Max-Min Durrmeyer Exponential Sampling Operators

• Max–Min Durrmeyer-type exponential sampling operators are nonlinear approximation processes that use Mellin kernels and supremum–minimum algebra on bounded, log-uniformly continuous functions.
• They employ discrete and continuous moment conditions to ensure robust pointwise, uniform, and modular convergence in generalized Orlicz spaces.
• Numerical experiments validate these operators with improved error bounds and stability, supporting applications in signal processing and function reconstruction.

Max–Min Durrmeyer-type exponential sampling operators constitute a nonlinear class of approximation processes acting on bounded, log-uniformly continuous functions and integrable functions on positive real intervals. These operators utilize Mellin-type kernels with discrete and continuous moment conditions and are designed to exhibit robust pointwise, uniform, and modular convergence properties, including in generalized Orlicz spaces equipped with the Haar measure. Their formulation involves supremum–minimum algebra, enabling output stability and improved accuracy under certain regularity conditions of the approximated function and kernel pairings. Numerical experiments confirm theoretical convergence rates and highlight practical strengths in both smooth and piecewise regimes (Pradhan et al., 6 Dec 2025, Pradhan et al., 16 Oct 2025).

1. Underlying Functional Framework

The domain of Max–Min Durrmeyer-type exponential sampling operators is structured around Mellin analysis and Orlicz space functional settings.

• Log-uniformly continuous and bounded functions: Let $\mathcal{I}\subset\mathbb{R}_+$ be compact. Functions $h:\mathcal{I}\rightarrow\mathbb{R}$ belong to $\mathcal{U}_{bl}(\mathcal{I})$ if $\forall\,\epsilon>0 \ \exists\,\delta>0$ such that $$|\log w_1 - \log w_2| < \delta \Rightarrow |h(w_1) - h(w_2)| < \epsilon$$ for all $w_1, w_2\in\mathcal{I}$, with bounded sup-norm $\|h\|_\infty = \sup_{w\in\mathcal{I}}|h(w)|$.
• Mellin–Orlicz Spaces: For measurable $h:\mathbb{R}_+\rightarrow\mathbb{R}$, the Orlicz modular is $$I_\zeta[h] = \int_0^\infty \zeta(|h(w)|)\,d\mathfrak{h}(w) = \int_0^\infty \zeta(|h(w)|)\,\frac{dw}{w}$$, with $\zeta$ convex and satisfying growth and continuity constraints. The space $L^\zeta_{\mathfrak{h}}(\mathbb{R}_+)$ comprises functions with $I_\zeta[\lambda h]<\infty$ for some $\lambda>0$. The Luxemburg norm is defined as $$\|h\|_\zeta = \inf\{\lambda > 0 : I_\zeta[h/\lambda] \leq 1\}$$. The $\Delta_2$-condition on $\zeta$ ensures equivalence of modular and norm convergence.
• Kernels and Moment Conditions: Mellin kernels $\Phi: \mathbb{R}_+ \rightarrow [0,\infty)$ and weights $\psi: \mathbb{R}_+$ are chosen to satisfy discrete and continuous moment conditions:
  • Discrete moments: $$M_r(\Phi) = \sup_{w > 0} \sum_{k \in \mathbb{Z}}|\Phi(e^{-k}w)|\,|k - \log w|^r < \infty$$ for $r = 0,1,2$.
  • Positivity: For $w\in [1,e]$, $\theta := \inf_{w\in[1,e]}\Phi(w)>0$.
  • Integrability: $\int_1^e \psi(w)\,\frac{dw}{w} = K > 0$, $$\sup_{v > 0} \sum_{k \in \mathbb{Z}} \psi(ve^{-k}) < \infty$$.

