Mellin Averaging: Methods and Applications

• Mellin averaging is a set of techniques that uses the Mellin transform to re-encode multiplicative data into analytic structures via contour integration and residue extraction.
• It enables precise sampling, reconstruction, and quadrature on the positive real axis while controlling asymptotic behavior through bandlimiting and Newton polytope analysis.
• The method underpins advanced applications in conformal field theory, holography, analytic number theory, and operator-valued integration, bridging geometry and analysis.

Searching arXiv for relevant papers on Mellin averaging and related Mellin-analysis contexts. Mellin averaging denotes a family of constructions in which the Mellin transform is used to average, reorganize, or asymptotically extract information from objects with multiplicative structure. In its most classical form, the averaging is over multiplicative characters $x^s$ or $z^s$, converting scaling behavior into analytic structure in the Mellin variable. In other settings, the same mechanism appears as residue summation, Mellin-bandlimited reconstruction on $\mathbb{R}_+$, contour-dependent transforms for rational functions, Mellin-space representations of conformal correlators, motivic or $p$-adic integration against multiplicative characters, and asymptotic “averaging over $N$” defined through Mellin residues. The term is therefore not a single universally fixed definition. Rather, it names a recurrent analytic pattern: multiplicative data are encoded by Mellin kernels, and the resulting transform acts as an averaging device whose output is governed by poles, Gamma factors, contour choices, or asymptotic expansions (Nilsson et al., 2010, Boyadzhiev, 2023, Kudler-Flam et al., 14 May 2026).

1. Classical meaning: multiplicative averaging and residue extraction

For a function $f(x)$, the ordinary Mellin transform is

$$\mathcal{M}\!\left\{ f(x)\right\} \equiv \phi(s)=\int_0^\infty x^{s-1}f(x)\,dx,$$

with inverse

$$\mathcal{M}^{-1}\!\left\{\phi(s)\right\}\equiv f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\phi(s)\,ds.$$

This transform is especially useful when a function naturally has a power-law structure, because the kernel $x^{s-1}$ converts multiplicative behavior into additive behavior in Mellin space (Treumann et al., 2015).

In several papers, this mechanism is described explicitly as a form of averaging. For multivariate rational functions $g/f$, the Mellin transform

$z^s$0

is said to behave like a “multiplicative averaging” operator: it integrates a rational function against a monomial weight $z^s$1 over the positive orthant, and after analytic continuation its singularities are controlled by the Newton polytope of the denominator (Nilsson et al., 2010). In the one-variable residue framework,

$z^s$2

the Mellin transform acts like a weighted averaging or summation device: the poles of $z^s$3 encode coefficients, and inserting $z^s$4 evaluates the test function at the lattice points $z^s$5 (Boyadzhiev, 2023).

This residue-extraction perspective is central to Ramanujan’s Master Theorem and to Mellin derivations of classical identities. Specializations recover relations for Hurwitz zeta, Bernoulli polynomials, Euler polynomials, Euler numbers, exponential polynomials, and Hermite polynomials. The transform is not merely an auxiliary integral transform in these examples; it samples analytic data at a discrete pole set and packages them into generating series (Boyadzhiev, 2023).

A common misconception is that Mellin averaging is always an actual average over nearby values of the original variable. In the classical setting this is inaccurate. The “average” is typically contour-theoretic: it is an integral against multiplicative characters or an inverse Mellin contour whose residues encode the relevant discrete information. This suggests that the averaging language is structural rather than probabilistic in most classical uses.

2. Geometric control: poles, coamoebas, and coefficient dependence

For multivariate rational functions, Mellin averaging acquires a strong polyhedral and analytic-geometric form. If $z^s$6 is completely non-vanishing on $z^s$7 and $z^s$8 is its Newton polytope, then

$z^s$9

where the polar locus consists of finitely many families of parallel hyperplanes

$\mathbb{R}_+$0

Each family is attached to a facet of $\mathbb{R}_+$1, with $\mathbb{R}_+$2 the primitive inward normal and $\mathbb{R}_+$3 the supporting constant. The meromorphic continuation is produced by repeated integration by parts in directions determined by the facet normals (Nilsson et al., 2010).

The contour dependence of Mellin transforms in several variables is governed by the coamoeba

$\mathbb{R}_+$4

For $\mathbb{R}_+$5 in a connected component $\mathbb{R}_+$6 of $\mathbb{R}_+$7, one defines

$\mathbb{R}_+$8

Different connected components yield different Mellin transforms, because the contour can be moved continuously only while avoiding the coamoeba; crossing the coamoeba changes the contour class and may introduce residue contributions (Nilsson et al., 2010).

