We outline an extension of paraproduct decompositions for compositions of the form $A(f)$ where $A \in C^{d}(\mathbb{R}), f \in Λ_α([0,1]^d)$ developed in [arXiv:2503.12629] and [arXiv:2508.13322] to settings where $(A \in C^1(\mathbb{R}),f \in Λ_α(X))$ and $ (A \in C^2(\mathbb{R}),f \in Λ_α(X \times Y))$. To do so, we construct partition trees on $X$ and $X \times Y$ such that analysis with respect to scale is sensible. We obtain results resembling those of [arXiv:2503.12629] and [arXiv:2508.13322], but with the finite sets $X$ and $X \times Y $ as support. In particular we construct the paraproduct $Π_{A',A''}^{L,S}: f \to \tilde{A}_{L,S}(f) + Δ_{L,S}(A,f)$ such that $Δ_{L,S}(A,f) \in Λ_{2α}(X \times Y)$ and $\lVert Δ_{L,S}(A,f) \rVert_{Λ_{2α}(X \times Y)} \leq C_A \lVert f \rVert_{Λ_α(X \times Y)}$. Analogous results are obtained when the support is just one finite set, $X$. This extension is motivated by situations where one wishes to separate the singular and smooth components of such compositions in graph signal processing environments.
Recent endeavors in some areas of graph signal processing [10] have focused on translating tools from harmonic analysis on Euclidean domains [12,9] to graphs and manifolds [11,3,6,7]. In particular, [11,3,7,6] construct wavelet bases for graphs and [7,6] provide theoretical results concerning the relationship between sparsity and wavelet coefficients in the graph setting. Outlined in this note is an analogous enterprise centered on extending the results of (tensor) paraproducts [2,5,4] for α-Hölder functions supported on [0, 1] d to hierarchical paraproducts for α-Hölder functions supported on X or X × Y where X and X × Y are finite sets with no explicit structure amongst their members. In particular, we obtain the following results on hierarchical paraproducts and hierarchical tensor paraproducts:
Theorem 1.1. Suppose A ∈ C 1 (R), f ∈ Λ α (X), 0 < α < 1 2 , then for the operator T : f → A(f ) we can approximate A(f ) with (1) such that the hierarchical paraproduct transforms T : f → A(f ) to
where ∆ L (A, f ) := A(f ) -ÃL (f ) ∈ Λ 2α (X) is the residual with twice the regularity of f and 1 2 , then for the operator T : f → A(f ) we can approximate A(f ) with
such that the hierarchical tensor paraproduct transforms T : f → A(f ) to Date: January 2026.
A ′ ,A ′′ : f → ÃL,S (f ) + ∆ (L,S) (A, f ) (5) where ∆ L,S (A, f ) = A(f ) -ÃL,S (f ) ∈ Λ 2α (X × Y ) is a residual with twice the regularity of f and ∥∆ L,S (A, f )∥ Λ2α(X×Y ) ≤ C A ∥f ∥ Λα(X×Y ) (6) Theorems 1.1 and 1.2 enumerated above are spiritually similar to the main results of [5,4], but their explicit constructions are technically distinct. Without the extension included in this note, computing paraproduct decompositions of compositions supported on a finite sets of points not imbued with a relational structure is intractable with the technology developed in [5,4]. Consequently, we (1) show how to construct the hierarchical operators needed to build the approximations ÃL , ÃL,S in Theorems 1.1 and 1.2 (2) include explicit estimates on the regularity of the residual as the regularity arguments are technically distinct from those in [5,4], though they remain morally compatible.
We mention in passing that hierarchical paraproducts for the setting (f ∈ Λ α (X), A ∈ C 1 (R)) can be useful in machine learning situations where f maps from a finite set of data points to labels (e.g. MNIST) and a C 1 function is applied to f , but one wishes to extract the latent features of f in the presence of this composition c.f. [7] and hierarchical tensor paraproducts can be useful in matrix organization settings as detailed by [6] where, again, knowledge of singularity invariances of the matrix with respect to its organized geometry built on the space of partition trees is obtainable in the presence of C 2 functions. Of note, one could consider an extension to this paper for the setting
, analogous to the generalization developed in [4].
The author wishes to thank Ronald R. Coifman for his insight.
We begin by defining a multiscale partition on a finite set X = {x 1 , x 2 , . . . , x N }. The relevant objects are itemized below, following the same notation as [7,6].
(
We say X l k is the kth node of X l which is a tree of X generated at scale l such that X l := ∪ n(l) k=1 X l k where k = 1, . . . , n(l) indexes the number of nodes in tree X l (4) Let j = 1, . . . , H(l, k) index the set of nodes at the next finest scale whose elements are contained in X l k such that
We require that all the nodes k = 1, . . . , n(l) of a partition tree at scale l are mutually disjoint i.e. X l 1 ∩ X l 2 ∩ . . . ∩ X l n(l) = ∅ Here we associate the construction of the partition tree, T X , in steps (1-5) above with the map ϕ(X, l, k, n(l), j, H(l, k)) → T X . We can use ϕ to construct an analogous partition tree for the finite set Y = {y 1 , y 2 , . . . , y N } for each y i ∈ R m such that the partition tree for Y is obtained by ϕ(Y, s, r, n(s), i, H(s, r)) → T Y . We refer the reader to [1,6] for a more detailed exposition on explicit constructions of multiscale partition trees on finite sets, but here we use the map ϕ to build T Y for the sake of readability. The following definitions are used to define α-Hölder regularity in this context: Definition 3.1. Dyadic distance between two points in a set
Definition 3.2. We say f ∈ Λ α (X), Λ α (X × Y ), respectively, if the following conditions are satisfied, respectively:
where C, α > 0 It now becomes profitable to define the Haar-like family (see [7,6] for more details) that we wish to process functions in Λ α (X), Λ α (X × Y ) with. Definition 3.3. We now define the Haar-like family supported on X and X × Y following [7]. Let the jth wavelet ψ l k,j (x) and scaling ϕ l k,j (x) functions associated with the node X l k be defined by:
where X l+1 m , X l+1 n ⊂ X l k and m, n ∈ {1, 2, . . . , H(l, k)} i.e. m, n are arbitrary indices associated with the nodes in the subtree of X l k used to define the jth Haar function associa
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