Further spectral properties of the weighted finite Fourier transform operator and approximation in weighted Sobolev spaces

In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWF). This set of special functions have been defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super‐exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.     

Isotherme à 25°C du Système Quaternaire Réciproque H3PO4?K2SO4?H2SO4?K3PO4?H2O Etude des Nappes du Liquidus

25°C Isotherm of the Reciprocal Quaternary System H₃PO₄? K₂SO₄? H₂SO₄? K₃PO₄? H₂O Study of the Liquidus Curves The liquidus of the reciprocal system H₃PO₄? K₂SO₄? H₂SO₄? K₃PO₄–H₂O has been investigated as a last step in the determination of the equilibrium diagram at 25°C and 1 atm pressure. Solubility has been detected by conductometric measurements in isopletic sections selected in order to complete the data obtained from the limiting ternary systems and the previously found boundary curves. Four isopletic sections have been completely established: K₂SO₄? KH₂PO₄, K₂SO₄? K₂HPO₄, KHSO₄? K₂HPO₄ and KHSO₄? KH₂PO₄. The two first ones are quasi ternary systems. Nine portions have been studied and the curves of equal water content plotted. The liquidus of KHSO₄, K₈(H₂PO₄)_1+x(HSO₄)_7?x and KH₂PO₄ · KHSO₄, show a rotationel shape which reveals a retrosolubility depending on the water content.