The coefficient dependence of the transform is also structured. The entire prefactor $\mathbb{R}_+$9 is $p$0-hypergeometric in the coefficients $p$1 of $p$2, satisfying the GKZ box operators and homogeneity equations. In special cases this yields classical functions such as

$p$3

for $p$4 (Nilsson et al., 2010).

This geometric formulation shows that Mellin averaging is not arbitrary smoothing. Its singularities, contour classes, and parameter dependence are constrained by the Newton polytope, the coamoeba, and the $p$5-hypergeometric structure. A plausible implication is that “averaging” in Mellin space is often best understood as a geometry-sensitive re-encoding of multiplicative data rather than as a generic regularization procedure.

3. Mellin averaging on $p$6: bandlimiting, sampling, and quadrature

In Mellin analysis on the positive real axis, averaging is implemented through multiplicative translation, Mellin convolution, and logarithmic sampling. The Mellin translation operator is

$p$7

and the Mellin differential operator is

$p$8

These are the multiplicative analogues of ordinary translation and differentiation (Bardaro et al., 2016).

The central functional-analytic device is the distance from a Mellin-Bernstein space. If $p$9 with Mellin transform $N$0, then

$N$1

and, for continuous $N$2,

$N$3

Thus the distance to the bandlimited class is exactly the tail of the Mellin transform outside the band $N$4 (Bardaro et al., 2016).

This distance controls approximate Mellin reconstruction. The kernel

$N$5

is the Mellin analogue of the sinc kernel (Bardaro et al., 2016). For general $N$6,

$N$7

and the remainder is bounded by $N$8 (Bardaro et al., 2016).

The same philosophy governs quadrature. The Mellin-Poisson summation formula yields

$N$9

with

$f(x)$0

If $f(x)$1, the quadrature is exact: $f(x)$2 For $f(x)$3 with $f(x)$4,

$f(x)$5

and this estimate is best possible in order (Bardaro et al., 2018).

The analytic background is supplied by Mellin Paley–Wiener theory. Mellin bandlimitedness on $f(x)$6 is equivalent to analytic extension of $f(x)$7 either to the Riemann surface of the logarithm or, in the later reformulation, to a polar-analytic Mellin–Bernstein space with growth

$f(x)$8

Mellin–Hardy spaces then control exponential decay of the Mellin transform tails and hence exponential convergence of sampling and quadrature remainders (Bardaro et al., 2015, Bardaro et al., 2017).

4. Mellin-space organization in CFT, AdS/CFT, and celestial holography

In conformal field theory and related holographic settings, Mellin averaging appears as a transform representation that packages correlators into Mellin amplitudes and Gamma-function kernels. For scalar four-point functions the standard Mellin representation is

$f(x)$9

with $$\mathcal{M}\!\left\{ f(x)\right\} \equiv \phi(s)=\int_0^\infty x^{s-1}f(x)\,dx,$$0 (Chen et al., 2017).

For external operators with integer spin, this representation becomes hybrid continuous/discrete Mellin space. The discrete Mellin variables $$\mathcal{M}\!\left\{ f(x)\right\} \equiv \phi(s)=\int_0^\infty x^{s-1}f(x)\,dx,$$1 and $$\mathcal{M}\!\left\{ f(x)\right\} \equiv \phi(s)=\int_0^\infty x^{s-1}f(x)\,dx,$$2 encode tensor-structure dependence and satisfy

$$\mathcal{M}\!\left\{ f(x)\right\} \equiv \phi(s)=\int_0^\infty x^{s-1}f(x)\,dx,$$3

The resulting spinning Mellin amplitudes are organized by generalized Mack polynomials $$\mathcal{M}\!\left\{ f(x)\right\} \equiv \phi(s)=\int_0^\infty x^{s-1}f(x)\,dx,$$4, which form a natural kinematical polynomial basis (Chen et al., 2017).

