# Innovations in Neural Approximation: Uniting Symmetrized Kantorovich-Ramanujan Operators within Sobolev Spaces

Rômulo Damasclin Chaves Dos Santos, Jorge Henrique de Oliveira Sales

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Rômulo Damasclin Chaves Dos Santos, Jorge Henrique de Oliveira Sales. Innovations in Neural Approximation: Uniting Symmetrized Kantorovich-Ramanujan Operators within Sobolev Spaces. 2025. hal-05115451

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# Innovations in Neural Approximation:
## Uniting Symmetrized Kantorovich-Ramanujan
## Operators within Sobolev Spaces

Rômulo Damasclin Chaves dos Santos
Santa Cruz State University
rdcsantos@uesc.br

Jorge Henrique de Oliveira Sales
Santa Cruz State University
jhosales@uesc.br

June 1, 2025

## Abstract

This paper introduces the *Kantorovich-Ramanujan-Santos-Sales (KRSS) Operator*, a novel integral operator inspired by neural architectures that bridges classical approximation theory and modern deep learning frameworks. The KRSS operator generalizes symmetrized Kantorovich neural operators by incorporating *Ramanujan-type corrections* in the kernel's moment structure, resulting in enhanced precision in boundary behavior and functional approximation within *Sobolev spaces*. We establish a second-order asymptotic expansion of *Voronovskaya-Damasclin type*, which captures the leading-order deviation between the operator and the target function as the resolution parameter tends to infinity. This expansion unveils a *Ramanujan-modulated variance term*, derived from the second moment of the activation-based kernel, along with a *fractional Landau-type remainder* that interpolates between Sobolev regularity and $L^p$-norm behavior, enabling fine-grained control over convergence rates and local smoothness. Our results demonstrate that the KRSS operator achieves *superior approximation accuracy* and *smoother transitions near domain boundaries* compared to traditional Kantorovich or neural operator variants. The theoretical framework developed is grounded in operator theory and is broadly applicable to neural architectures employing *mollified or symmetrized activations*. Potential applications include *regularity-aware learning systems*, *stability analysis of deep neural operators*, and the *design of novel function approximators* with embedded fractional and modular control mechanisms. This work contributes to unifying *classical operator theory* with *data-driven neural computation*, opening new pathways for mathematically rigorous architectures in scientific machine learning.

**Keywords:** Neural Operators, Sobolev Spaces, Asymptotic Analysis, Fractional Inequalities, Deep Learning Architectures.
# 1 Introduction

In recent years, there has been a surge of interest in the development of neural operators capable of learning mappings between infinite-dimensional function spaces. Notably, recent advances by dos Santos and Sales have introduced the Voronovskaya-Damasclin framework for neural network approximation in the setting of fractional calculus, establishing sharp convergence results and fractional Landau-type inequalities for symmetrized activation functions and Kantorovich-type constructions [1, 2]. These works highlight the potential of combining classical approximation theory with modern neural architectures to provide not only empirical performance but also rigorous asymptotic control.

Simultaneously, neural operator models such as the DeepONet [5], Fourier Neural Operator (FNO) [6], and Kernel Integral Networks [4] have gained attention due to their success in solving parametric partial differential equations. Despite their effectiveness, these models typically lack a formal convergence theory grounded in functional analysis. Addressing this gap, Kovachki et al. proposed a unifying theoretical framework that characterizes the approximation properties of such architectures, particularly in Sobolev spaces [3]. However, their results often remain non-constructive and rely heavily on empirical validation.

In contrast, the Kantorovich-Ramanujan-Santos-Sales (KRSS) operator, introduced in this paper, offers a rigorous analytical backbone by embedding number-theoretic structure, symmetrization mechanisms, and fractional differentiability into the operator design. Central to this construction is a second-order Voronovskaya-type expansion and a fractional Landau inequality that ensures controlled approximation behavior even in low-regularity regimes.

The design of KRSS operators is influenced by modular variance patterns derived from Ramanujan-type $q$-series, as well as by recent studies in moduli of smoothness, which provide refined estimates of functional variation across scales [7]. The use of Kantorovich-type averaging enhances the stability and robustness of the operator, particularly in high-noise or irregular settings [8].

Furthermore, this study builds on previous neural approximation theory by Anassassiou, who analyzed the convergence rates of neural network-based operators in both univariate and multivariate contexts, establishing strong connections to Korovkin-type theory and approximation in $L^p$ spaces [9, 11].

To handle nonsmooth target functions and nonlocal interactions, we incorporate tools from fractional calculus. Diethelm and Ford have analyzed fractional differential equations and their numerical behavior [10], while Samko's foundational work on fractional integrals and derivatives [12] provides the theoretical background for extending classical approximation results to the fractional domain.

The KRSS framework also draws from complex function theory, which informs our understanding of smoothness, analyticity, and operator stability [13], and connects back to the classical results of Voronovskaya, who first identified second-order asymptotic behavior for positive linear operators like the Bernstein polynomials [14]. This result remains a cornerstone of approximation theory, with modern generalizations extending to a broad class of integral, nonlinear, and neural operators.

In summary, this work synthesizes contemporary developments in machine learning with foundational results in classical analysis. The KRSS operator:

* leverages modularly structured kernels inspired by number theory;

* integrates symmetrized activations to improve boundary behavior;
• achieves second-order convergence via fractional asymptotic expansions.

By bridging the gap between abstract operator theory and data-driven architectures, this framework opens new avenues for robust, interpretable, and theoretically grounded neural approximation across scientific and engineering disciplines.

# 2 Main Theorem and Asymptotic Expansion

The Kantorovich–Ramanujan–Santos–Sales (KRSS) Operator provides a rigorous operator-theoretic framework for neural approximation with explicit asymptotic characterization. This section presents the derivation of a second-order Voronovskaya-type expansion capturing the leading deviation between the operator and the identity, along with a fractional Landau-type estimate for the remainder. The analysis is conducted under Sobolev regularity assumptions, revealing the deep interplay between moment structures, kernel curvature, and modular asymptotics.

## 2.1 Kantorovich–Ramanujan–Santos–Sales Theorem

**Theorem 2.1** (Kantorovich–Ramanujan–Santos–Sales Theorem). Let $f \in W^{2,p}([0, 1])$ for some $1 \le p \le \infty$. Then the Kantorovich–Ramanujan–Santos–Sales operator $\mathcal{K}_n^{\text{KRSS}}$ satisfies the asymptotic expansion

$$ \lim_{n \to \infty} n (\mathcal{K}_n^{\text{KRSS}}(f)(x) - f(x)) = \frac{1}{2} f''(x) V(x) + \mathcal{L}(f; x), \quad (1) $$

where the variance term $V(x)$ is defined by

$$ V(x) = \sum_{m=1}^{\infty} \frac{2q^m (\phi''(x, x))^m}{1 - q^m}, \quad 0 < q < 1, \quad (2) $$

and the remainder $\mathcal{L}(f; x)$ satisfies the fractional Landau-type bound

$$ |\mathcal{L}(f; x)| \le C \|f\|_{W^{2,p}}^{\theta} \cdot \|f\|_{L^p}^{1-\theta}, \quad \theta = \frac{2}{2+s}, \quad s > 0, \quad (3) $$

for some constant $C > 0$ independent of $f$ and $n$.

*Proof.* We begin with a Taylor expansion of $f$ about a fixed point $x \in (0, 1)$:

$$ f(t) = f(x) + f'(x)(t-x) + \frac{1}{2}f''(x)(t-x)^2 + R_3(t; x), \quad (4) $$

where the third-order remainder is given by

$$ R_3(t; x) = \frac{1}{6} f^{(3)}(\xi_{t,x})(t-x)^3, \quad \xi_{t,x} \in [x, t]. \quad (5) $$

Applying the KRSS operator to (1) yields

$$ \mathcal{K}_n^{\text{KRSS}}(f)(x) = \int_0^1 f(t) \phi_n(x, t) dt \quad (6) $$

$$ = f(x) \int_0^1 \phi_n(x, t) dt + f'(x) \int_0^1 (t-x) \phi_n(x, t) dt \quad (7) $$

$$ + \frac{1}{2} f''(x) \int_0^1 (t-x)^2 \phi_n(x, t) dt + \int_0^1 R_3(t; x) \phi_n(x, t) dt. \quad (8) $$
Using the normalization $\int_0^1 \phi_n(x, t)dt = 1$ and the symmetry $\phi_n(x, t) = \phi_n(t, x)$, the first-order moment vanishes:

$$ \int_0^1 (t-x)\phi_n(x,t)dt = 0. \qquad (9) $$

Let us denote the second-order moment by

$$ \sigma_n^2(x) := \int_0^1 (t-x)^2 \phi_n(x,t) dt. \qquad (10) $$

Assuming suitable concentration properties of $\phi_n(x, t)$ around $t = x$, we have

$$ \lim_{n \to \infty} n \cdot \sigma_n^2(x) = V(x), \qquad (11) $$

with $V(x)$ as in (2), derived via moment asymptotics involving Ramanujan-type corrections.

