# Anchored Accumulation Calculus: Mellin Diagonalization and Operator-Theoretic Classification of Scale-Covariant Memory

## Abstract

We formalize Anchored Accumulation Calculus (AAC), an operator-theoretic framework for scale-covariant memory on the multiplicative group R+. By defining memory as causal accumulation along exponential scale trajectories, we establish that AAC operators on a dense core are unitarily equivalent to translation-invariant Fourier multipliers. This construction allows us to define the maximal closed extension of these operators, avoiding ad hoc integration-by-parts arguments. We subsequently classify their generators: completely monotone memory kernels are rigorously shown to generate scale-invariant L´evy subordinators via Bernstein’s theorem, while heavy-tailed kernels are unitarily equivalent to Mellinbased Hadamard fractional derivatives. Finally, to demonstrate that AAC is not merely a reformulation of Markovian semigroups, we analyze oscillatory memory kernels, proving they generate dispersive, non-dissipative unitary groups on the scale domain.

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## Full Text

Anchored Accumulation Calculus:
Mellin Diagonalization and Operator-Theoretic Classification
of Scale-Covariant Memory

Rich C. Young

www.youtube.com/GuruOne

April 14, 2026

Abstract

We formalize Anchored Accumulation Calculus (AAC), an operator-theoretic framework
for scale-covariant memory on the multiplicative group R+. By defining memory as causal
accumulation along exponential scale trajectories, we establish that AAC operators on a
dense core are unitarily equivalent to translation-invariant Fourier multipliers. This con-
struction allows us to define the maximal closed extension of these operators, avoiding ad
hoc integration-by-parts arguments. We subsequently classify their generators: completely
monotone memory kernels are rigorously shown to generate scale-invariant L´evy subordina-
tors via Bernstein’s theorem, while heavy-tailed kernels are unitarily equivalent to Mellin-
based Hadamard fractional derivatives. Finally, to demonstrate that AAC is not merely
a reformulation of Markovian semigroups, we analyze oscillatory memory kernels, proving
they generate dispersive, non-dissipative unitary groups on the scale domain.

1
Introduction

Memory effects in systems governed by scale invariance—such as anomalous diffusion on fractal
geometries, polymer chain dynamics, and viscoelastic log-price models—are naturally modeled
on the multiplicative group R+ = (0, ∞). While the Mellin transform is the standard analytical
tool for such geometries [3, 6], the operator-theoretic treatment of scale-covariant memory (often
framed heuristically via Hadamard fractional calculus [4]) frequently lacks rigorous grounding
in domain closure, sectoriality, and complete semigroup generation.
In this paper, we construct Anchored Accumulation Calculus (AAC), a rigorous functional-
analytic framework that parameterizes memory as a causal Volterra-type accumulation along a
backward logarithmic scale trajectory.
To resolve the functional-analytic ambiguities present in informal fractional calculi, we pro-
vide three main contributions:

1. Graph Closure via Unitary Equivalence: We define the AAC operator on a dense
core and formally identify its maximal closed domain via continuous Fourier multipliers.

2. L´evy–Khintchine Derivation: Utilizing Bernstein function theory, we rigorously prove
that completely monotone kernels generate conditionally negative definite symbols and
explicit scale-invariant L´evy measures.

3. Spectral Classification Beyond Semigroups: We prove that heavy-tailed kernels cor-
respond to sectorial normal operators (recovering Hadamard calculus via Balakrishnan’s
spectral theorem), while oscillatory kernels escape the L´evy framework entirely to generate
dispersive scale-dynamics.

2
Functional Setting and the Dense Core

Let H = L2(R+, dx/x) denote the Hilbert space equipped with the invariant Haar measure.

Definition 2.1 (Core Domain and Unitary Map). Let D0 = C∞
c (R+) be the space of smooth
functions with compact support in (0, ∞), which is dense in H.
We define the logarithmic
unitary map U : H →L2(R, dy) by (Uf)(y) = f(ey). U is an isometric isomorphism mapping
D0 bijectively onto C∞
c (R).

Definition 2.2 (Mellin Transform). The Mellin transform M : H →L2(R, dξ) is defined via
M = F ◦U, where F is the standard unitary Fourier transform on L2(R).

