<p>Fractional calculus has become a strong mathematical tool for modelling dynamic systems with memory and hereditary effects. Fourier series, on the other hand, is still a key tool for studying periodic signals. Classical Fourier analysis is intrinsically local and fails to account for long-range correlations or history-dependent phenomena. In this paper, we present and thoroughly analyse a Memory-Weighted Fractional Derivative featuring a power-law kernel, specifically tailored for the Fourier domain. The suggested operator combines fractional differentiation with memory weighting, which lets Fourier coefficients and harmonics change over time because of long-term memory. We demonstrate its essential theoretical attributes, encompassing linearity, boundedness, spectral scaling characteristics, and mapping relations across Sobolev spaces, along with evidence of well-posedness and stability in the presence of perturbations. A spectral technique utilizing the Fast Fourier transform is formulated, attaining computational complexity of O(NlogN) with spectral precision for smooth periodic functions. Error analysis shows that for analytic inputs, the convergence is exponential, and for signals with limited smoothness, the convergence is algebraic. This is in line with Sobolev regularity. Numerical experiments validate the theoretical predictions and demonstrate amplitude scaling, phase shifting, and memory-induced modulation of harmonic modes. Secure communications, biological signal processing, anomalous diffusion, and oscillatory systems with hereditary dynamics are some of the possible uses that are emphasized. This paper establishes a robust theoretical framework and an effective computational tool for modelling intricate periodic events exhibiting long-range memory effects by integrating fractional calculus and harmonic analysis inside a memory-weighted Fourier framework.</p>