Ozbek U.2026-10-162026-10-162026Ozbek U / Moduli of smoothness and growth properties of Hartley transform / SDU University / Department of Mathematics and Natural Scienceshttps://repository.sdu.edu.kz/handle/123456789/2678This study analyzes how the moduli of smoothness of functions influence the behavior of their Hartley transforms. While harmonic analysis has historically centered on the Fourier transform, the Hartley transform offers a computationally efficient, real-valued alternative for processing real data. The analysis establishes key results, including variants of the Riemann–Lebesgue lemma adapted to the Hartley transform context, the Parseval identity, and the proof of the Hausdorff–Young inequality applicable in Euclidean spaces. By employing the symmetric difference operator, we derive weighted estimates that link the smoothness properties of a function, measured by higher-order moduli of smoothness, to the decay rates of its Hartley transform. Additionally, the investigation extends to Hartley series on the torus, where conditions for convergence and decay of coefficients are formulated through the corresponding moduli of smoothness..enHartley Series on the TorusFourier Analysis PreliminariesHartley Multiplier for Iterated Symmetric DifferencesThesis