Outlier-Robust Convergence of Integer- and Fractional-Order Difference Operators in Fuzzy-Paranormed Spaces: Diagnostics and Engineering Applications

We develop a convergence framework for Grünwald–Letnikov (GL) fractional and classical integer difference operators acting on sequences in fuzzy-paranormed (fp) spaces, motivated by data that are imprecise and contain sporadic outliers. Fuzzy paranorms provide a resolution-dependent notion of proximity, while statistical and lacunary statistical convergence downweight sparse deviations by natural density; together, they yield robust criteria for difference-filtered signals. Within this setting, we establish uniqueness of fp–Δm statistical limits; an equivalence between fp-statistical convergence of Δm (and its GL extension Δα) and fp-strong p-Cesàro summability; an equivalence between lacunary fp-Δm statistical convergence and blockwise strong p-Cesàro summability; and a density-based decomposition into a classically convergent part plus an fp-null remainder. We also show that GL binomial weights act as an ℓ1 convolution, ensuring continuity of Δα in the fp topology, and that nabla/delta forms are transferred by the discrete Q–operator. The usefulness of the criteria is illustrated on simple engineering-style examples (e.g., relaxation with memory, damped oscillations with bursts), where the fp-Cesàro decay of difference residuals serves as a practical diagnostic for Cesàro compliance. Beyond illustrative mathematics, we report engineering-style diagnostics where the fuzzy Cesàro residual index correlates with measurable quantities (e.g., vibration amplitude and energy surrogates) under impulsive disturbances and missing data. We also calibrate a global decision threshold τglob via sensitivity analysis across (α,p,m), where m∈N is the integer difference order, α>0 is the fractional order, and p≥1 is the Cesàro exponent, and provide quantitative baselines (median/M-estimators, ℓ1 trend filtering, Gaussian Kalman filtering, and an α-stable filtering structure) to show complementary gains under bursty regimes. The results are stated for integer m and lifted to fractional orders α>0 through the same binomial structure and duality.