Power Fourier Decay: Theory and Applications

• Power Fourier decay is a quantitative measure describing how Fourier coefficients or transforms decay at a polynomial rate, indicating underlying regularity.
• It distinguishes polynomial frequency decay from the Rajchman, rapid, or exponential cases, linking measure-theoretic and analytic properties.
• Applications include the analysis of self-similar measures, hyperbolic dynamics, and spectral theory, demonstrating its impact on fractal and analytic structures.

Power Fourier decay is the quantitative high-frequency decay of Fourier coefficients or Fourier transforms at a polynomial rate. In the most common measure-theoretic form, it means that for some $\alpha>0$ one has $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$ as $|\xi|\to\infty$; for periodic functions it appears as decay of Fourier coefficients $c_k(f)=O(|k|^{-\alpha})$; and in other settings it is formulated through averaged Fourier norms or oscillatory integrals. The literature also distinguishes this notion from the weaker Rajchman property $\widehat{\nu}(t)\to 0$, from rapid decay faster than every power, and from exponential decay. In several measure-theoretic settings on $\mathbb R^d$, power Fourier decay is equivalent to positive Fourier dimension, whereas in tame analytic categories exponential decay is tied to holomorphic extension to a strip (Solomyak, 2019, Solomyak, 20 Aug 2025, Piro, 2014, Comte et al., 8 Jul 2025).

1. Definitions and principal variants

The precise meaning of power Fourier decay depends on the underlying object, but the common feature is a bound by a negative power of frequency.

Setting	Fourier quantity	Typical decay statement
Finite Borel measure	$\widehat{\nu}(t)$ or $\widehat{\mu}(\xi)$	$O(|t|^{-\alpha})$ or $O(|\xi|^{-\alpha})$ for some $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$0
Periodic or holomorphic function	Fourier coefficients or Taylor/Fourier coefficients	$|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$1, or stronger $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$2 for every $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$3
Fractal average-decay setting	$|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$4	$|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$5

For a finite positive Borel measure $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$6 on $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$7, one standard definition is

$|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$8

and “power Fourier decay” means membership in $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$9 (Solomyak, 2019). In $|\xi|\to\infty$0, homogeneous self-affine and self-similar papers use the analogous condition $|\xi|\to\infty$1, often denoting the corresponding class by $|\xi|\to\infty$2 or $|\xi|\to\infty$3 (Solomyak, 2021, Solomyak, 20 Aug 2025).

Several adjacent notions are important. Rapid decay is stronger than power decay: for cusp functions on the upper half-plane, the coefficients satisfy $|\xi|\to\infty$4 for every integer $|\xi|\to\infty$5, which is stronger than any single power law (Piro, 2014). Rajchman decay is weaker: it asks only for $|\xi|\to\infty$6 as $|\xi|\to\infty$7 (Solomyak, 2019). Exponential decay is stronger again, and for $|\xi|\to\infty$8 it is characterized by analyticity and holomorphic strip extension, with the optimal decay rate equal to the maximal strip width (Comte et al., 8 Jul 2025). There are also intermediate formulations: “polynomially decaying on average,” where large Fourier coefficients occur only on a sparse exceptional set of frequencies, and polylogarithmic decay, of the form $|\xi|\to\infty$9, which is weaker than any power law (Baker et al., 2024).

2. Function-theoretic formulations

In classical holomorphic and function-space settings, power Fourier decay is closely tied to regularity, boundary behavior, and oscillation structure.

For cusp functions $c_k(f)=O(|k|^{-\alpha})$0 on the upper half-plane

$c_k(f)=O(|k|^{-\alpha})$1

periodicity $c_k(f)=O(|k|^{-\alpha})$2 and the cusp condition $c_k(f)=O(|k|^{-\alpha})$3 allow one to pass through the map $c_k(f)=O(|k|^{-\alpha})$4 to a holomorphic function on the disk. The resulting Fourier expansion

$c_k(f)=O(|k|^{-\alpha})$5

has coefficients satisfying, for every $c_k(f)=O(|k|^{-\alpha})$6, constants $c_k(f)=O(|k|^{-\alpha})$7 with

$c_k(f)=O(|k|^{-\alpha})$8

Thus the decay is not merely polynomial for one exponent but rapidly decreasing in the Schwartz-type sense (Piro, 2014).

