chui and E

C. Chui and E. Quak. Wavelets on a bounded interval. Numer. Meth. Approx. Theory , 9:53–75, 1992. 19

1992
[14]

Coifman and M

R. Coifman and M. Maggioni. Diﬀusion wavelets. Appl. Comput. Harmon. Anal. , 21(1):53–94, 2006

2006
[15]

L. Crone. A characterization of matrix operator on l2. Math. Z. , 123:315–317, 1971

1971
[16]

W. Dahmen. Wavelet and multiscale methods for operator equations. Acta Numer. , 6:55–228, 1997

1997
[17]

Dahmen, A

W. Dahmen, A. Kunoth, and K. Urban. Biorthogonal spline wavelets on the interval – stability and moment conditions. Appl. Comp. Harm. Anal. , 6(2):132–196, 1999

1999
[18]

Dahmen and R

W. Dahmen and R. Schneider. Composite wavelet basis for operator equations. Math. Comp., 68:1533–1567, 1999

1999
[19]

Dahmen and R

W. Dahmen and R. Stevenson. Element-by-element constr uction of wavelets satisfying stability and moment conditions. SIAM J. Numer. Anal. , 37(1):319–352, 1999

1999
[20]

Daubechies

I. Daubechies. Ten Lectures on Wavelets . Society of Industrial and Applied Mathe- matics, Philadelphia, 1992

1992
[21]

R. J. Duﬃn and A. C. Schaeﬀer. A class of nonharmonic Fouri er series. Trans. Amer. Math. Soc. , 72:341–366, 1952

1952
[22]

M. Fiedler. Algebraic connectivity of graphs. Czechoslov. Math. J. , 23(2):298–305, 1973

1973
[23]

Gavish, B

M. Gavish, B. Nadler, and R. Coifman. Multiscale wavele ts on trees, graphs and high dimensional data: theory and applications to semi supervis ed learning. In ICML, volume 10, pages 367–74, 2010

2010
[24]

Gohberg, S

I. Gohberg, S. Goldberg, and M. A. Kaashoek. Classes of Linear Operators , volume I of Operator Theory: Advances and Applications . Birkh¨ auser, Basel, 1990

1990
[26]

Harbrecht and R

H. Harbrecht and R. Schneider. Biorthogonal wavelet ba ses for the boundary element method. Math. Nachr. , 269(1):167–188, 2004

2004
[27]

J. B. Kruskal and M. Wish. Multidimensional Scaling . Sage, Newbury Park, 1978

1978
[28]

S. Mallat. A Wavelet Tour of Signal Processing . Academic Press, San Diego, 1999

1999
[29]

V. I. Meleshko. Pseudoinverse operators in banach spac es. Ukr. Math. J. , 29(545-553), 1977

1977
[30]

Micchelli and T

C. Micchelli and T. Rivlin. A survey of optimal recovery . In C. Micchelli and T. Rivlin, editors, Optimal Estimation in Approximation Theory , pages 1–54, New York, 1977. Springer. 20

1977
[31]

B. Mohar. The Laplacian spectrum of graphs. In Graph theory, Combinatorics, and Applications. Vol. 2 (Kalamazoo, MI, 1988) , pages 871–898, New York, 1991. John Wiley & Sons

1988
[32]

Owhadi and C

H. Owhadi and C. Scovel. Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization. Cambridge University Press, Cambridge, 2019

2019
[33]

I. Ram, M. Elad, and I. Cohen. Generalized tree-based wa velet transform. IEEE Trans. Signal Process., 59(9):4199–4209, 2011

2011
[34]

W. Rudin. Functional Analysis. McGraw-Hill, New York, 1991

1991
[35]

Sauter and C

S. Sauter and C. Schwab. Boundary Element Methods . Springer Series in Computa- tional Mathematics. Springer Berlin Heidelberg, 2010

2010
[36]

Stevenson

R. Stevenson. Adaptive solution of operator equations using wavelet frames. SIAM J. Numer. Anal. , 41(3):1074–1100, 2003

2003
[37]

D. T. Stoeva. Generalization of the frame operator and t he canonical dual frame to banach spaces. Asian-Eur. J. Math. , 1(4):631–643., 2008

2008
[38]

D. T. Stoeva. Xd-frames in Banach spaces and their duals. Int. J. Pure Appl. Math. , 52(1):1–14, 2009

2009
[39]

Tausch and J

J. Tausch and J. White. Multiscale bases for the sparse r epresentation of boundary integral operators on complex geometry. SIAM J. Sci. Comput. , 24(5):1610–1629, 2003

2003
[40]

Thomann, I

P. Thomann, I. Steinwart, and N. Schmid. Towards an axio matic approach to hier- archical clustering of measures. J. Mach. Learn. Res. , 16:1949–2002, 2015

1949
[41]

Von Luxburg

U. Von Luxburg. A tutorial on spectral clustering. Statist. Comput. , 17(4):395–416, 2007

2007
[42]

von Luxburg, M

U. von Luxburg, M. Belkin, and O. Bousquet. Consistency of spectral clustering. Ann. Stat. , 36(2):555–586, 2008

2008
[43]

von Petersdorﬀ, C

T. von Petersdorﬀ, C. Schwab, and R. Schneider. Multiwav elets for second-kind integral equations. SIAM J. Numer. Anal. , 34(6):2212–2227, 1997. 21

1997

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [25]

Harbrecht and M

H. Harbrecht and M. Multerer. Samplets: Construction a nd scattered data compres- sion. J. Comput. Phys. , 471:111616, 2022

work page 2022

[2] [1]

D. Alm, H. Harbrecht, and U. Kr¨ amer. The H2-wavelet method. J. Comput. Appl. Math., 267:131–159, 2014

work page 2014

[3] [2]

B. Alpert. A class of bases in L2 for the sparse representation of integral operators. SIAM J. Math. Anal. , 24(1):246–262, 1993

work page 1993

[4] [3]

P. Balazs. Matrix-representation of operators using fr ames. Sampl. Theory Signal Image Process., 7(1):39–54, January 2008

work page 2008

[5] [4]

Balazs and K

P. Balazs and K. Gr¨ ochenig. A guide to localized frames a nd applications to Galerkin- like representations of operators. In I. Pesenson, H. Mhask ar, A. Mayeli, Q. T. L. Gia, and D.-X. Zhou, editors, Frames and Other Bases in Abstract and Function Spaces , Applied and Numerical Harmonic Analysis series (ANHA). Bir khauser/Springer, 2017

work page 2017

[6] [5]

Balazs and H

P. Balazs and H. Harbrecht. Frames for the solution of ope rator equations in Hilbert spaces with ﬁxed dual pairing. Numerical Functional Analysis and Optimization , 40(1):65–84, 2019

work page 2019

[7] [6]

Balazs, N

P. Balazs, N. Holighaus, T. Necciari, and D. Stoeva. Fram e theory for signal process- ing in psychoacoustics. In R. Balan, J. J. Benedetto, W. Czaj a, and K. Okoudjou, editors, Excursions in Harmonic Analysis Vol. 5, , pages 225–268. Springer, 2017

work page 2017

[8] [7]

Baroli, H

D. Baroli, H. Harbrecht, and M. Multerer. Samplet basis p ursuit: Multiresolution scattered data approximation with sparsity constraints. IEEE Trans. Sig. Proc. , 72:1813–1823, 2024

work page 2024