Multivariate Weighted Wiener Spaces

• Multivariate weighted Wiener spaces are families of function spaces defined via weighted Fourier coefficient summability and local oscillation control across diverse settings.
• They include formulations such as weighted Fourier algebras, quasi-Beurling algebras, Wiener amalgam/modulation spaces, and weighted Paley–Wiener spaces, each with distinct geometric and operator-theoretic approaches.
• Their applications span approximation theory, time-frequency analysis, and operator calculus, influencing sparse sampling, spectral invariance, and tractability in high-dimensional integration.

Multivariate weighted Wiener spaces are families of several-variable function and sequence spaces in which Fourier coefficients, local oscillations, or time-frequency representations are controlled by weights. In contemporary arXiv literature, the label covers at least five distinct but related settings: weighted Fourier algebras on locally compact abelian groups and on $\mathbb T^d$, weighted quasi-Beurling algebras on $\mathbb Z^d$, weighted Wiener amalgam and modulation spaces on $\mathbb R^n$, mixed-smoothness weighted Wiener classes on the torus, and weighted Paley–Wiener spaces of bandlimited entire functions. These theories share a weighted summability principle, but they differ in ambient geometry, admissible weights, and operator calculus (Bhimani et al., 22 Jul 2025, Dabhi et al., 23 Feb 2026, Guo et al., 2016, Moeller et al., 10 Aug 2025, Carneiro et al., 2023).

1. Terminological scope and principal models

A common starting point is the weighted Fourier algebra on a locally compact abelian group $G$ with dual $\Gamma$. If $\omega:G\to[1,\infty)$ is submultiplicative and $X$ is a unital commutative complex Banach algebra, the weighted Lebesgue norm is

$$\|f\|_{L^q_\omega(G,X)}=\left(\int_G \|f(x)\|^q\omega(x)^q\,dx\right)^{1/q},$$

and the associated weighted Fourier algebra is

$$A^q_\omega(\Gamma,X)=\{\widehat f:\Gamma\to X:\ f\in L^q_\omega(G,X)\}.$$

In the discrete multivariate case $G=\mathbb Z^d$, $\mathbb Z^d$0, this becomes

$\mathbb Z^d$1

so the multivariate structure is encoded directly in weighted $\mathbb Z^d$2-summability of Fourier coefficients (Bhimani et al., 22 Jul 2025).

A second model is provided by weighted quasi-Beurling algebras on $\mathbb Z^d$3,

$\mathbb Z^d$4

with convolution as multiplication. Here the weight is again submultiplicative, but the functional-analytic setting is quasi-Banach rather than Banach when $\mathbb Z^d$5 (Dabhi et al., 23 Feb 2026).

A third model is time-frequency based. For a nonzero window $\mathbb Z^d$6, the weighted Wiener amalgam space $\mathbb Z^d$7 is defined by

$\mathbb Z^d$8

with $\mathbb Z^d$9 the short-time Fourier transform. The local component is frequency-side $\mathbb R^n$0, the global component is spatial $\mathbb R^n$1, and the weights split as $\mathbb R^n$2 (Guo et al., 2016).

A fourth model is the mixed-smoothness torus class

$\mathbb R^n$3

together with related hybrid spaces

$\mathbb R^n$4

These spaces combine dominating mixed smoothness and isotropic smoothness in one Fourier norm (Moeller et al., 10 Aug 2025, Kolomoitsev et al., 2021).

A fifth model is the weighted Paley–Wiener space

$\mathbb R^n$5

which is a weighted $\mathbb R^n$6-space of bandlimited entire functions (Carneiro et al., 2023).

These constructions are not equivalent definitions of a single space. Rather, they form a family of weighted Wiener-type frameworks linked by Fourier summability, multivariate geometry, and operator-theoretic applications.

2. Fourier-algebra and Beurling formulations

In weighted Fourier algebras on LCA groups, admissibility of the weight is central. A weight $\mathbb R^n$7 is called admissible when it satisfies the Beurling–Domar condition

$\mathbb R^n$8

and for $\mathbb R^n$9 the paper also uses the notion of a $G$0-algebra weight, characterized by

$G$1

These hypotheses ensure algebra properties, existence of localized elements, and a workable Fourier-transform theory (Bhimani et al., 22 Jul 2025).