A different line of work characterizes generalized Lipschitz and Besov spaces through weighted Fourier-tail conditions. For a $c_k(f)=O(|k|^{-\alpha})$9-admissible family of multiplier operators $\widehat{\nu}(t)\to 0$0, membership in $\widehat{\nu}(t)\to 0$1 or $\widehat{\nu}(t)\to 0$2 is equivalent to annular Fourier conditions weighted by the majorant $\widehat{\nu}(t)\to 0$3; when $\widehat{\nu}(t)\to 0$4, these become classical Titchmarsh-type power-decay statements (Jordão, 2019). The same theme appears in a more pointwise form for absolutely continuous, uniformly Hölder continuous functions whose finite differences have a uniformly bounded number of local extrema: under those assumptions, one gains one full power and obtains

$\widehat{\nu}(t)\to 0$5

while the converse implication recovers $\widehat{\nu}(t)\to 0$6 regularity, and for $\widehat{\nu}(t)\to 0$7 also absolute continuity of the next derivative (Nissilä, 2018).

Hardy-Morrey theory furnishes another explicit scale of pointwise decay estimates. The Fourier transform of $\widehat{\nu}(t)\to 0$8 in the homogeneous Hardy-Morrey space satisfies

$\widehat{\nu}(t)\to 0$9

and the localizable version gives the global weighted bound

$\mathbb R^d$0

These estimates are derived from atomic decompositions and moment conditions (Almeida et al., 2020).

At the analytic end of the spectrum, $\mathbb R^d$1-functions exhibit a sharp dichotomy between polynomial, rapid, and exponential behavior. For $\mathbb R^d$2, exponential decay of $\mathbb R^d$3 is equivalent to analyticity, rapid decay is equivalent to $\mathbb R^d$4 and in fact again to analyticity, and the optimal exponential rate is exactly the maximal width of a horizontal strip to which $\mathbb R^d$5 extends holomorphically (Comte et al., 8 Jul 2025). This shows that power Fourier decay is substantial regularity, but it is still strictly weaker than strip-analyticity.

3. Self-similar and self-affine measures

A major modern domain for power Fourier decay is the theory of self-similar and self-affine measures. The basic self-similar measure on the line is the unique probability measure satisfying

$\mathbb R^d$6

with positive weights $\mathbb R^d$7 (Solomyak, 2019). In the homogeneous case there is an infinite convolution structure; in the non-homogeneous case this factorization is absent, and the lack of convolution is the main new difficulty (Solomyak, 2019).

A sharp generic theorem for the line states that for every $\mathbb R^d$8 there exists an exceptional set $\mathbb R^d$9 with $\widehat{\nu}(t)$0 such that for every contraction vector $\widehat{\nu}(t)$1, every non-trivial translation vector $\widehat{\nu}(t)$2, and every positive probability vector $\widehat{\nu}(t)$3, the self-similar measure $\widehat{\nu}(t)$4 has power Fourier decay (Solomyak, 2019). The proof is a multi-parameter generalization of the Erdős–Kahane argument and replaces product structure with a recursive tree analysis of “good” edges in frequency space.

In higher-dimensional homogeneous self-affine settings, the Fourier transform admits an infinite product representation

$\widehat{\nu}(t)$5

and typical-parameter theorems again give power decay. For diagonal $\widehat{\nu}(t)$6 with $\widehat{\nu}(t)$7, Lebesgue-a.e. parameter vector yields power Fourier decay for every homogeneous affinely irreducible self-affine measure (Solomyak, 2021).