2. Operator Definitions and Algebraic Structure

• Max–Product Durrmeyer Operator (Editor’s term: $𝒟^{M}$): For $h \in L^1(\mathcal{I}) \cap [0,1]$,

  $$\mathcal{D}^{(M)}_{n,\Phi,\psi}(h)(w) = \frac{ \max_{k \in J_n} \Phi(e^{-k}w^n)\,n\int_a^b \psi(e^{-k}v^n)\,h(v)\frac{dv}{v} }{ \max_{k \in J_n} \Phi(e^{-k}w^n)\,n\int_a^b \psi(e^{-k}v^n)\frac{dv}{v} }$$

  This operator is monotone, subadditive, and positively homogeneous, due to the max–product structure.

• Max–Min Durrmeyer Operator (Editor’s term: $𝒟^{m}$):

  $$\mathcal{D}^{(m)}_{n,\Phi,\psi}(h)(w) = \max_{k \in J_n} \left[ n\int_a^b \psi(e^{-k}v^n)\,h(v)\frac{dv}{v} \wedge \frac{ \Phi(e^{-k}w^n) }{ \max_{j \in J_n} \Phi(e^{-j}w^n)\,n\int_a^b \psi(e^{-j}v^n)\frac{dv}{v} } \right]$$

  The symbol $\wedge$ denotes minimum. Both operators leverage supremum and minimum operations over discrete index sets defined by the scaling and kernel localization properties.

3. Convergence Theorems and Quantitative Rates

• Pointwise and Uniform Convergence: For $h \in \mathcal{U}_{bl}(\mathcal{I})$, both operators satisfy uniform convergence:

  $$\lim_{n\to\infty} \mathcal{D}^{(M)}_{n,\Phi,\psi}(h)(w) = h(w),\quad \lim_{n\to\infty} \mathcal{D}^{(m)}_{n,\Phi,\psi}(h)(w) = h(w)$$

  The convergence is uniform on compact subintervals of $\mathbb{R}_+$ (Pradhan et al., 6 Dec 2025, Pradhan et al., 16 Oct 2025).

• Modular Convergence in Orlicz Space:

  If $h\in L^\zeta_{\mathfrak{h}}(\mathcal{I})$, then for any $\lambda > 0$,

  $$\lim_{n\to\infty} I_\zeta[\lambda(\mathcal{D}^{(M)}_{n,\Phi,\psi}(h) - h)] = 0$$

  $$\lim_{n\to\infty} I_\zeta[\lambda(\mathcal{D}^{(m)}_{n,\Phi,\psi}(h) - h)] = 0$$

  When $\zeta$ satisfies the $\Delta_2$-condition, modular convergence implies convergence in Luxemburg norm.

• Rates and Quantitative Error Bounds:

  Error estimates are obtained via localization:

  $$|\mathcal{D}^{(M)}_{n,\Phi,\psi}(h)(w) - h(w)| \leq C\,\omega(h; O(1/n)) + O(n^{-\nu})$$

  Where $\omega$ is the modulus of log-continuity and $\nu>0$ relates to kernel decay. For max–min, analogous bounds hold with an additional term due to the min-composition.

  Quantitative rates in terms of the logarithmic modulus $\mho_h(\delta)$ are given for $h \in LU_b(\mathbb{R}_+)$ by:

  $$|\mathscr D^{\,m}_{n,\Phi,\Psi}[h](z)-h(z)| \leq \frac{\mho_h(\delta)}{\theta\,m_0(\Phi)\,M_0(\Psi)} + \frac{\mho_h(\delta)}{n\,\theta\,\delta}[m_0(\Phi)\,M_1(\Psi)+m_1(\Phi)\,M_0(\Psi)]$$

  For log-Hölder continuous functions ($h\in \mathcal{L}_{\log}^\alpha$), convergence rates are $O(n^{-\frac{\alpha}{1+\alpha}})$ as $n\to\infty$.