Assumption 2.1 (Kernel Regularity). The memory kernel ϕ is a tempered distribution in S′(R)
with supp(ϕ) ⊆[0, ∞). We assume its distributional Fourier–Laplace transform bϕ(ξ) exists such
that the multiplier function m(ξ) = iξbϕ(ξ) is measurable and belongs to L∞
loc(R).

3
The AAC Operator and its Maximal Extension

We define the AAC operator initially on the core domain, where boundary conditions trivially
vanish.

Definition 3.1 (AAC Operator on the Core). For f ∈D0, the AAC operator T 0
ϕ is defined via
the causal exponential scale trajectory γx(s) = xe−s:

T 0
ϕ f(x) = −
Z ∞

0
ϕ(s) ∂

∂s

f(xe−s)

ds.

Because s 7→f(xe−s) is smooth with compact support in [0, ∞), the action of the distribution ϕ
is strictly well-defined.

Lemma 3.1 (Distributional Convolution). On the core D0, UT 0
ϕ U−1 acts as the additive con-
volution operator g 7→ϕ ∗g′ in S′(R).

Proof. Let g = Uf ∈C∞
c (R). By the chain rule, −∂

∂s[f(xe−s)] = −g′(ln x −s). The integral
against ϕ(s) evaluates the distribution ϕ on the test function s 7→−g′(ln x−s), which is exactly
the definition of the convolution (ϕ ∗g′)(ln x). Applying U maps ln x 7→y.

Theorem 3.1 (Maximal Closed Extension). Under Assumption 2.1, T 0
ϕ is closable in H. Its
unique closed extension, denoted Tϕ, has the maximal domain:

D(Tϕ) =

f ∈H : m(ξ)(Mf)(ξ) ∈L2(R, dξ)
	
,

equipped with the graph norm ∥f∥2
Tϕ = ∥f∥2
H + ∥m · Mf∥2
L2(R). Furthermore, Tϕ is unitarily
equivalent to the densely defined Fourier multiplier Tm = F−1MmF.

Proof. By Lemma 3.1, UT 0
ϕ f = ϕ ∗(Uf)′. Taking the Fourier transform over C∞
c (R), we find

F(ϕ ∗g′)(ξ) = bϕ(ξ)(iξbg(ξ)) = m(ξ)bg(ξ). Thus, on D0, T 0
ϕ = U−1F−1MmFU. Because m(ξ)
is measurable, the maximal multiplier Mm is a closed, densely defined operator on L2(R, dξ).
Since U and F are unitary, the pull-back operator Tϕ is closed and densely defined on H. The
domain D(Tϕ) follows directly from the domain of Mm.

4
L´evy–Khintchine Representation

We consider the Cauchy problem ∂tf = −Tϕf. For this to generate a C0-contraction semigroup,
the multiplier m(ξ) must be conditionally negative definite (CND) with Re(m(ξ)) ≥0.

Theorem 4.1 (L´evy–Khintchine Derivation). Assume ϕ(s) is completely monotone, admitting
the Bernstein representation ϕ(s) =
R ∞
0 e−sτν(dτ) where
R ∞
0 (1 ∧τ −1)ν(dτ) < ∞. Then m(ξ)
is CND, and there exists a unique positive L´evy measure µ such that:

m(ξ) =
Z ∞


1 −e−iξy
µ(dy).

0

Consequently, −Tϕ generates a scale-invariant L´evy subordinator on H.

Proof. Substituting the Bernstein representation into the symbol m(ξ) = iξ
R ∞
0 e−iξsϕ(s)ds
yields:

m(ξ) =
Z ∞

iξ
τ + iξ ν(dτ).

0

Using the integral identity
iξ
τ+iξ =
R ∞
0 (1−e−iξy)τe−τydy, we apply Fubini–Tonelli to the double
integral.
The modulus of the integrand is bounded by |1 −e−iξy|τe−τy ≤(2 ∧|ξ|y)τe−τy.
Integrating with respect to y gives a bound proportional to τ
R ∞
0 ye−τydy = τ −1 for small τ,
and 2 for large τ. Because ν integrates 1 ∧τ −1, the Fubini swap is rigorously justified, allowing
us to define the measure:

µ(dy) =
Z ∞

0
τe−τyν(dτ)

dy.