For homogeneous self-similar measures in $\widehat{\nu}(t)$8, $\widehat{\nu}(t)$9, two generic results are available. If the digit set spans $\widehat{\mu}(\xi)$0, then for every fixed orthogonal part $\widehat{\mu}(\xi)$1 and weights $\widehat{\mu}(\xi)$2, all contraction ratios except a set of Hausdorff dimension $\widehat{\mu}(\xi)$3 produce power Fourier decay. In even dimensions $\widehat{\mu}(\xi)$4, affine irreducibility yields a stronger almost-everywhere statement in the full parameter space of contracting similitudes (Solomyak, 20 Aug 2025). Combined with the Corso–Shmerkin $\widehat{\mu}(\xi)$5-dimension theorem, these decay results imply absolute continuity and even $\widehat{\mu}(\xi)$6 densities in the super-critical regime (Solomyak, 20 Aug 2025).

A distinct phenomenon appears for nonlinear images. A homogeneous self-similar measure may itself have poor decay, but if $\widehat{\mu}(\xi)$7 satisfies $\widehat{\mu}(\xi)$8, then the pushforward $\widehat{\mu}(\xi)$9 has genuine power Fourier decay: $O(|t|^{-\alpha})$0 This was proved in a quantitative form for homogeneous self-similar measures without any separation condition (Mosquera et al., 2017). A related theorem for homogeneous self-similar measures satisfying the strong separation condition and $O(|t|^{-\alpha})$1 gives polynomial decay for oscillatory integrals

$O(|t|^{-\alpha})$2

whenever $O(|t|^{-\alpha})$3 with $O(|t|^{-\alpha})$4 and $O(|t|^{-\alpha})$5 (Chang et al., 2017).

4. Dynamical and hyperbolic mechanisms

Power Fourier decay also arises for invariant measures in hyperbolic and symbolic dynamics, where the central mechanisms are stationarity, nonlinearity, temporal distance, sum-product phenomena, and flattening.

For convex-cocompact hyperbolic surfaces, $O(|t|^{-\alpha})$6-Patterson–Sullivan densities associated to a Hölder potential $O(|t|^{-\alpha})$7 with $O(|t|^{-\alpha})$8 are stationary measures for a random walk on $O(|t|^{-\alpha})$9 with exponential moment. As a consequence, when $O(|\xi|^{-\alpha})$0 is a surface, the associated equilibrium state $O(|\xi|^{-\alpha})$1 satisfies power decay for a broad class of oscillatory integrals

$O(|\xi|^{-\alpha})$2

for Hölder amplitudes and $O(|\xi|^{-\alpha})$3 phases with nonvanishing differential. The non-wandering set of the geodesic flow then has positive intrinsic Fourier dimension (Leclerc, 2023).

A broader Axiom A result treats nonlinear, area-preserving diffeomorphisms on surfaces. If $O(|\xi|^{-\alpha})$4 is a $O(|\xi|^{-\alpha})$5 Axiom A diffeomorphism on a surface, area-preserving on a basic set $O(|\xi|^{-\alpha})$6, and at least one of the invariant distributions $O(|\xi|^{-\alpha})$7 or $O(|\xi|^{-\alpha})$8 is not $O(|\xi|^{-\alpha})$9, then the equilibrium state has positive lower Fourier dimension. The proof reduces Fourier decay to non-concentration of a temporal distance function through a sum-product argument and a suspension-flow construction. A corollary gives power Fourier decay for the density of states measure of the Fibonacci Hamiltonian and hence positivity of the lower Fourier dimension for its spectrum in the small-coupling regime (Leclerc, 31 Jul 2025).

A unified methodological framework is provided by $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$00-flattening. The three-step scheme is averaging, flattening, and separation. In this framework, polynomial decay is obtained for Patterson–Sullivan measures of convex cocompact hyperbolic manifolds, Gibbs measures associated to non-integrable $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$01 conformal systems, and stationary measures for carpet-like non-conformal iterated function systems, while Diophantine self-similar measures yield only polylogarithmic decay (Baker et al., 2024). This contrast is central: nonlinear and hyperbolic systems supply stronger frequency separation than rigid self-similar ones.