The main structural theorem in this setting is a weighted converse Wiener–Lévy result. If the composition operator $G$2 maps $G$3 to $G$4, or acts on $G$5, then $G$6 must be real analytic on its domain; the paper states the argument for $G$7 and remarks that the same method works for $G$8 (Bhimani et al., 22 Jul 2025). In this sense, mapping properties on multivariate weighted Wiener algebras characterize analyticity of nonlinear superposition operators.

The discrete quasi-Beurling theory on $G$9 emphasizes spectral invariance and inverse-closedness. For fixed $\Gamma$0 and $\Gamma$1, the cited Wiener–Domar–Żelazko–GRS criterion states that

$\Gamma$2

The same paper develops countable projective and inductive limits,

$\Gamma$3

and proves that their inverse-closedness is governed by the GRS and extended GRS conditions, respectively (Dabhi et al., 23 Feb 2026).

For finite-dimensional multivariate theory, the explicit two-variable case $\Gamma$4 is especially detailed. The paper introduces coordinatewise growth indicators $\Gamma$5, constructs auxiliary weights $\Gamma$6, and proves that pointwise invertibility of $\Gamma$7 on $\Gamma$8 yields inverses in corresponding projective or inductive limit spaces. This places weighted Wiener spaces inside a hierarchy ranging from polynomially weighted to exponentially weighted and subexponentially weighted algebras (Dabhi et al., 23 Feb 2026).

3. Periodic mixed-smoothness spaces, sparse approximation, and tractability

On the torus, one major branch studies weighted Wiener norms adapted to dominating mixed smoothness. For

$\Gamma$9

the geometry of the frequency set is organized through step hyperbolic layers

$\omega:G\to[1,\infty)$0

on which the product weight is essentially constant. This hyperbolic-cross structure drives nonlinear approximation rates (Moeller et al., 10 Aug 2025).

The main $\omega:G\to[1,\infty)$1-term trigonometric approximation theorem states that if $\omega:G\to[1,\infty)$2 and

$\omega:G\to[1,\infty)$3

then for $\omega:G\to[1,\infty)$4,

$\omega:G\to[1,\infty)$5

while for $\omega:G\to[1,\infty)$6, $\omega:G\to[1,\infty)$7, the upper bound acquires an extra factor $\omega:G\to[1,\infty)$8. The same paper proves the Wiener-to-Wiener estimate

$\omega:G\to[1,\infty)$9

and uses these widths in nonlinear sampling recovery via compressed sensing (Moeller et al., 10 Aug 2025).

A closely related sparse-grid theory is developed for the hybrid torus spaces $X$0. Using univariate quasi-interpolation operators $X$1, tensorized detail operators

$X$2

and generalized sparse index sets

$X$3

the paper proves a Littlewood–Paley-type norm equivalence

$X$4

and derives sparse-grid error bounds in both isotropic and mixed Wiener norms (Kolomoitsev et al., 2021). This replaces exact Fourier projection by a broad quasi-interpolation calculus that includes interpolation, sampling, Kantorovich-type operators, and related constructions.

A different computational direction concerns integration in subspaces of the Wiener algebra

$X$5

For the standard Wiener algebra $X$6, the minimal deterministic worst-case error of linear algorithms equals $X$7 for every $X$8, so $X$9 is unbounded for $$\|f\|_{L^q_\omega(G,X)}=\left(\int_G \|f(x)\|^q\omega(x)^q\,dx\right)^{1/q},$$0. By contrast, for the support-width weight

$$\|f\|_{L^q_\omega(G,X)}=\left(\int_G \|f(x)\|^q\omega(x)^q\,dx\right)^{1/q},$$1

the paper proves

$$\|f\|_{L^q_\omega(G,X)}=\left(\int_G \|f(x)\|^q\omega(x)^q\,dx\right)^{1/q},$$2

hence deterministic strong polynomial tractability; for the product-log weight

$$\|f\|_{L^q_\omega(G,X)}=\left(\int_G \|f(x)\|^q\omega(x)^q\,dx\right)^{1/q},$$3

it proves polynomial tractability; and for Goda’s space

$$\|f\|_{L^q_\omega(G,X)}=\left(\int_G \|f(x)\|^q\omega(x)^q\,dx\right)^{1/q},$$4

it proves randomized strong polynomial tractability with

$$\|f\|_{L^q_\omega(G,X)}=\left(\int_G \|f(x)\|^q\omega(x)^q\,dx\right)^{1/q},$$5

(Dick et al., 2024). These results show that the exact choice of multivariate weight can change the information-based complexity of integration from complete deterministic failure to dimension-independent tractability.

