The fractional sine series (FRSS) and the fractional cosine series (FRCS) were defined. Three types of discrete convolution operations for FRCS and FRSS were introduced, along with a detailed investigation into their corresponding convolution theorems. The interrelationship between these convolution operations was also discussed. Additionally, as an application of the presented results, two forms of discrete convolution equations based on the proposed convolution theorems were examined, resulting in explicit solutions for these equations. Furthermore, numerical simulations were provided to demonstrate that our proposed solution can be easily implemented with low computational complexity.

Convolution plays a pivotal role in the domains of signal processing and optics. This paper primarily focuses on studying the weighted convolution for quaternion linear canonical cosine transform (QLCcT) and its application in multiplicative filter analysis. Firstly, we propose QLCcT by combining quaternion algebra with linear canonical cosine transform (LCcT), which extends LCcT to Hamiltonian quaternion algebra. Secondly, we introduce weighted convolution and correlation operations for QLCcT, accompanied by their corresponding theorems. We also explore the properties of QLCcT. Thirdly, we utilize these proposed convolution structures to analyze multiplicative filter models that offer lower computational complexity compared to existing methods based on quaternion linear canonical transform (QLCT). Additionally, we discuss the rationale behind studying such transforms using quaternion functions as an illustrative example.

Multiobjective programming refers to a mathematical problem that requires the simultaneous optimization of multiple independent yet interrelated objective functions when solving a problem. It is widely used in various fields, such as engineering design, financial investment, environmental planning, and transportation planning. Research on the theory and application of convex functions and their generalized convexity in multiobjective programming helps us understand the essence of optimization problems, and promotes the development of optimization algorithms and theories. In this paper, we firstly introduces new classes of generalized $(F,\alpha ,\rho ,d)-I$ functions for semi-preinvariant convex multiobjective programming. Secondly, based on these generalized functions, we derive several sufficient optimality conditions for a feasible solution to be an efficient or weakly efficient solution. Finally, we prove weak duality theorems for mixed-type duality.

In the fields of signal processing and applied mathematics, the uncertainty principle plays a significant role. Motivated by the theoretical significance of the octonion quadratic-phase Fourier transform (OQPFT), we present a systematic investigation of uncertainty relations for this transform. Firstly, we derive the differential properties of the OQPFT. Secondly, we systematically derive four corresponding uncertainty principles for the OQPFT, covering both deterministic and random octonion signal scenarios. Finally, three sets of simulation experiments using 3D octonion chirped signals are performed, including two Gaussian-modulated cases with different complexity and a constant-coefficient rectangular-window non-Gaussian case for comparative verification. Full three-dimensional spectral observation shows that the OQPFT achieves superior frequency-domain aggregation compared with the octonion Fourier transform (OFT). Further comparative analysis demonstrates that this spectral aggregation advantage is not limited to Gaussian signals, but also holds for non-Gaussian windowed signals. This paper enriches the uncertainty theory in the hypercomplex domain and provides theoretical support for the frequency-domain analysis of high-dimensional quadratic-phase signals.