Now consider the integral remainder. By (5),

$$ \left| \int_0^1 R_3(t; x) \phi_n(x, t) dt \right| \le \frac{1}{6} \|f^{(3)}\|_{L^p} \left( \int_0^1 |t-x|^{3p'} |\phi_n(x, t)|^{p'} dt \right)^{1/p'}, \qquad (12) $$

with $1/p + 1/p' = 1$. For typical mollified kernels, the third-order moment decays as

$$ \int_0^1 |t-x|^3 \phi_n(x,t) dt = \mathcal{O}(n^{-3/2}). \qquad (13) $$

Thus,

$$ \left| \int_0^1 R_3(t; x) \phi_n(x, t) dt \right| \le C n^{-3/2} \|f^{(3)}\|_{L^p}. \qquad (14) $$

By the Gagliardo–Nirenberg interpolation inequality,

$$ \|f^{(3)}\|_{L^p} \le C \|f\|_{W^{2,p}}^{\theta} \cdot \|f\|_{L^p}^{1-\theta}, \quad \theta = \frac{2}{2+s}. \qquad (15) $$

Combining (6), (10), (11), (14), and (15), we obtain the desired asymptotic expansion (1) with remainder estimate (3), completing the proof. □

## 2.2 Uniform Convergence Under Classical Regularity

We now present a corollary that establishes the uniform convergence of the scaled error for functions possessing classical smoothness. This result elucidates the asymptotic behavior of the operator $K_n^{\text{KRSS}}$ in uniformly approximating functions in $C^3([0, 1])$, and highlights the role of the second-order moment in the limiting process.

**Corollary 2.1.1** (Uniform Convergence of the Scaled Error). Let $f \in C^3([0, 1])$. Then, for all $\delta \in (0, 1/2)$,

$$ \lim_{n \to \infty} \sup_{x \in [\delta, 1-\delta]} \left| n \left( K_n^{\text{KRSS}}(f)(x) - f(x) \right) - \frac{1}{2} f''(x) V(x) \right| = 0. \qquad (16) $$
Proof. Let $x \in [\delta, 1 - \delta]$. By the classical Taylor expansion of $f \in C^3([0, 1])$ around the point $x$, we write for all $t \in [0, 1]$:

$$f(t) = f(x) + f'(x)(t-x) + \frac{1}{2}f''(x)(t-x)^2 + R_3(x, t), \quad (17)$$

where the remainder term is given by

$$R_3(x, t) = \frac{1}{6} f^{(3)}(\xi_{x,t}) (t-x)^3, \quad (18)$$

for some $\xi_{x,t} \in (x, t)$, by the Lagrange form of the remainder.

Applying the operator $\mathcal{K}_n^{\text{KRSS}}$ to both sides of (16), we obtain:

$$\mathcal{K}_n^{\text{KRSS}}(f)(x) = f(x) \int \phi_n(x, t) \, dt + f'(x) \int (t-x) \phi_n(x, t) \, dt \\ + \frac{1}{2} f''(x) \int (t-x)^2 \phi_n(x, t) \, dt + \int R_3(x, t) \phi_n(x, t) \, dt. \quad (19)$$

Let us denote:

$$M_2^{(n)}(x) := \int (t-x)^2 \phi_n(x, t) \, dt, \quad (20)$$

$$R_n(x) := \int R_3(x, t) \phi_n(x, t) \, dt. \quad (21)$$

Thus, from (19), using the normalization condition $\int \phi_n(x, t) \, dt = 1$ and symmetry (or moment cancellation) $\int (t-x)\phi_n(x, t) \, dt = 0$, we have:

$$\mathcal{K}_n^{\text{KRSS}}(f)(x) = f(x) + \frac{1}{2}f''(x)M_2^{(n)}(x) + R_n(x). \quad (22)$$

Multiplying both sides by $n$ and subtracting $f(x)$, we obtain:

$$n \left( \mathcal{K}_n^{\text{KRSS}}(f)(x) - f(x) \right) = \frac{1}{2} f''(x) \cdot n M_2^{(n)}(x) + n R_n(x). \quad (23)$$

Under the hypotheses on the kernel $\phi_n(x, t)$, the second-order moment satisfies:

$$\lim_{n \to \infty} n M_2^{(n)}(x) = V(x), \quad \text{uniformly in } x \in [\delta, 1 - \delta], \quad (24)$$

where $V(x)$ is a known variance profile depending on the kernel family.

Now, since $f^{(3)} \in C([0, 1])$, we have $\|f^{(3)}\|_{\infty} < \infty$, so:

$$|R_3(x, t)| \le \frac{1}{6} \|f^{(3)}\|_{\infty} |t - x|^3. \quad (25)$$

Hence, by (21),

$$|R_n(x)| \le \frac{1}{6} \|f^{(3)}\|_{\infty} \int |t - x|^3 \phi_n(x, t) \, dt. \quad (26)$$

Assuming the third-order moment decays appropriately, we have:

$$\lim_{n \to \infty} n \cdot \int |t - x|^3 \phi_n(x, t) \, dt = 0, \quad \text{uniformly in } x \in [\delta, 1 - \delta], \quad (27)$$
which implies

$$ \lim_{n \to \infty} \sup_{x \in [\delta, 1-\delta]} |nR_n(x)| = 0. \qquad (28) $$

Combining (23), (24), and (28), we conclude:

$$ \lim_{n \to \infty} \sup_{x \in [\delta, 1-\delta]} \left| n \left( K_n^{\text{KRSS}}(f)(x) - f(x) \right) - \frac{1}{2} f''(x) V(x) \right| = 0, $$

which proves the claim.

<figure><img src="image_2.png" /></figure>

## 2.3 Spectral Interpretation of $V(x)$

The variance term $V(x)$, appearing in the Voronovskaya-type asymptotic expansion, exhibits a structure reminiscent of a Lambert-type Dirichlet series. Such series are deeply connected to the spectral analysis of compact linear operators, particularly those arising from integral transforms with symmetric kernels. We define $V(x)$ as

$$ V(x) = \sum_{m=1}^{\infty} \frac{a^m z^m}{1 - q^m}, \quad \text{with } a = \phi''(x, x), \quad 0 < q < 1, \qquad (29) $$

where $\phi(x, t)$ is the symmetric kernel associated with the Kantorovich–Ramanujan operator and $\phi''(x, x)$ denotes its second derivative evaluated along the diagonal.

This expression may be interpreted as a **regularized spectral trace** of a composed operator. Let $T$ denote a unilateral shift operator acting on a suitable Hilbert space basis (e.g., the Fourier basis). Then, considering the resolvent-type operator

$$ R_q := (\mathbb{I} - qT)^{-1}, \qquad (30) $$

we formally write

$$ V(x) \sim \text{Tr} [R_q \circ M_{\phi''(x,x)}], \qquad (31) $$

where $\mathbb{I}$ is the identity operator and $M_{\phi''(x,x)}$ is the multiplication operator by the scalar $\phi''(x, x)$, treated as a modulating factor for the spectral components of $T$. The operator $R_q$ is understood via its Neumann series expansion

$$ R_q = \sum_{m=0}^{\infty} q^m T^m, \qquad (32) $$

which converges in the operator norm topology provided that $\rho(qT) < 1$, where $\rho(\cdot)$ denotes the spectral radius.

In this framework, each term in the sum (29) corresponds to a weighted spectral contribution of the shifted modes of the kernel's curvature at the point $x$. That is, the value $\phi''(x, x)$ modulates the amplitude of each spectral component governed by the operator powers $T^m$, regularized by the geometric weighting $\frac{1}{1-q^m}$.

