On approximation of lattice-valued functions using lattice integral transforms

This paper examines the approximation capabilities of lattice integral transforms and their compositions in reconstructing lattice-valued functions. By introducing an integral kernel Q on the function domain, we define the concept of a Q-inverse integral kernel, which generalizes the traditional inverse kernel defined as a transposed integral kernel. Leveraging these Q-inverses, we establish upper and lower bounds for a transformed version of the original function induced by the integral kernel Q. The quality of approximation is analyzed using a lattice-based modulus of continuity, specifically designed for functions valued in complete residuated lattices. Additionally, under specific conditions, we demonstrate that the approximation quality for extensional functions with respect to the kernel Q can be estimated through the integral of the square of Q, and in certain cases, these extensional functions can be perfectly reconstructed. The theoretical findings, illustrated through examples, provide a strong foundation for further theoretical advancement and practical applications.