OICC Press Mathematical Analysis and its Contemporary Applications 6 4 2024 12 24 Uncertainty principles and extremal functions for generalized Hartley-Gabor transform 10.30495/maca.2025.2046685.1117 EN Ahmed Chana Laboratory of Fundamental and Applied Mathematics, Department of Mathematics and Informatics, Faculty of Sciences Ain Chock, University of Hassan II, B.P 5366 Maarif, Casablanca, Morocco Abdellatif Akhlidj Laboratory of Fundamental and Applied Mathematics, Department of Mathematics and Informatics, Faculty of Sciences Ain Chock, University of Hassan II, B.P 5366 Maarif, Casablanca, Morocco Souhir Arhilas Laboratory of Fundamental and Applied Mathematics, Department of Mathematics and Informatics, Faculty of Sciences Ain Chock, University of Hassan II, B.P 5366 Maarif, Casablanca, Morocco Journal Article 2024 12 24 The main crux of this paper is to introduce a new integral transform called the generalized Hartley-Gabor transform which generalizes the classical Gabor Fourier transform and to give some new results related to this transform as Plancherel's, Parseval's, inversion and Calderon's reproducing formulas. Next, we analyse the concentration of this transform on sets of finite measures and we give the uncertainty principle for orthonormal sequences. Last, using the best approximations and the theory of reproducing kernels, we study the extremal functions related to this transform and we give an integral representation, band estimates of these functions on weighted Sobolev spaces.