Vectorial Ball Prolate Spheroidal Wave Functions With The Divergence Free Constraint

The prolate spheroidal wave functions(PSWFs)are a family of orthogonal ban-dlimited functions,originated from the investigation of time-frequency concentration problem in the 1960s,and by a remarkable coincidence,they are also the eigenfunc-tions of a Sturm-Liouville problem,arisen from the context of separation of variables for solving the Helmholtz equation in spheroida.l coordinates.The heart of this disser-tation is to solve the divergence free constrained maximum concentration problem in three dimensions,i.e.,answer the question:to what extent can the total energy of a band-limited divergence free vectorial function can be optimally concentrated within the unit ball?For this purpose we define one family of vectorial prolate spheroidal wave functions(PSWFs)of real order ?>-1 on the unit ball in three dimension R3,which satisfy the divergence free constraint,and derive their analytic and asymptotic formulas.The first aim of this thesis is to demonstrate that the second order Sturm-Liouville operator only have one kind of divergence free vectorial eigen-functions which are divergence free vectorial ball PSWFs.Simultaneously,it turns out that they solve another second order Sturm-Liouville eigen equation defined through the curl operator(?)×instead of the gradient operator(?).Then,we show that these divergence free vec-torial ball PSWFs are exactly the band-limited vectorial eigen-functions which satisfy the finite Fourier integral equation.Moreover,we derive that the divergence freevec-torial ball PSWFs possess a simple and close relation with the scalar ball PSWFs such that they acquire the same merits of being the eigenfunctions of a differential opera-tor and an integral operator simultaneously,as well as forming a complete orthogonal system.More importantly,we verify that any optimally concentrated divergence free vec-torial functions when represented in series in vector spherical harmonics,shall be also concentrated in one of the three vectorial spherical harmonics modes.Finally,we shall illustrate an optimal Bouwkamp Spectral algorithm to evaluate the divergence free vec-torial ball PSWFS and their associated eigenvalues and describe the connection with existing works.