4. Time-frequency, Wiener amalgam, and operator calculus

Weighted Wiener amalgam spaces on $$\|f\|_{L^q_\omega(G,X)}=\left(\int_G \|f(x)\|^q\omega(x)^q\,dx\right)^{1/q},$$6 are defined through the short-time Fourier transform

$$\|f\|_{L^q_\omega(G,X)}=\left(\int_G \|f(x)\|^q\omega(x)^q\,dx\right)^{1/q},$$7

For separated moderate weights $$\|f\|_{L^q_\omega(G,X)}=\left(\int_G \|f(x)\|^q\omega(x)^q\,dx\right)^{1/q},$$8, the norm

$$\|f\|_{L^q_\omega(G,X)}=\left(\int_G \|f(x)\|^q\omega(x)^q\,dx\right)^{1/q},$$9

defines the weighted Wiener amalgam space $$A^q_\omega(\Gamma,X)=\{\widehat f:\Gamma\to X:\ f\in L^q_\omega(G,X)\}.$$0, whereas the reversed order defines the corresponding modulation space $$A^q_\omega(\Gamma,X)=\{\widehat f:\Gamma\to X:\ f\in L^q_\omega(G,X)\}.$$1. The Fourier transform exchanges the two: $$A^q_\omega(\Gamma,X)=\{\widehat f:\Gamma\to X:\ f\in L^q_\omega(G,X)\}.$$2 with $$A^q_\omega(\Gamma,X)=\{\widehat f:\Gamma\to X:\ f\in L^q_\omega(G,X)\}.$$3 (Guo et al., 2016).

A principal theorem of this branch states that product, convolution, and embedding properties of weighted Wiener amalgam spaces are exactly reducible to weighted Lebesgue and weighted sequence conditions. For example,

$$A^q_\omega(\Gamma,X)=\{\widehat f:\Gamma\to X:\ f\in L^q_\omega(G,X)\}.$$4

holds if and only if

$$A^q_\omega(\Gamma,X)=\{\widehat f:\Gamma\to X:\ f\in L^q_\omega(G,X)\}.$$5

equivalently

$$A^q_\omega(\Gamma,X)=\{\widehat f:\Gamma\to X:\ f\in L^q_\omega(G,X)\}.$$6

for compact Fourier support $$A^q_\omega(\Gamma,X)=\{\widehat f:\Gamma\to X:\ f\in L^q_\omega(G,X)\}.$$7 (Guo et al., 2016). In the same framework, the quotient-weight criterion

$$A^q_\omega(\Gamma,X)=\{\widehat f:\Gamma\to X:\ f\in L^q_\omega(G,X)\}.$$8

gives a compact summary of the embedding problem.

Operator-theoretic structure was developed through weighted mixed-norm $$A^q_\omega(\Gamma,X)=\{\widehat f:\Gamma\to X:\ f\in L^q_\omega(G,X)\}.$$9 spaces, Gabor frame representations, and Grothendieck theory. Weighted mixed-norm spaces $G=\mathbb Z^d$0 have the metric approximation property under the factorization bound

$G=\mathbb Z^d$1

and this transfers to weighted modulation and Wiener amalgam spaces. As a consequence, $G=\mathbb Z^d$2-nuclearity criteria and trace formulae hold for operators on these spaces (Delgado et al., 2014).

The ultradistributional extension replaces classical local components by a translation–modulation invariant Banach space $G=\mathbb Z^d$3 and global components by $G=\mathbb Z^d$4 or $G=\mathbb Z^d$5. The resulting spaces

$G=\mathbb Z^d$6

admit window-independent norms, discrete characterizations via uniformly concentrated partitions of unity, duality

$G=\mathbb Z^d$7

and complex interpolation formulae (Dimovski et al., 2022). This branch shows that weighted Wiener amalgam theory persists even in the quasianalytic ultradistribution regime.