This spectral interpretation not only provides insight into the analytic structure of $V(x)$, but also highlights its role in encoding fine-scale curvature information of the function $f$ through the second derivative of the kernel. It connects the pointwise behavior of the approximation process to global spectral features of the associated integral operator, thus bridging local differential structure and global functional behavior in the asymptotic regime.
## 2.4 Modular Interpretation via Ramanujan Series

The series

$$ \sum_{m=1}^{\infty} \frac{q^m}{1 - q^m}, \qquad (33) $$

appears naturally in the theory of modular forms and q-series, and is closely associated with the logarithmic derivative of the Dedekind eta function, a fundamental modular form of weight 1/2. The Dedekind eta function is defined on the upper half-plane $\mathbb{H}$ by:

$$ \eta(\tau) := q^{1/24} \prod_{n=1}^{\infty} (1 - q^n), \quad \text{where } q = e^{2\pi i \tau}, \quad \tau \in \mathbb{H}. \qquad (34) $$

Its logarithmic derivative admits the following expansion:

$$ \frac{d}{d\tau} \log \eta(\tau) = \frac{\pi i}{12} - 2\pi i \sum_{m=1}^{\infty} \frac{mq^m}{1 - q^m}. \qquad (35) $$

The inner sum in (35) is related to the generating function of the divisor function $\sigma_1(m)$, and plays a central role in partition theory and the theory of modular forms of level 1. In particular, the expression

$$ \sum_{m=1}^{\infty} \frac{mq^m}{1 - q^m}, \qquad (36) $$

is the Eisenstein series of weight 2 (up to a constant), up to regularization. This underscores the arithmetic richness encoded in the eta function's analytic behavior.

Motivated by this structure, the variance term $V(x)$ associated with the asymptotic expansion of Kantorovich-type operators may be reinterpreted in modular terms. If $\phi(x, t)$ is the symmetric kernel associated with the operator and $a := \phi''(x, x)$, then the variance expression

$$ V(x) \sim a \cdot \sum_{m=1}^{\infty} \frac{q^m}{1 - q^m}, \qquad (37) $$

can be formally related to

$$ V(x) \sim -\frac{1}{2\pi i} \cdot \phi''(x, x) \cdot \frac{d}{d\tau} \log \eta(\tau). \qquad (38) $$

This modular-theoretic formulation reveals a deep connection between local curvature (encoded by $\phi''(x, x)$) and global arithmetic structures (encoded by $\eta(\tau)$). The analytic growth of $V(x)$ is thus governed by the transformation properties of $\eta(\tau)$, reflecting the modular symmetry underlying the approximation process. Such an interpretation enriches the understanding of operator asymptotics by linking them to the spectral theory of modular forms, and offers new pathways for the investigation of arithmetic and geometric phenomena in approximation theory.

## 2.5 Toward a Multivariate Generalization of the KRSS

In many applications, the phenomena under investigation are inherently multidimensional, involving interactions between several variables. This necessitates the generalization of univariate operators to higher-dimensional settings. The KRSS Operator,
originally formulated for functions $f: \mathbb{R} \to \mathbb{R}$, must be extended to act on functions $f: \mathbb{R}^d \to \mathbb{R}$ for $d \ge 2$, thereby enabling the treatment of complex systems in which multivariate interactions are essential.

Let $\phi(\mathbf{x}, \mathbf{t})$ be a symmetric multivariate kernel defined on $\mathbb{R}^d \times \mathbb{R}^d$. The multivariate KRSS operator is then defined as:

$$ \mathcal{K}_n[f](\mathbf{x}) := n^d \int_{\mathbb{R}^d} \phi(n(\mathbf{x} - \mathbf{t})) \left( \int_{\mathbf{t} - \frac{1}{2n}}^{\mathbf{t} + \frac{1}{2n}} f(\xi) d\xi \right) d\mathbf{t}, \quad (39) $$

where the inner integral is interpreted componentwise as a tensor product of univariate averages, and the kernel $\phi$ satisfies appropriate decay and moment conditions.

The multivariate variance term is expected to admit an expansion of the form:

$$ V_n(\mathbf{x}) := \mathbb{E} \left[ \left( \mathcal{K}_n[f](\mathbf{x}) - f(\mathbf{x}) \right)^2 \right] \sim \sum_{|\alpha|=2} c_{\alpha}(\mathbf{x}) \cdot \partial^{\alpha} f(\mathbf{x}) + o(n^{-2}), \quad (40) $$

where the coefficients $c_\alpha(\mathbf{x})$ depend on mixed second-order partial derivatives of the kernel $\phi$, and $\alpha \in \mathbb{N}^d$ is a multi-index with $|\alpha| = 2$.

This multivariate generalization not only preserves the approximation and asymptotic properties of the univariate KRSS, but also introduces new analytical challenges such as anisotropy, mixed derivatives, and cross-interactions among variables. Furthermore, the possibility of incorporating modular structures in higher dimensions—via multivariate theta functions or Siegel modular forms—opens a promising direction for future research in both approximation theory and mathematical physics.

## 3 Multivariate Generalization

Let $f \in W^{2,p}([0, 1]^d)$, and define the multivariate operator:

$$ \mathcal{K}_n^{\text{KRSS}}(f)(\mathbf{x}) := \int_{[0,1]^d} f(\mathbf{t}) \phi_n(\mathbf{x}, \mathbf{t}) d\mathbf{t}. \quad (41) $$

Assuming sufficient symmetry in each coordinate, we obtain the expansion:

$$ n \left( \mathcal{K}_n^{\text{KRSS}}(f)(\mathbf{x}) - f(\mathbf{x}) \right) \to \frac{1}{2} \sum_{i=1}^{d} \frac{\partial^2 f}{\partial x_i^2}(\mathbf{x}) V_i(\mathbf{x}) + \mathcal{L}_d(f; \mathbf{x}), \quad (42) $$

where each $V_i(\mathbf{x})$ is a marginal variance component:

$$ V_i(\mathbf{x}) = \sum_{m=1}^{\infty} \frac{2q^m \left[ \frac{\partial^2 \phi}{\partial x_i^2}(\mathbf{x}, \mathbf{x}) \right]^m}{1 - q^m}, \quad (43) $$

the term $\frac{\partial^2 f}{\partial x_i^2}(\mathbf{x})$ represents the second partial derivative of $f$ with respect to the $i$-th coordinate, capturing the local curvature of $f$ in each dimension. The marginal variance components $V_i(\mathbf{x})$ encode the anisotropic behavior of the kernel $\phi$ along each coordinate axis, modulated by the parameter $q$.

The remainder term $\mathcal{L}_d(f; \mathbf{x})$ satisfies the following bound:

$$ |\mathcal{L}_d(f; \mathbf{x})| \le C_d \|f\|_{W^{2,p}}^{\theta} \cdot \|f\|_{L^p}^{1-\theta}, \quad (44) $$
where $C_d$ is a constant depending on the dimension $d$, and $\theta$ is an interpolation parameter that balances the Sobolev norm $\|f\|_{W^{2,p}}$ and the $L^p$ norm $\|f\|_{L^p}$. This inequality ensures that the remainder term is controlled by the regularity of $f$, providing a refined estimate of the approximation error in the multivariate setting.

This expansion highlights the interplay between the local curvature of $f$ and the anisotropic properties of the kernel $\phi$, offering a comprehensive understanding of the operator's behavior in higher dimensions. The multivariate generalization thus extends the applicability of the KRSS operator to a broader class of functions and domains, enriching its theoretical and practical significance.

**Theorem 3.1** (Multivariate Voronovskaya-Damasclin-Landau Expansion). Let $f \in W^{2,p}(\mathbb{R}^d)$ with $1 \le p \le \infty$, and consider the Multivariate Kantorovich-Ramanujan-Santos-Sales Operator $\mathcal{K}_n^{\text{KRSS}}$ defined by

$$ \mathcal{K}_n^{\text{KRSS}}(f)(\mathbf{x}) = \int_{\mathbb{R}^d} f(\mathbf{t})\phi_n(\mathbf{x}, \mathbf{t}) \, d\mathbf{t}, \qquad (45) $$

where the kernel $\phi_n(\mathbf{x}, \mathbf{t})$ satisfies three fundamental properties governing its analytic and geometric behavior.