By construction,
R ∞
0 (1 ∧y)µ(dy) < ∞, satisfying the strict criteria for a L´evy measure [1].
Because Re(1 −e−iξy) = 1 −cos(ξy) ≥0, Lumer–Phillips theorem guarantees −Tϕ generates a
C0-contraction semigroup.

5
Fractional Powers and Sectoriality

Theorem 5.1 (Sectoriality and Normal Operators). Let ϕ(s) =
1
Γ(1−β)s−β for β ∈(0, 1). The
resulting operator Tϕ is a closed normal operator and is sectorial of angle ω = βπ/2.

Proof. The distributional transform of ϕ(s) yields the multiplier m(ξ) = (iξ)β. To ensure m(ξ)
is single-valued and measurable, we define it via the principal branch: m(ξ) = |ξ|βei sgn(ξ)βπ/2

for ξ ∈R \ {0}. Because m(ξ) is measurable, the maximal multiplier Tm is a normal operator
[2], and thus Tϕ is normal on H. Its spectrum σ(Tϕ) is the essential range of m(ξ), specifically
the rays Γ = {re±iβπ/2 : r ≥0}.
This spectrum is entirely contained in the sector Σω =
{z ∈C : | arg z| ≤βπ/2}. By the spectral theorem for normal operators, the resolvent satisfies
∥(λI −Tϕ)−1∥≤dist(λ, σ(Tϕ))−1 for λ /∈Σω, satisfying the definition of a sectorial operator.

Corollary 5.1 (Mellin-Hadamard Equivalence). Because Tϕ is unitarily equivalent to the mul-
tiplier |ξ|βei sgn(ξ)βπ/2, it is identically equivalent to the Mellin-based definition of the Hadamard
fractional derivative (x d

dx)β evaluated via the Balakrishnan spectral calculus. Thus, under this
specific functional calculus, AAC provides a rigorous maximal realization of the Hadamard
derivative.

6
Beyond L´evy Semigroups: Oscillatory Kernels

To demonstrate that AAC is a fundamentally broader spectral framework than translation-
invariant L´evy semigroups, we examine non-monotone kernels.

Let the memory kernel be purely oscillatory: ϕ(s) = sin(ωs) for ω > 0. The associated
multiplier is m(ξ) = iξ
R ∞
0 e−iξs sin(ωs)ds. Evaluated distributionally, the symbol is strictly
real:
m(ξ) =
ωξ
ω2 −ξ2 .

Because m(ξ) is real-valued, Re(m(ξ)) changes sign, meaning −Tϕ is not conditionally negative
definite and cannot generate a contraction semigroup. However, iTϕ is self-adjoint. By Stone’s
Theorem [2], Tϕ generates a strongly continuous unitary group G(t) = exp(itTϕ). Thus, the
Cauchy problem ∂tf = iTϕf yields a dispersive, Schr¨odinger-type unitary evolution on the scale
domain. Consequently, AAC serves as a unified spectral classification framework encompassing
both dissipative Markovian memory (L´evy dynamics) and dispersive, conservative memory.

7
Conclusion

Anchored Accumulation Calculus rigorously embeds multiplicative convolutions into the theory
of continuous Fourier multipliers. By establishing closed maximal extensions via unitary maps,
we provided a mathematically airtight derivation of the scale-invariant L´evy–Khintchine formula
for completely monotone kernels. Furthermore, by formalizing the sectoriality of heavy-tailed
kernels and the dispersive unitary groups generated by oscillatory kernels, AAC transcends
heuristic fractional calculus, offering a complete spectral taxonomy of scale-covariant memory
operators.

References

[1] D. Applebaum, L´evy Processes and Stochastic Calculus, Cambridge University Press, 2nd
Edition, 2009.

[2] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-
Verlag, 2000.

[3] P. Flajolet, X. Gourdon, P. Dumas, “Mellin transforms and asymptotics: Harmonic sums,”
Theoretical Computer Science, 144(1-2), 3–58, 1995.

[4] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Appli-
cations, Gordon and Breach Science Publishers, 1993.

[5] R. L. Schilling, R. Song, Z. Vondraˇcek, Bernstein Functions: Theory and Applications, De
Gruyter, 2nd Edition, 2012.

[6] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford University Press,
2nd Edition, 1948.


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