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This paper introduces the fractional Fourier series on the fractional torus and proceeds to investigate several fundamental aspects. Our study includes topics such as fractional convolution, fractional approximation, fractional Fourier inversion, and the Poisson summation formula. We also explore the relationship between the decay of its fractional Fourier coefficients and the smoothness of a function. Additionally, we establish the convergence of the fractional Féjer means and Bochner–Riesz means. Finally, we demonstrate the practical applications of the fractional Fourier series, particularly in the context of solving fractional partial differential equations with periodic boundary conditions, and showcase the utility of approximation methods on the fractional torus for recovering non-stationary signals. [ABSTRACT FROM AUTHOR]

In this study, we propose an advanced category of a fuzzy adaptive neural network (FANN) based on a feature self-enhancement unit (FSU) and statistical selection methods (SSMs). Undoubtedly, the raw data contain a large amount of information with varying importance. One of the most important tasks for regression model design is to avoid losing these important details. However, the raw data cannot participate in the whole training process due to the data fuzzification unit in the structure of conventional fuzzy neural networks (FNNs). Meanwhile, the polynomial-based neuron also has its limitations as a common node in FNNs. For example, in the polynomial neuron, the complexity of the neurons increases exponentially with the increase in network size. Consequently, overfitting and insufficient raw data information are two primary drawbacks in the structure of conventional FNNs. To address these limitations, we designed the FSU and the SSM as effective vehicles to reduce data dimensionality and select significant raw information. The proposed FANN also demonstrates the capability to improve modeling accuracy in neural networks. Moreover, this is the first instance of integrating statistical methods and feature self-enhancement techniques into a fuzzy model. To validate and showcase the superiority of the proposed FANN, the model is applied to 16 machine learning datasets, outperforming other comparative models in 81.25% of the datasets utilized. Additionally, the proposed FANN model outperformed the latest FNN models, achieving an average 5.1% increase in modeling accuracy. The comparison experiment section not only includes classical machine learning models but also references the experimental results from two recent related studies. [ABSTRACT FROM AUTHOR]

We establish one-sided weighted endpoint estimates for the ϱ -variation ( ϱ > 2 ) operators of one-sided singular integrals under certain priori assumption by applying one-sided Calderón–Zygmund argument. Using one-sided sharp maximal estimates, we further prove that the ϱ -variation operators of related commutators are bounded on one-sided weighted Lebesgue and Morrey spaces. In addition, we also show that these operators are bounded from one-sided weighted Morrey spaces to one-sided weighted Campanato spaces. As applications, we obtain some results for the λ -jump operators and the numbers of up-crossings. Our main results represent one-sided extensions of many previously known ones. [ABSTRACT FROM AUTHOR]