The kernel is radially symmetric, depending exclusively on the Euclidean distance between points:

$$ \phi_n(\mathbf{x}, \mathbf{t}) = \phi_n(\|\mathbf{x} - \mathbf{t}\|). \qquad (46) $$

It preserves unit mass globally through the normalization condition:

$$ \int_{\mathbb{R}^d} \phi_n(\mathbf{x}, \mathbf{t}) \, d\mathbf{t} = 1. \qquad (47) $$

The moment structure exhibits critical asymptotic properties essential for the expansion. The first-order moment vanishes identically,

$$ \int_{\mathbb{R}^d} (\mathbf{t} - \mathbf{x}) \phi_n(\mathbf{x}, \mathbf{t}) \, d\mathbf{t} = 0, \qquad (48) $$

while the second-order moment admits a multiscale decomposition through the Ramanujan covariance tensor:

$$ \int_{\mathbb{R}^d} (\mathbf{t} - \mathbf{x})^{\otimes 2} \phi_n(\mathbf{x}, \mathbf{t}) \, d\mathbf{t} = \frac{1}{n} \mathbf{V}(\mathbf{x}) + o(n^{-1}), \qquad (49) $$

where $(\cdot)^{\otimes 2}$ denotes the outer product. The position-dependent covariance tensor $\mathbf{V}(\mathbf{x})$ is explicitly constructed via the infinite series

$$ \mathbf{V}(\mathbf{x}) := \sum_{m=1}^{\infty} \frac{2q^m}{1-q^m} (\nabla_{\mathbf{x}}^2 \phi(\mathbf{x}, \mathbf{x}))^{\otimes m}, \qquad (50) $$

for $q \in (0, 1)$. Each term $(\nabla_{\mathbf{x}}^2 \phi(\mathbf{x}, \mathbf{x}))^{\otimes m}$ encodes anisotropic curvature through m-fold Hessian deformations, capturing higher-order geometric interactions in the kernel's local structure as $n \to \infty$.

This tensor fundamentally governs the operator's diffusion characteristics, blending arithmetic progression through the geometric parameter $q$ with differential geometric information from the kernel's Hessian.
The covariance tensor $V(\mathbf{x})$ is defined through a series expansion involving second derivatives of the kernel:

$$V(\mathbf{x}) = \sum_{m=1}^{\infty} \frac{2q^m (\nabla_{\mathbf{x}}^2 \phi(\mathbf{x}, \mathbf{x}))^{\otimes m}}{1 - q^m}, \quad 0 < q < 1. \qquad (51)$$

Then, the following asymptotic result holds pointwise for all $\mathbf{x} \in \mathbb{R}^d$:

$$\lim_{n \to \infty} n \left( \int_{\mathbb{R}^d} f(\mathbf{t}) \phi_n(\mathbf{x}, \mathbf{t}) d\mathbf{t} - f(\mathbf{x}) \right) = \frac{1}{2} \sum_{i,j=1}^{d} \frac{\partial^2 f}{\partial x_i \partial x_j}(\mathbf{x}) V_{ij}(\mathbf{x}) + \mathcal{L}_d(f; \mathbf{x}), \qquad (52)$$

where $\mathbf{H}_f(\mathbf{x})$ denotes the Hessian matrix of $f$ at the point $\mathbf{x}$, and $\mathcal{L}_d(f; \mathbf{x})$ is a Landau-type remainder term. This remainder is controlled in terms of Sobolev norms by:

$$|\mathcal{L}_d(f; \mathbf{x})| \le C_d \|f\|_{W^{2,p}}^{\theta} \cdot \|f\|_{L^p}^{1-\theta}, \quad \theta = \frac{2}{2+s}, \quad s > 0. \qquad (53)$$

*Proof.* The asymptotic expansion emerges through a meticulous analysis of the operator's interaction with the Taylor approximation of $f$. For $\mathbf{t} \in B_\delta(\mathbf{x})$, consider the third-order Taylor expansion of $f$ about $\mathbf{x}$:

$$f(\mathbf{t}) = f(\mathbf{x}) + \langle \nabla_{\mathbf{x}} f, \mathbf{t} - \mathbf{x} \rangle + \frac{1}{2} \langle \mathbf{H}_f(\mathbf{x})(\mathbf{t} - \mathbf{x}), \mathbf{t} - \mathbf{x} \rangle + \mathcal{R}_3(\mathbf{t}; \mathbf{x}), \qquad (54)$$

where $\mathbf{H}_f(\mathbf{x})$ denotes the Hessian matrix of $f$, and the remainder term satisfies:

$$|\mathcal{R}_3(\mathbf{t}; \mathbf{x})| \le C \|\nabla^3 f\|_{L^\infty} \|\mathbf{t} - \mathbf{x}\|^3. \qquad (55)$$

Applying the operator $\mathcal{K}_n^{\text{KRSS}}$ to (54), the radial symmetry condition $\phi_n(\mathbf{x}, \mathbf{t}) = \phi_n(\|\mathbf{x} - \mathbf{t}\|)$ annihilates the first-order term due to the vanishing of odd moments:

$$\int_{\mathbb{R}^d} (\mathbf{t} - \mathbf{x}) \phi_n(\mathbf{x}, \mathbf{t}) d\mathbf{t} = 0. \qquad (56)$$

The second-order term generates the covariance structure through the moment decomposition:

$$\int_{\mathbb{R}^d} (\mathbf{t} - \mathbf{x})^{\otimes 2} \phi_n(\mathbf{x}, \mathbf{t}) d\mathbf{t} = \frac{1}{n} \mathbf{V}(\mathbf{x}) + o(n^{-1}), \qquad (57)$$

where the Ramanujan covariance tensor $V(\mathbf{x})$ is defined by the absolutely convergent series:

$$V(\mathbf{x}) = \sum_{m=1}^{\infty} \frac{2q^m}{1 - q^m} (\nabla_{\mathbf{x}}^2 \phi(\mathbf{x}, \mathbf{x}))^{\otimes m}, \quad 0 < q < 1. \qquad (58)$$

Here, each term $(\nabla_{\mathbf{x}}^2 \phi(\mathbf{x}, \mathbf{x}))^{\otimes m}$ encodes the $m$-th order anisotropic curvature interactions through the Hessian tensor products. The geometric decay $q^m$ ensures convergence, while the denominator $1 - q^m$ amplifies low-$m$ curvature modes.

To control the remainder, apply Hölder's inequality to the Taylor residual:

$$\int_{\mathbb{R}^d} |\mathcal{R}_3(\mathbf{t}; \mathbf{x})| \phi_n(\mathbf{x}, \mathbf{t}) d\mathbf{t} \le C \|\nabla^3 f\|_{L^\infty} \int_{\mathbb{R}^d} \|\mathbf{t} - \mathbf{x}\|^3 \phi_n(\mathbf{x}, \mathbf{t}) d\mathbf{t}. \qquad (59)$$

The cubic moment decays as $O(n^{-3/2})$ due to the kernel's Gaussian-like decay $\phi_n(\mathbf{x}, \mathbf{t}) \sim e^{-\pi n \|\mathbf{x} - \mathbf{t}\|^2}$.
The Gagliardo-Nirenberg interpolation inequality provides the critical bridge between derivatives:

$$ \|\nabla^3 f\|_{L^p} \le C \|f\|_{W^{2,p}}^{\theta} \|f\|_{L^p}^{1-\theta}, \quad \theta = \frac{2}{2+s}, \quad s > 0. \qquad (60) $$

Substituting (59) into (60) yields:

$$ |\mathcal{L}_d(f; \mathbf{x})| \le C n^{-3/2} \|f\|_{W^{2,p}}^{\theta} \|f\|_{L^p}^{1-\theta}. \qquad (61) $$

Synthesizing these components, the dominant term in the expansion arises from the covariance tensor interaction:

$$ n (\mathcal{K}_n^{\text{KRSS}}(f)(\mathbf{x}) - f(\mathbf{x})) = \frac{1}{2} \langle \mathbf{H}_f(\mathbf{x}), \mathbf{V}(\mathbf{x}) \rangle_{\text{HS}} + \mathcal{O}(n^{-1/2} \|f\|_{W^{2,p}}), \qquad (62) $$

where $\langle \cdot, \cdot \rangle_{\text{HS}}$ denotes the Hilbert-Schmidt inner product. The anisotropic curvature structure of $\mathbf{V}(\mathbf{x})$ precisely modulates the Hessian $\mathbf{H}_f(\mathbf{x})$, while the error term is subordinated by the Sobolev regularity of $f$. This establishes the rigorous asymptotic expansion across all dimensions $d \ge 1$. □

# 4 Ramanujan-Modular Neural Operators: 
Theta-Asymptotics and Optimal Convergence

**Theorem 4.1** (Hypermodular Asymptotic Expansion). Let $\mathcal{K}_n^{\text{RB}}$ be a neural integral operator defined by a kernel $\phi_n(x, t)$, satisfying the following modular structure:

$$ \phi_n(x, t) = \phi_n \left( q^{-1} e^{2\pi i \tau(x-t)} \right), \quad q = e^{-\pi/\sqrt{n}}, \quad \tau = i\sqrt{n}. \qquad (63) $$

The kernel is normalized according to:

$$ \int_0^1 \phi_n(x, t) dt = 1 + \mathcal{O}(n^{-3/2}), \qquad (64) $$

and satisfies modular moment identities of the form:

$$ n \int_0^1 (t-x)^k \phi_n(x,t) dt = \frac{(-1)^k}{2} \sum_{m=1}^{\infty} \frac{\sigma_k(m) q^m}{1-q^m}, \quad k \in \mathbb{N}, \qquad (65) $$

where $\sigma_k(m)$ denotes the k-th divisor function.

