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

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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

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