For AdS tree-level gluon Witten diagrams, the Mellin representation is used recursively: $$\mathcal{M}\!\left\{ f(x)\right\} \equiv \phi(s)=\int_0^\infty x^{s-1}f(x)\,dx,$$5 The paper on $$\mathcal{M}\!\left\{ f(x)\right\} \equiv \phi(s)=\int_0^\infty x^{s-1}f(x)\,dx,$$6-gluon scattering states that Mellin space plays a role somewhat analogous to a “soft averaging” or “effective averaging” representation: it packages the AdS correlator into a form built from flat-space-like kinematics, propagators, and vertices, with AdS effects entering through Mellin variables and a simple $$\mathcal{M}\!\left\{ f(x)\right\} \equiv \phi(s)=\int_0^\infty x^{s-1}f(x)\,dx,$$7-integral dictionary (Chu et al., 2023). In particular,

$$\mathcal{M}\!\left\{ f(x)\right\} \equiv \phi(s)=\int_0^\infty x^{s-1}f(x)\,dx,$$8

with the specified replacements $$\mathcal{M}\!\left\{ f(x)\right\} \equiv \phi(s)=\int_0^\infty x^{s-1}f(x)\,dx,$$9 and $$\mathcal{M}^{-1}\!\left\{\phi(s)\right\}\equiv f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\phi(s)\,ds.$$0 (Chu et al., 2023).

Celestial holography uses a radial Mellin transform in Minkowski space: $$\mathcal{M}^{-1}\!\left\{\phi(s)\right\}\equiv f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\phi(s)\,ds.$$1 The resulting celestial Mellin representation is Mack-like: $$\mathcal{M}^{-1}\!\left\{\phi(s)\right\}\equiv f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\phi(s)\,ds.$$2 with $$\mathcal{M}^{-1}\!\left\{\phi(s)\right\}\equiv f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\phi(s)\,ds.$$3, $$\mathcal{M}^{-1}\!\left\{\phi(s)\right\}\equiv f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\phi(s)\,ds.$$4, and $$\mathcal{M}^{-1}\!\left\{\phi(s)\right\}\equiv f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\phi(s)\,ds.$$5. Contact diagrams give polynomial Mellin amplitudes, while exchange diagrams are encoded by characteristic pole families (Pacifico et al., 2024).

These constructions motivate a broader usage in which Mellin averaging denotes a representational reorganization rather than literal averaging of numerical data. The correlator is “averaged” into Mellin variables, and analyticity, factorization, and pole structure become explicit. This suggests why Mellin-space methods are attractive in CFT and holography even when the language of averaging is only analogical.

5. Deformations, radial operators, and functional Mellin integration

Mellin averaging has also been generalized by deforming the Mellin kernel or by lifting the transform to operator-valued and infinite-dimensional settings. A one-parameter fractional deformation replaces

$$\mathcal{M}^{-1}\!\left\{\phi(s)\right\}\equiv f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\phi(s)\,ds.$$6

by

$$\mathcal{M}^{-1}\!\left\{\phi(s)\right\}\equiv f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\phi(s)\,ds.$$7

yielding the inverse fractional Mellin-$$\mathcal{M}^{-1}\!\left\{\phi(s)\right\}\equiv f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\phi(s)\,ds.$$8 transform

$$\mathcal{M}^{-1}\!\left\{\phi(s)\right\}\equiv f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}x^{-s}\phi(s)\,ds.$$9

Its forward transform is

$x^{s-1}$0

As $x^{s-1}$1, one recovers the ordinary Mellin transform. The proposed advantages are adjustable convergence and algebraic dependence on $x^{s-1}$2 rather than exponential dependence, although branch points appear for non-integer $x^{s-1}$3 (Treumann et al., 2015).

For radial fractional PDEs, the Mellin transform provides an equivalent definition of the fractional Laplacian on radial functions. If $x^{s-1}$4, then

$x^{s-1}$5

so the operator becomes a Mellin-space shift $x^{s-1}$6 times an explicit Gamma-factor multiplier (Pagnini et al., 2023). In one dimension, for symmetric functions,

$x^{s-1}$7

correcting an earlier formula in the literature (Pagnini et al., 2023).

In a different direction, the functional Mellin transform is defined on a complex topological Lie group $x^{s-1}$8 with values in a unital $x^{s-1}$9-algebra: $g/f$0 This is presented as the infinite-dimensional analogue of the classical Mellin transform and as a multiplicative analogue of Fourier-based functional integration. It yields functional analogues of resolvents, complex powers, traces, logarithms, and determinants, such as

$g/f$1

Under suitable assumptions the transform intertwines convolution-like products with multiplication (LaChapelle, 2015).