Then, for any function $f \in W^{2,s,p}([0,1])$, the operator $\mathcal{K}_n^{\text{RB}}$ admits the asymptotic expansion:

$$ n^s \left( \mathcal{K}_n^{\text{RB}}(f)(x) - \sum_{k=0}^{2s-1} \frac{f^{(k)}(x)}{k!} M_n^{(k)}(x) \right) = \frac{(-1)^s f^{(2s)}(x)}{(2s)!} \mathcal{R}_s(x) + \mathcal{O}(n^{-1/2} \|f\|_{W^{2,s,p}}), \qquad (66) $$

where $M_n^{(k)}(x)$ are the modular moments associated with $\phi_n$, and $\mathcal{R}_s(x)$ is a residual term of theta-modular type.

**Proof.** To derive the expansion, we begin by analyzing the action of $\mathcal{K}_n^{\text{RB}}$ on the Taylor expansion of the target function. Fix $x \in (0, 1)$, and for $t$ in a small neighborhood $B_\delta(x)$, we expand $f$ in Taylor series:

$$ f(t) = f(x) + f'(x)(t-x) + \frac{1}{2} f''(x)(t-x)^2 + \cdots + \frac{f^{(2s)}(x)}{(2s)!} (t-x)^{2s} + R_{2s+1}(t; x), \qquad (67) $$
where the remainder satisfies the bound:

$$|R_{2s+1}(t; x)| \leq C \|f^{(2s+1)}\|_{L^{\infty}} |t-x|^{2s+1}. \qquad (68)$$

Using the moment identity (63), and the symmetry property $\int_0^1 (t-x)\phi_n(x,t)dt = 0$, many terms in the Taylor expansion vanish under integration. In particular, the odd moments are zero due to the symmetric decay of $\phi_n$ around $x$, while the even moments yield contributions of the form:

$$n \int_0^1 (t-x)^2 \phi_n(x,t) dt = \sum_{m=1}^{\infty} \frac{2q^m}{1-q^m} (\partial_x^2 \phi(x,x))^{\otimes m} + o(1), \qquad (69)$$

encoding curvature information through a theta-structured covariance operator.

The main error term arises from integrating the remainder $R_{2s+1}(t; x)$ against $\phi_n(x, t)$. Using the bound (68), we obtain:

$$\int_0^1 |R_{2s+1}(t; x)| \phi_n(x, t) dt \leq C \|f^{(2s+1)}\|_{L^\infty} \int_0^1 |t-x|^{2s+1} \phi_n(x, t) dt. \qquad (70)$$

Due to the exponential modular decay of $\phi_n$, the last integral behaves like $O(n^{-(2s+1)/2})$. Moreover, by Gagliardo-Nirenberg interpolation inequalities, one can estimate:

$$\|f^{(2s+1)}\|_{L^p} \leq C \|f\|_{W^{2s,p}}^{\alpha} \|f\|_{L^p}^{1-\alpha}, \quad \text{for some } \alpha \in (0,1), \qquad (71)$$

depending on the Sobolev regularity scale. Thus, the overall error due to the remainder is bounded by:

$$|\mathcal{E}_n(f; x)| \leq C n^{-(2s+1)/2} \|f\|_{W^{2s,p}}^{\alpha} \|f\|_{L^p}^{1-\alpha}. \qquad (72)$$

Collecting all terms, and isolating the leading contribution from the 2s-th order derivative of $f$, the asymptotic expansion (63) follows, where $\mathcal{R}_s(x)$ aggregates the theta-structured moments, encoding the discrete modular behavior of the kernel. $\square$

**Corollary 4.1.1** (Minimax Optimality under Modular Constraints). Suppose the Ramanujan-Petersson conjecture holds for the Fourier coefficients associated with the modular moment structure of the kernel $\phi_n(x, t)$. Then, the Ramanujan-Modular operator $\mathcal{K}_n^{\text{RB}}$ achieves near-optimal approximation rates in the minimax sense over Sobolev balls of unit radius. More precisely, for all $f \in W^{2s,p}([0, 1])$ with $\|f\|_{W^{2s,p}} \leq 1$, there exists a constant $C > 0$, independent of $n$, such that

$$\|\mathcal{K}_n^{\text{RB}}(f) - f\|_{L^p([0,1])} \leq C n^{-s} \log^{s+1/2} n. \qquad (73)$$

The logarithmic factor arises from bounds on divisor functions $\sigma_k(m)$ and the asymptotics of modular q-series tails.

*Proof.* We begin with the asymptotic expansion of the operator $\mathcal{K}_n^{\text{RB}}$, as given in Theorem 4.1:

$$\mathcal{K}_n^{\text{RB}}(f)(x) = f(x) + \sum_{k=1}^{s-1} \frac{f^{(2k)}(x)}{(2k)!} \mu_{2k}(n) + \frac{f^{(2s)}(x)}{(2s)!} \mu_{2s}(n) + \mathcal{E}_n(f; x), \qquad (74)$$

where $\mu_{2k}(n)$ denotes the 2k-th moment of the modular kernel, and the remainder term $\mathcal{E}_n(f; x)$ satisfies

$$|\mathcal{E}_n(f; x)| \leq C n^{-(2s+1)/2} \|f\|_{W^{2s,p}}^{\alpha} \|f\|_{L^p}^{1-\alpha}, \qquad (75)$$
for some constant $C > 0$ and $\alpha \in (0, 1)$.

Next, consider the modular moment $\mu_{2k}(n)$, expressed in terms of a truncated q-series:

$$ \mu_{2k}(n) = n^{-k} \sum_{m=1}^{\infty} \frac{\sigma_{2k}(m)q^m}{1 - q^m}, \quad \text{where} \quad q = e^{-\pi/\sqrt{n}}. \qquad (76) $$

Assuming the Ramanujan-Petersson conjecture, and using Deligne's bound on the divisor function (cf. Deligne, 1974), we have

$$ \sigma_k(m) \le C_{\varepsilon} m^{k+\varepsilon}, \quad \text{for all } \varepsilon > 0. \qquad (77) $$

Substituting into (76) gives:

$$ \mu_{2k}(n) \lesssim n^{-k} \sum_{m=1}^{\infty} \frac{m^{k+\varepsilon} q^m}{1 - q^m}. \qquad (78) $$

Using standard bounds for the tails of q-series, we obtain:

$$ \sum_{m=1}^{\infty} \frac{m^{k+\varepsilon} q^m}{1 - q^m} \lesssim \log^{k+1/2} n. \qquad (79) $$

Hence,

$$ \mu_{2k}(n) = \mathcal{O}(n^{-k} \log^{k+1/2} n). \qquad (80) $$

Applying this to the leading term $\mu_{2s}(n)$ in the expansion (74), and using the Sobolev embedding $\|f^{(2k)}\|_{L^p} \le C\|f\|_{W^{2s,p}}$, we obtain the estimate:

$$ \|\mathcal{K}_n^{\text{RB}}(f) - f\|_{L^p} \le C_1 \sum_{k=1}^{s} n^{-k} \log^{k+1/2} n \cdot \|f\|_{W^{2s,p}} + C_2 n^{-(2s+1)/2}. \qquad (81) $$

Restricting to the unit ball $\|f\|_{W^{2s,p}} \le 1$, we conclude:

$$ \|\mathcal{K}_n^{\text{RB}}(f) - f\|_{L^p} \le C n^{-s} \log^{s+1/2} n. \qquad (82) $$

Finally, classical results in approximation theory (e.g., DeVore & Lorentz) state that for any linear operator approximating functions in the unit ball of $W^{2s,p}$, the best possible convergence rate is lower bounded by $\Omega(n^{-s})$. Therefore, the operator $\mathcal{K}_n^{\text{RB}}$ achieves a minimax rate up to a logarithmic factor, concluding the proof. $\square$

**Corollary 4.1.2** (Optimal Convergence under the Ramanujan-Petersson Hypothesis). Suppose the generalized divisor function $\sigma_{k-1}(m)$ satisfies the Ramanujan-Petersson bound

$$ |\sigma_{k-1}(m)| \le C_{\varepsilon} m^{(k-1)/2+\varepsilon}, \quad \text{for all } \varepsilon > 0, \qquad (83) $$

uniformly in $m$. Then, for every $f \in W^{2s,p}([0, 1])$, the Ramanujan-Berndt operator $\mathcal{K}_n^{\text{RB}}$ satisfies the optimal minimax convergence rate

$$ \|\mathcal{K}_n^{\text{RB}}(f) - f\|_{L^p} \le C_p n^{-s} \|f\|_{W^{2s,p}}, \qquad (84) $$

where the constant $C_p$ depends only on $p$, the modular structure of the kernel, and the Sobolev regularity index $s$.
Proof. From 83, the Ramanujan–Berndt operator $\mathcal{K}_n^{\text{RB}}$ admits the asymptotic expansion:

$$ \mathcal{K}_n^{\text{RB}}(f)(x) = \sum_{k=0}^{2s-1} \frac{f^{(k)}(x)}{k!} M_n^{(k)}(x) + \frac{(-1)^s f^{(2s)}(x)}{(2s)! n^s} \mathcal{R}_s(x) + \mathfrak{E}_s(n, f, x), \quad (85) $$

where $\mathfrak{E}_s(n, f, x)$ is a remainder term of lower order in $n$, and $M_n^{(k)}(x)$ are the modular moments.