Weighted Wiener amalgam spaces also serve as symbol classes for pseudodifferential operators. For $G=\mathbb Z^d$8, the paper on $G=\mathbb Z^d$9-pseudodifferential operators proves that

$\mathbb Z^d$00

is equivalent to the existence of $\mathbb Z^d$01 such that

$\mathbb Z^d$02

which is an almost-diagonalization along a $\mathbb Z^d$03-dependent graph in phase space. This yields boundedness, algebra properties, and a Wiener property for the corresponding operator calculus (Cordero et al., 2018).

5. Weighted Paley–Wiener spaces and bandlimited variants

A distinct Hilbertian branch studies multivariate weighted Paley–Wiener spaces

$\mathbb Z^d$04

By the multidimensional Paley–Wiener theorem cited there, such functions have Fourier transform supported in the closed ball

$\mathbb Z^d$05

The basic embedding problem is encoded by

$\mathbb Z^d$06

which equals the inverse squared norm of the embedding $\mathbb Z^d$07 (Carneiro et al., 2023).

The central structural theorem is the dimension-shift identity

$\mathbb Z^d$08

proved by radial symmetrization and lift operators. Thus the optimal embedding constant is dimension-independent, despite the ambient multivariate entire-function setting (Carneiro et al., 2023). The same paper gives asymptotics for $\mathbb Z^d$09 and, in the arithmetic case $\mathbb Z^d$10, exact sharp constants and extremizers in terms of Bessel/de Branges functions $\mathbb Z^d$11 and $\mathbb Z^d$12.

This Hilbert-space theory has a one-variable de Branges precursor. Weighted Paley–Wiener spaces in the sense of Lyubarskii–Seip are de Branges spaces whose majorant-weight $\mathbb Z^d$13 satisfies

$\mathbb Z^d$14

and the model spaces $\mathbb Z^d$15 are defined through a logarithmic potential $\mathbb Z^d$16. The exposition of this theory shows that for mountain-chain de Branges spaces the majorant takes the generic form

$\mathbb Z^d$17

with $\mathbb Z^d$18, and weighted Paley–Wiener spaces are precisely the corresponding model spaces $\mathbb Z^d$19 up to equivalence of norms (Poulin, 2013). That work is one-dimensional rather than multivariate, but it supplies a precise analytic model for the bandlimited weighted theory.

6. Foundational precursors and distinct usages of “Wiener space”

Not all multivariate Wiener-type spaces in the literature are weighted. Brudnyi’s space

$\mathbb Z^d$20

is a multivariate Wiener–L. Young type space defined via local best polynomial approximation over cube packings. It satisfies the nonlinear piecewise polynomial approximation estimate

$\mathbb Z^d$21

but the paper explicitly states that it is entirely unweighted (Brudnyi, 2015). Likewise, the Jordan–Wiener space

$\mathbb Z^d$22

is defined through higher-order oscillation or local polynomial approximation and is presented as an unweighted multivariate generalization of the classical Jordan and Wiener spaces (Brudnyi et al., 2018).

These unweighted theories are nevertheless conceptually important. They show that in several variables the correct local object is higher-order oscillation or local polynomial approximation on cube packings, not naive first-order cube oscillation. This suggests a template for weighted Jordan–Wiener extensions, although neither paper develops such a theory (Brudnyi, 2015, Brudnyi et al., 2018).

A separate source of ambiguity comes from stochastic analysis. “Cubature on Wiener space” with weighted spaces studies Banach spaces of test functions

$\mathbb Z^d$23

whose derivatives are controlled by weights $\mathbb Z^d$24, in order to treat SPDEs with unbounded coefficients and payoffs. Here the multivariate aspect comes from $\mathbb Z^d$25-dimensional Brownian motion and iterated Lie derivatives $\mathbb Z^d$26, but the resulting weighted spaces are not Fourier-algebraic Wiener spaces (Doersek et al., 2012). Likewise, abstract Wiener model spaces are enhanced Gaussian objects built from graded ambient spaces, nonlinear lifts, and homogeneous norms

$\mathbb Z^d$27

again representing a different usage of the term “Wiener space” (Chiusole et al., 2023).

The broad conclusion is that multivariate weighted Wiener spaces do not form a single canonical category. In Fourier analysis and approximation theory they are weighted coefficient or time-frequency spaces; in bandlimited function theory they are weighted Paley–Wiener spaces; in stochastic analysis they are weighted test-function or enhanced Gaussian model spaces. What unifies these branches is the use of a weight to control multivariate complexity, regularity, or growth, but the underlying notions of geometry and approximation are fundamentally different.