4. Kernel Selection and Explicit Error Performance

• Classical Kernel Choices:
  • Mellin B-spline of order 2: $\Phi(z) = \max\{0,1-|\log z|\}$, supported on $[e^{-1},e]$.
  • Mellin B-spline of order 3: $B_3(w)$.
  • Mellin–Fejér kernel: $$\Psi(z) = \tfrac{1}{2}[\text{sinc}(\tfrac{\log z}{2})]^2$$.
  • Mellin–Jackson kernel: $J_{1.05,1}(w)$.

  Moment values and positivity constants for these kernels determine existence, bounds, and localization properties of the operators in implementation, and impact numerical stability.

• Error Table Illustration:

  Empirical error data for the smooth oscillatory test function $h_1$ with $B_2(w)$ and $J_{1.05,1}(w)$:

  n	$|\mathscr{D}^{M}_{n}-h_1|$ at $w=0.8$	$|\mathscr{D}^{m}_{n}-h_1|$ at $w=0.8$	$|\mathscr{D}^{M}_{n}-h_1|$ at $w=2.0$	$|\mathscr{D}^{m}_{n}-h_1|$ at $w=2.0$
  17	0.00916	0.01972	0.18534	0.21016
  26	0.00679	0.01368	0.10942	0.12752
  35	0.00497	0.00993	0.07972	0.09501
  53	0.00321	0.00639	0.05126	0.06193

  There is empirical confirmation of $O(n^{-\alpha})$ decay of error and operator-specific differences in smooth versus non-smooth regimes. Higher-order kernels enhance convergence and minimize Gibbs-type oscillations (Pradhan et al., 6 Dec 2025).

5. Analytical Techniques for Proof and Localization

The convergence theory for the Max–Min and Max–Product operators leverages several techniques:

• Local–nonlocal decomposition: Splitting sums or maxima into neighborhoods near $k\sim n\log w$ (where the kernel is localized) and elsewhere (where kernel contributions decay as $n^{-\nu}$).

• Triangle-type estimates for extremal functionals: Used for error control in supremum and minimum constructions.

• Discrete moment and lower bound lemmas: Guarantee the existence of non-negligible kernel values in requisite intervals, precluding denominator degeneration.

• Passing convexity via Fubini–Tonelli integration: Essential for modular convergence results in Orlicz spaces.

• Density arguments: Approximating general $L^\zeta_{\mathfrak{h}}$ functions via $\mathcal{U}_{bl}$ functions for controlling the modular difference.

6. Practical Implications, Stability, and Extensions

• Operator Stability and Range: Max–Min Durrmeyer-type operators enforce outputs in $[0,1]$ when $h$ is bounded in $[0,1]$, crucial for probabilistic or fuzzy system applications (Pradhan et al., 16 Oct 2025).

• Nonlinear versus Linear Durrmeyer variants: The supremum–minimum structure often yields higher accuracy than linear summation counterparts, specifically under finite kernel support and regularity constraints.

• Application Contexts: These operators suit Mellin-transform signal processing, numerical reconstruction of functions on stretched intervals, and scenarios requiring robust nonlinear approximation schemes.

• Limitations and Open Questions:

  • The requirement $\theta > 0$ (strict positiveness of kernel) restricts endpoints for compact intervals.
  • Saturation theorems and inverse results remain open for generalizations.
  • Multivariate extensions, adaptive kernel selection, and data-driven sampling in the Mellin context are promising directions.
  • Extensions to variable-exponent and weighted Orlicz spaces are natural and have initial study foundations (Pradhan et al., 14 Aug 2025).

7. Summary and Outlook

Max–Min Durrmeyer-type exponential sampling operators represent a nonlinear, kernel-driven, Mellin-analytic framework for function approximation, with thoroughly characterized convergence in log-uniform and modular topologies. They offer complementary advantages—max-product for smooth accuracy, max-min for stability near singularities—with provable rates and robust empirical performance. Ongoing research is focused on generalizing to broader function spaces, saturation and multivariate questions, and adaptive algorithmic deployment (Pradhan et al., 6 Dec 2025, Pradhan et al., 16 Oct 2025).