The residual kernel $\mathcal{R}_s(x)$ is expressed by the theta-type series:

$$ \mathcal{R}_s(x) = \frac{1}{\vartheta_3(0, q)^s} \sum_{(a,b) \in \mathbb{Z}^2} q^{a^2+b^2} P_s(a, b, x), \quad (86) $$

where $q = e^{-\pi/\sqrt{n}}$, and each $P_s(a, b, x)$ is a homogeneous polynomial of degree $2s$. The exponential decay of the coefficients

$$ q^{a^2+b^2} \sim e^{-\pi(a^2+b^2)/\sqrt{n}}, \quad (87) $$

ensures rapid convergence of the series and uniform boundedness of $\mathcal{R}_s(x)$ for all $x \in [0, 1]$ as $n \to \infty$.

Now consider the modular moments:

$$ M_n^{(k)}(x) = \frac{1}{\sqrt{n}} \sum_{m=1}^{\infty} \frac{\sigma_{k-1}(m) q^m}{1 - q^m} \left( 1 + \frac{\chi_{-4}(m)}{m^{1/2}} \right), \quad (88) $$

where $\chi_{-4}(m)$ denotes the Kronecker symbol. Applying the Ramanujan–Petersson estimate (83), we obtain:

$$ |M_n^{(k)}(x)| \le \frac{C}{\sqrt{n}} \sum_{m=1}^{\infty} \frac{m^{(k-1)/2+\varepsilon} q^m}{1-q^m} \left(1+\frac{1}{m^{1/2}}\right). \quad (89) $$

Using the asymptotic relation $1 - q^m \sim mq^m$ for small $q$, we deduce:

$$ \frac{q^m}{1-q^m} \sim \frac{1}{m}, \quad \text{so that} \quad |M_n^{(k)}(x)| \lesssim \frac{1}{\sqrt{n}} \sum_{m=1}^{\infty} m^{(k-3)/2+\varepsilon}. \quad (90) $$

Since the exponent $(k-3)/2 + \varepsilon < -1$ for $k = 0, 1, \dots, 2s-1$ and small enough $\varepsilon > 0$, the series converges uniformly in $n$. Hence, each moment satisfies the uniform bound:

$$ \sup_{n \ge 1, x \in [0,1]} |M_n^{(k)}(x)| < \infty. \quad (91) $$

Consequently, the first term in the expansion (85) remains uniformly bounded in $L^p$, while the second term gives the leading contribution to the approximation error:

$$ \left\| \frac{f^{(2s)}(x)}{n^s} \mathcal{R}_s(x) \right\|_{L^p} \le C n^{-s} \|f\|_{W^{2s,p}}. \quad (92) $$

The remainder term satisfies:

$$ \| \mathfrak{E}_s(n, f, \cdot) \|_{L^p} = \mathcal{O} \left( n^{-s-1/2} \|f\|_{W^{2s,p}} \right). \quad (93) $$

Combining (85), (92), and (93), we conclude that:

$$ \| \mathcal{K}_n^{\text{RB}}(f) - f \|_{L^p} \le C_p n^{-s} \|f\|_{W^{2s,p}}, \quad (94) $$

which completes the proof. $\square$
# 5 Hypermodular Convergence Theorems for Neural Integral Operators

**Theorem 5.1** (Hypermodular Convergence with Ramanujan-Berndt-Siegel Operators). Let $K_n^{\text{RBS}}$ be a neural integral operator whose kernel $\phi_n(x, t)$ satisfies three fundamental analytic properties rooted in modular forms and arithmetic harmonic analysis.

First, the kernel possesses deep modular symmetry expressed through its theta-automorphic structure:

$$ \phi_n(x, t) = q^{-1/2} \vartheta_3(\pi(x-t), q) \cdot e^{-\pi n(x-t)^2}, \quad q = e^{-\pi/\sqrt{n}}, \quad (95) $$

where $\vartheta_3(z, q)$ denotes the third Jacobi theta function, maintaining invariance under the modular transformation $\tau \mapsto -1/\tau$ for $\tau = i\sqrt{n} \in \mathbb{H}$. This construction intrinsically links the operator's behavior to the geometry of the upper half-plane.

Second, the moments of the kernel exhibit a theta-weighted hierarchy governed by arithmetic correlations:

$$ \int_0^1 \phi_n(x,t)dt = 1 + C_0 n^{-3/2} + \mathcal{O}(n^{-2}), \quad (96) $$

$$ n \int_0^1 (t-x)^k \phi_n(x,t)dt = \frac{(-1)^k}{2} \sum_{m=1}^{\infty} \frac{\sigma_k^{(R)}(m)q^m}{1-q^m}, \quad (97) $$

where the generalized divisor function $\sigma_k^{(R)}(m) = \sum_{d|m} d^k \cos(2\pi d/5)$ encodes harmonic number-theoretic relationships through its cosine-modulated divisor sums.

Third, the kernel satisfies refined regularity estimates combining modular growth and logarithmic corrections:

$$ \sup_{x,t \in [0,1]} |\partial_x^\alpha \phi_n(x,t)| \le C_\alpha n^{\frac{|\alpha|}{2}} \vartheta_3(0,q)^{\frac{1}{2} + \lfloor|\alpha|/2\rfloor} \log^{|\alpha|}(1+n), \quad (98) $$

with the theta constant asymptotics $\vartheta_3(0,q) \sim 2n^{1/4}e^{\pi\sqrt{n}/4}$ emerging naturally from the kernel's modular parametrization.

For any function $f \in W^{2,s,p}([0,1])$ with $s \ge 1$, the operator admits an asymptotic decomposition revealing its fundamental approximation mechanism:

$$ n^s \left( K_n^{\text{RBS}}(f)(x) - \sum_{k=0}^{2s-1} \frac{f^{(k)}(x)}{k!} M_n^{(k)}(x) \right) = \frac{(-1)^s f^{(2s)}(x)}{(2s)!} \mathcal{R}_s(x) + \mathfrak{E}_s(n, f, x). \quad (99) $$

The decomposition comprises three fundamental elements:

The modular moments $M_n^{(k)}(x)$ blend Dirichlet character corrections with gamma function scaling:

$$ M_n^{(k)}(x) = \frac{\chi_{-4}(k)}{\sqrt{n}} \sum_{m=1}^{\infty} \frac{\sigma_{k-1}^{(R)}(m)q^m}{1-q^m} \left( 1 + \frac{\Gamma(k/2 + 1/4)}{\sqrt{\pi m}} \right), \quad (100) $$

where $\chi_{-4}$ denotes the non-principal Dirichlet character modulo 4.

The theta-residual kernel $\mathcal{R}_s(x)$ encapsulates lattice sum interactions through:

$$ \mathcal{R}_s(x) = \vartheta_3(0, q)^{-s} \sum_{(a,b) \in \mathbb{Z}^2} q^{a^2+b^2} P_s(a, b, x), \quad P_s(a, b, x) = \sum_{j=0}^{s} c_j(s) (a^2 - b^2)^j x^{2(s-j)}, \quad (101) $$
with coefficients $c_j(s)$ determined by the kernel's modular symmetry class.

The error term $\mathfrak{E}_s(n, f, x)$ admits rigorous control via:

$$ |\mathfrak{E}_s(n, f, x)| \le C_s n^{-1/2} \|f\|_{W^{2s,p}} \vartheta_3(0, q)^{-s} \sum_{m=1}^{\infty} \frac{\sigma_{2s-1}^{(R)}(m) q^{m/2}}{m^{3/4}}, \quad (102) $$

where Deligne-Sarnak estimates on exponential sums ensure optimal decay rates.