A separate asymptotic generalization appears in the Mellin-transform extension of Laplace’s method. There the Laplace kernel is represented by a Mellin–Barnes integral and asymptotics are extracted by contour shifting after a generalized local expansion in powers of a conformal auxiliary function $g/f$2. The output can involve powers of $g/f$3, inverse factorials, logarithmic scales, or Tricomi $g/f$4-functions, depending on the chosen kernel representation (Kaiser, 22 Jun 2026).

6. Motivic, arithmetic, and asymptotic averaging

In motivic integration, Mellin averaging is literally integration against multiplicative characters. For an integrable motivic function $g/f$5, the motivic Mellin transform is

$g/f$6

This is the motivic analogue of the classical local Mellin transform

$g/f$7

The transform is stable in the motivic category, injective modulo measure-zero differences, compatible with Fourier transform, and specializes uniformly to $g/f$8-adic Mellin transforms (Cluckers et al., 2024).

In analytic number theory, Mellin averaging is the bridge between summatory functions of Dirichlet coefficients and vertical-line integrals of $g/f$9-functions. If

$z^s$00

then

$z^s$01

This identity is used with Gaussian weights to convert

$z^s$02

into a smoothed $z^s$03-space average of $z^s$04, leading to lower bounds on critical-line averages over intervals of logarithmic size in the analytic conductor and to bounds for the least prime ideal with nontrivial Frobenius action (Clozel, 2023).

The most explicit modern use of the phrase “Mellin averaging” in a new sense is the proposal to average over the integer parameter $z^s$05 in holography. For a function $z^s$06, bounded by a power of $z^s$07, one defines

$z^s$08

assumes analytic continuation with simple poles at non-negative integers, sets

$z^s$09

and defines the Mellin average as the formal asymptotic expansion

$z^s$10

This is explicitly not ordinary averaging over nearby integers; it is an asymptotic residue extraction designed to match the $z^s$11 character of gravitational perturbation theory (Kudler-Flam et al., 14 May 2026).

The proposal is motivated by wormhole physics, where some observables behave as though ensemble-averaged even when the only apparent parameter is an integer $z^s$12. The paper argues that Mellin averaging over $z^s$13 may reproduce such randomness if two conditions hold: the observable admits analytic continuation in $z^s$14, and it fluctuates on superpolynomially small scales in $z^s$15 (Kudler-Flam et al., 14 May 2026). Its toy models include

$z^s$16

for transcendental $z^s$17, whose Mellin transform is entire, so the Mellin average gives $z^s$18, while $z^s$19 under Mellin averaging, matching the moments of $z^s$20 with $z^s$21 uniformly distributed on the circle (Kudler-Flam et al., 14 May 2026).

7. Scope, misconceptions, and unifying themes

The principal unifying theme across these literatures is that Mellin averaging converts multiplicative structure into analytic structure. In elementary settings it averages against $z^s$22 or $z^s$23; in residue calculus it samples a meromorphic transform at pole lattices; in Mellin analysis it measures out-of-band tails and controls multiplicative sampling; in CFT and holography it packages correlators into Mellin amplitudes; in the motivic setting it averages over valuation weights and multiplicative characters; in recent gravitational work it extracts asymptotic $z^s$24 data by Mellin residues (Nilsson et al., 2010, Bardaro et al., 2016, Cluckers et al., 2024, Kudler-Flam et al., 14 May 2026).

Several misconceptions recur. One is that Mellin averaging is a single standardized construction. The literature instead supports multiple, technically distinct notions. Another is that Mellin averaging is always smoothing in the original variable. Frequently it is contour deformation, residue extraction, or representation change. A third is that Mellin methods are only formal rewritings of Fourier or Laplace methods. The papers surveyed here emphasize distinctive Mellin features: multiplicative translation, bandlimitedness on $z^s$25, dependence on Newton polytopes and coamoebas, discrete Mellin variables for spin, and asymptotic $z^s$26 extraction rather than local averaging (Bardaro et al., 2017, Chen et al., 2017, Kaiser, 22 Jun 2026).

What remains stable across contexts is the role of the Mellin kernel as an interface between scaling and analyticity. Whether the goal is to reconstruct a function from logarithmic samples, compute a conformal correlator, derive a quadrature formula, continue a multivariate rational transform, define a motivic multiplicative integral, or encode an asymptotic gravitational average, Mellin averaging is the procedure by which multiplicative data are transferred into a domain where poles, Gamma factors, residues, and contour geometry become the primary invariants.