5. Arithmetic and combinatorial thresholds

Several results identify sharp arithmetic or dimensional thresholds beyond which power Fourier decay must occur.

For multiplicative convolutions, let $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$02 be probability measures on $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$03 with finite $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$04-energy and

$|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$05

Then the multiplicative convolution

$|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$06

has power Fourier decay: $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$07 for sufficiently large $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$08 and some $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$09 depending only on the $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$10 (Orponen et al., 2023). The threshold $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$11 is presented as the optimal one suggested by Bourgain in 2010 (Orponen et al., 2023).

Continued-fraction constructions show that power decay is not the generic endpoint. For a very broad class of recursively constrained sets of irrational numbers, one can construct measures supported on numbers with arbitrarily long blocks of partial quotients satisfying the given rule, and the resulting Fourier transform tends to zero. The general conclusion is Rajchman decay, not power decay, although explicit slow quantitative bounds are obtained in special cases such as $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$12 or $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$13 (Fraser, 21 Mar 2025). This sharpens the conceptual distinction between pointwise vanishing at infinity and polynomial-rate decay.

In a different Fourier-transform setting, the Fock–Krylov formalism writes the survival amplitude of an unstable quantum state as

$|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$14

If the density of states has threshold behavior $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$15, then the survival probability has the power-law tail

$|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$16

The same analysis shows that a purely exponential survival amplitude is impossible under the formalism’s stated assumptions (Jiménez et al., 2021). This is not a Fourier-decay theorem for $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$17 itself, but it is a direct power-law consequence of Fourier-transform structure.

6. Variants, limitations, and consequences

Power Fourier decay sits inside a wider hierarchy of decay phenomena, and the literature repeatedly emphasizes the need to separate them.

One important variant is average decay. For an $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$18-dimensional measure $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$19 on $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$20, the quantity

$|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$21

is defined as the supremum of $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$22 such that

$|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$23

Improved upper and lower bounds for $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$24 yield sharper fractal maximal estimates for the Schrödinger and wave equations, including the consequence that a wave solution with data in $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$25 cannot diverge on a $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$26-dimensional manifold (Lucà et al., 2015). This is a decay theory of spherical averages rather than pointwise power Fourier decay, but it shares the same oscillatory-cancellation core.

Another recurring limitation is that the strongest available theorem may stop short of polynomial decay. Homogeneous self-similar measures can exhibit sparse exceptional sets of large Fourier values rather than full power decay (Mosquera et al., 2017). The flattening approach produces polylogarithmic rather than polynomial decay for Diophantine self-similar measures (Baker et al., 2024). Continued-fraction constructions often yield only Rajchman behavior in full generality (Fraser, 21 Mar 2025). Conversely, some settings give more than power decay: cusp functions have coefficients decaying faster than any polynomial (Piro, 2014), and analytic $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$27-functions have exponential Fourier decay governed exactly by strip width (Comte et al., 8 Jul 2025).

The consequences of power Fourier decay are correspondingly diverse. In self-similar and self-affine geometry it is linked to positive Fourier dimension and, in super-critical parameter regimes, to absolute continuity and $|\widehat{\mu}(\xi)|=O(|\xi|^{-\alpha})$28 densities (Solomyak, 20 Aug 2025). For smooth images of self-similar measures it yields normality phenomena through Davenport–Erdős–LeVeque-type criteria (Chang et al., 2017). In hyperbolic dynamics it feeds into positive lower Fourier dimension, spectral gaps, and fractal uncertainty principles (Leclerc, 2023, Baker et al., 2024, Leclerc, 31 Jul 2025). A plausible implication is that power Fourier decay functions as a quantitative non-concentration principle in frequency space: when available, it converts structural information—holomorphy, curvature, transversality, Diophantine separation, or hyperbolic nonlinearity—into measurable regularity of the underlying object.