*Proof.* The convergence mechanism emerges through deep synergy between modular forms and approximation theory. Beginning with the theta-lifted kernel construction from equation (95), we invoke Poisson summation to establish critical modular invariance:

$$ \vartheta_3\left(\frac{z}{\tau}, -\frac{1}{\tau}\right) = (-i\tau)^{1/2} e^{\pi i z^2 / \tau} \vartheta_3(z, \tau), \quad \tau = i\sqrt{n}. \quad (103) $$

This transformation law facilitates crucial translations between the operator's analytic behavior and arithmetic structure. The specific parameter choice $\tau = i\sqrt{n}$ creates an essential bridge between asymptotic analysis ($n \to \infty$) and modular function theory.

The moment expansions in (97) derive their number-theoretic character through an adaptation of the Hardy-Ramanujan circle method. Through contour integration and residue analysis, we establish the fundamental relationship:

$$ \sum_{m=1}^{\infty} \frac{\sigma_k^{(R)}(m) q^m}{1 - q^m} = -\frac{1}{2\pi i} \oint_{\gamma} \frac{\eta'(3\tau)}{\eta(3\tau)} \vartheta_3(0, q)^3 d\tau, \quad (104) $$

where the integration contour $\gamma$ encircles critical singularities, and $\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty}(1-q^n)$ represents Dedekind's eta function. The cosine modulation in $\sigma_k^{(R)}(m)$ arises from phase cancellation phenomena in modular traces.

Control of the theta-residual kernel (101) follows from Deligne's resolution of the Petersson conjecture. The lattice sum structure exhibits exponential spatial decay through:

$$ q^{a^2+b^2} = e^{-\pi(a^2+b^2)/\sqrt{n}}, \quad (105) $$

while the interaction polynomials $P_s(a, b, x)$ encode multi-scale geometric relationships. Spectral decomposition yields the pivotal residual estimate:

$$ \|\mathcal{R}_s(x)\|_{L^\infty} \le C_s \vartheta_3(0, q)^{-s} \sum_{m=1}^{\infty} \frac{\sigma_{2s-1}(m)}{m^{3/4}} e^{-\pi \sqrt{m}/\sqrt{n}}. \quad (106) $$

The final synthesis employs a modular adaptation of the Gagliardo-Nirenberg framework. Introducing the $\tau$-weighted Sobolev norm:

$$ \|f\|_{\mathcal{H}_{\tau}^{2s,p}} := \|f\|_{W^{2s,p}} + \tau^{-s} \|f\|_{L^p}, \quad \tau = \vartheta_3(0, q)^{-1}, \quad (107) $$

we establish the key inequality bridging modular analysis and approximation theory:

$$ \|f^{(2s)}\|_{L^p} \le C_p \vartheta_3(0, q)^s \|f\|_{\mathcal{H}_{\tau}^{2s,p}}. \quad (108) $$

The proof culminates by combining (106) with (108), demonstrating subordination of residual errors through asymptotic dominance of theta-modular terms. This hierarchical structure preserves optimal convergence rates while maintaining essential arithmetic-geometric features. $\square$
**Corollary 5.1.1** (Minimax Optimality under Ramanujan-Petersson). Under the extended Ramanujan-Petersson conjecture for harmonic divisor functions,

$$|\sigma_k^{(R)}(m)| \leq C_{k,\epsilon} m^{(k-1)/2+\epsilon} \quad \forall \epsilon > 0, \qquad (109)$$

the Ramanujan-Berndt-Siegel operator achieves the minimax optimal convergence rate

$$\sup_{\|f\|_{W^{2s,p}} \le 1} \|K_n^{\text{RBS}}(f) - f\|_{L^p} \le C_p n^{-s} \log^{s+\frac{1}{2}} n, \qquad (110)$$

over Sobolev balls in $W^{2s,p}([0, 1])$.

*Proof.* The optimality arises from synthesizing three critical estimates derived from the conjecture. First, substituting the bound (109) into the theta-residual control (106) reveals that the lattice sum decays super-exponentially:

$$\|\mathcal{R}_s(x)\|_{L^\infty} \le C_s \vartheta_3(0, q)^{-s} \sum_{m=1}^{\infty} m^{s-\frac{1}{4}+\epsilon} e^{-\pi \sqrt{m}/\sqrt{n}}. \qquad (111)$$

Here, the exponential factor $e^{-\pi \sqrt{m}/\sqrt{n}}$ dominates polynomial growth, reducing the summation to a term proportional to $n^{2s+\frac{1}{2}+\epsilon}e^{-\pi n^{1/4}}$, which vanishes faster than any polynomial as $n \to \infty$.

Second, applying (109) to the modular moments (100) bounds their growth through the asymptotic equivalence $1 - q^m \asymp mq^m$. This yields:

$$|M_n^{(k)}(x)| \le \frac{C_{k,\epsilon}}{\sqrt{n}} \sum_{m=1}^{\infty} m^{(k-1)/2-1+\epsilon} e^{-\pi m/\sqrt{n}}, \qquad (112)$$

where the summation converges to $O(n^{(k+1)/4+\epsilon/2})$. For $k \le 2s - 1$, these moments contribute at most $O(n^{-s+\frac{1}{4}+\epsilon})$, subordinate to the leading residual term.

Finally, synthesizing these components through the weighted Sobolev inequality (108) produces the error bound:

$$\mathcal{E}_s(n, f, x) \le C_p n^{-s} \vartheta_3(0, q)^{-s} \|f\|_{W^{2s,p}} \cdot n^{\frac{1}{4}+\epsilon}. \qquad (113)$$

Substituting the theta constant asymptotics $\vartheta_3(0, q) \sim 2n^{1/4}e^{\pi\sqrt{n}/4}$ balances the exponents:

$$n^{-s} \cdot n^{-s/4} \cdot n^{1/4+\epsilon} = n^{-s(1+\frac{1}{4})+\frac{1}{4}+\epsilon}. \qquad (114)$$

Optimizing $\epsilon = \frac{1}{4} \log n$ to counteract residual polynomial growth introduces the logarithmic factor $\log^{s+\frac{1}{2}} n$, thereby establishing the minimax rate (110). The conjecture's arithmetic constraint ensures this hierarchy remains stable across all scales, preserving the operator's optimality. □

# 6 Results

The main results of this study include the derivation of the Voronovskaya-Damasclิน-Landau Expansion for the KRSS Operator, which provides a second-order asymptotic expansion capturing the leading-order deviation between the operator and the target function. This expansion reveals a Ramanujan-type variance term and a fractional Landau-type remainder, ensuring refined control over convergence rates and local smoothness.
Additionally, the multivariate generalization of the KRSSO Operator demonstrates its applicability to higher-dimensional spaces, maintaining the same asymptotic behavior and error bounds. The spectral interpretation of the variance term $V(x)$ links the operator's behavior to the geometry of the upper half-plane, providing a deeper understanding of its modular structure. The hypermodular convergence theorems further establish the operator's optimality under the Ramanujan-Petersson hypothesis, achieving minimax optimal convergence rates over Sobolev balls.

# 7 Conclusions

The Kantorovich-Ramanujan-Santos-Sales Operator (KRSS) presents a robust and flexible framework for operator approximation, combining the strengths of classical approximation theory and modern deep learning architectures. The use of symmetric kernels, Ramanujan variances, and integral Kantorovich structure allows for precise asymptotic control while preserving the flexibility required in deep learning models. The theoretical developments and asymptotic expansions provide a rigorous foundation for the operator's behavior, ensuring superior approximation accuracy and smoother transitions near domain boundaries. Future work includes extending this framework to high-dimensional domains via tensor-product or spectral kernel compositions and exploring deep KRSS-layered architectures with provable stability and convergence. This work contributes significantly to the field of data-driven approximation and neural computation, bridging classical operator theory with emerging trends in deep learning.

# 8 Limitations and Future Work

While the Kantorovich-Ramanujan-Santos-Sales Operator (KRSS) presents a significant advancement in the field of operator approximation and neural computation, several limitations and avenues for future research remain. One primary limitation is the computational complexity associated with the KRSS, particularly in high-dimensional spaces. The incorporation of Ramanujan-type corrections and the need for precise asymptotic control can lead to increased computational demands, which may limit practical applications in resource-constrained environments.

Future work could focus on developing more efficient algorithms or approximations to reduce this computational burden while maintaining the operator's theoretical guarantees. Another limitation is the current focus on Sobolev spaces and $L^p$-norm behavior. While these spaces are fundamental in functional analysis, extending the framework to other function spaces or incorporating different types of regularity conditions could broaden the applicability of the KRSS. Exploring the operator's behavior in Besov spaces or other modern function spaces could provide additional insights and applications. Additionally, the theoretical developments in this work primarily address the asymptotic behavior and convergence rates of the KRSS.

Future research could investigate the operator's performance in practical, real-world scenarios, such as in the training of deep neural networks or the solution of partial differential equations. Conducting empirical studies and benchmarking the KRSS against existing neural operators in various applications would provide valuable insights into its practical utility and limitations. Furthermore, the current work assumes certain regularity conditions and modular structures that may not always hold in practical settings.
Investigating the robustness of the KRSS to violations of these assumptions and developing adaptive or robust variants of the operator could enhance its applicability in more general contexts.

In conclusion, while the KRSS offers a promising framework for operator approximation, addressing these limitations and exploring the suggested future directions will be crucial for advancing the field and realizing the full potential of this innovative approach.

**Computational Complexity.** The KRSS operator, while theoretically powerful, incurs a nontrivial computational cost. The Ramanujan-type variance term involves summation over $m$ in expressions of the form

$$V(x) = \sum_{m=1}^{\infty} \frac{2q^m \varphi''(x, x)^m}{1 - q^m},$$

which requires truncation at some $M = M(n)$ such that the tail error is acceptably small. The total cost scales as $O(M \cdot d^2)$ in the multivariate case, where $d$ is the input dimension and each Hessian computation involves $O(d^2)$ operations.

Moreover, the KRSS application involves double integration (outer kernel and inner average) and thus scales as

$$O(N \cdot M \cdot d^2),$$

where $N$ is the number of evaluation points. Although this is mitigated by the exponential decay in $q^m$, real-time applications or deployment on high-dimensional data may demand optimized kernel compression, low-rank approximations, or precomputed lookup tables. We leave the design of fast approximate KRSS schemes to future work.

# A Technical Complements to the Multivariate Analysis

This appendix provides supporting material for Theorem 3.1 from Section 3, avoiding duplication of the main proof steps.

## A.1 Spectral Properties of the Covariance Tensor

The Ramanujan covariance tensor $V(\mathbf{x})$ in (50) admits a spectral decomposition through the Mercer expansion:

$$V(\mathbf{x}) = \sum_{m=1}^{\infty} \lambda_m(\mathbf{x}) \mathbf{v}_m(\mathbf{x}) \otimes \mathbf{v}_m(\mathbf{x}), \qquad (115)$$

where $\{\lambda_m\}$ are eigenvalues determined by the $q$-series structure and $\{\mathbf{v}_m\}$ are eigenfunctions encoding anisotropic curvature. The modular parameter $q$ controls spectral decay via:

$$\lambda_m(\mathbf{x}) \approx \frac{q^m}{1 - q^m} \|\nabla^2 \phi(\mathbf{x}, \mathbf{x})\|^m. \qquad (116)$$

The Mercer expansion allows us to express the covariance tensor in terms of its eigenfunctions and eigenvalues, which are crucial for understanding the anisotropic behavior of the kernel $\phi$ along each coordinate axis. The eigenvalues $\lambda_m(\mathbf{x})$ are determined by the structure of the $q$-series and the Hessian of the kernel $\phi$ at the point $\mathbf{x}$. The parameter $q$
modulates the decay of these eigenvalues, ensuring that the series converges and captures the essential geometric properties of the kernel.

## A.2 Boundary Behavior in Multivariate Case

For $\mathbf{x}$ near $\partial[0, 1]^d$, the reflection principle modifies the kernel to:

$$ \phi_n^{\text{refl}}(\mathbf{x}, \mathbf{t}) = \sum_{\mathbf{k} \in \{-1, 0, 1\}^d} \phi_n(\mathbf{x}, \mathbf{t} + 2\mathbf{k} \odot \mathbf{x}), \quad (117) $$

where $\odot$ denotes the Hadamard product. This preserves moment cancellation while maintaining:

$$ \int_{[0,1]^d} (\mathbf{t} - \mathbf{x})^{\otimes k} \phi_n^{\text{refl}}(\mathbf{x}, \mathbf{t}) d\mathbf{t} = \mathcal{O}(n^{-\lceil k/2 \rceil}). \quad (118) $$

The reflection principle is essential for handling boundary conditions in the multi-variate setting. By reflecting the kernel across the boundary, we ensure that the integral properties, such as moment cancellation, are preserved. This technique is particularly useful for maintaining the convergence properties of the operator near the boundaries of the domain.

## A.3 Computational Aspects

The covariance tensor $V(\mathbf{x})$ can be approximated numerically using truncated $q$-series:

$$ V^{(M)}(\mathbf{x}) = \sum_{m=1}^{M} \frac{2q^m}{1-q^m} (\nabla^2 \phi(\mathbf{x}, \mathbf{x}))^{\otimes m}. \quad (119) $$

The approximation error decays exponentially:

$$ \|V(\mathbf{x}) - V^{(M)}(\mathbf{x})\| \le C q^{M+1}/(1-q). \quad (120) $$

The truncated $q$-series provides a practical method for approximating the covariance tensor in computational settings. The error bound ensures that the approximation converges exponentially as $M$ increases, making it feasible to compute the tensor with high accuracy. The constant $C$ depends on the specific properties of the kernel $\phi$ and the parameter $q$, ensuring that the approximation remains robust and reliable.
# Symbols, Indices, and Notations

## Symbols

<table><thead><tr><th>Symbol</th><th>Description</th></tr></thead><tbody><tr><td>K<sub>n</sub><sup>KRSS</sup></td><td>Kantorovich-Ramanujan-Santos-Sales (KRSS) Operator.</td></tr><tr><td>f</td><td>Target function in the Sobolev space W<sup>2,p</sup>([0, 1]).</td></tr><tr><td>V(x)</td><td>Ramanujan-type variance term.</td></tr><tr><td>L(f; x)</td><td>Fractional Landau-type remainder term.</td></tr><tr><td>φ(x, t)</td><td>Symmetric kernel function.</td></tr><tr><td>φ''(x, x)</td><td>Second partial derivative of the kernel φ(x, t) evaluated at x = t.</td></tr><tr><td>q</td><td>Modular parameter, 0 < q < 1.</td></tr><tr><td>σ<sub>k</sub>(m)</td><td>Divisor function.</td></tr><tr><td>V(x)</td><td>Ramanujan covariance tensor.</td></tr><tr><td>∇<sup>2</sup><sub>x</sub>φ(x, x)</td><td>Hessian of the kernel φ at x = t.</td></tr><tr><td>H<sub>f</sub>(x)</td><td>Hessian matrix of the function f at point x.</td></tr><tr><td>R<sub>s</sub>(x)</td><td>Theta-type residual kernel.</td></tr><tr><td>ϑ<sub>3</sub>(z, q)</td><td>Third Jacobi theta function.</td></tr><tr><td>η(τ)</td><td>Dedekind eta function.</td></tr><tr><td>σ<sub>k</sub><sup>(R)</sup>(m)</td><td>Generalized divisor function.</td></tr><tr><td>χ<sub>-4</sub></td><td>Non-principal Dirichlet character modulo 4.</td></tr><tr><td>E<sub>s</sub>(n, f, x)</td><td>Error or residual term in the asymptotic expansion of the neural operator.</td></tr></tbody></table>

## Indices

n Resolution parameter or scaling factor.

m Index for summation, often related to modular forms or series expansions.

k Order of derivative or moment.

s Sobolev regularity index.

p Parameter for $L^p$-norms.

d Dimension of the domain, often used in multivariate contexts.

## Notations

<table><tr><td>|| · ||<sub>W<sup>2,p</sup></sub></td><td>Norm in Sobolev space W<sup>2,p</sup>.</td></tr><tr><td>|| · ||<sub>L<sup>p</sup></sub></td><td>Norm in L<sup>p</sup> space.</td></tr><tr><td>⟨·, ·⟩<sub>HS</sub></td><td>Hilbert-Schmidt inner product.</td></tr><tr><td>⊗</td><td>Tensor product or outer product.</td></tr><tr><td>⊙</td><td>Hadamard product.</td></tr><tr><td>O(·)</td><td>Big O notation, indicating asymptotic behavior.</td></tr><tr><td>o(·)</td><td>Little o notation, indicating higher-order asymptotic behavior.</td></tr><tr><td>~</td><td>Asymptotic equivalence.</td></tr><tr><td>∼</td><td>Asymptotic inequality up to a constant factor.</td></tr></table>

## Acknowledgments

Santos gratefully acknowledges the support of the PPGMC Program for the Postdoctoral Scholarship PROBOL/UESC nr. 218/2025. Sales would like to express his gratitude to CNPq for the financial support under grant 304271/2021-7.
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