Almansi Type Decompositions And Their Applications

Being the direct generalization of polynomial functions,polyharmonic functions have many applications in partial differential equations,numerical computation, wavelet analysis,several complex variables,elasticity theory and radar image theory. The classical Almansi decomposition theorem is the core theorem of the theory of polyharmonic functions and it is the generalization of the Fischer decomposition theorem,which is the foundation of spherical harmonic function theory.The Fischer decomposition is closely related to the Bargmann transform by the Fischer inner product and the Bargmann transform has many applications in the representation theory of Heisenberg groups([59,36]).The classical Almansi decomposition reduces polyharmonic functions theory to harmonic ones.The early stage results were given in the monograph Polyharmonic Functions(1983)[3].In this thesis,we will systematically study the Almansi type decomposition theorems,constructing the finite type and infinite type Almansi decompositions and setting up corresponding Almansi type decompositions in Clifford analysis,Dunkl-Clifford analysis and Umbral analysis.For the finite type Almansi decompositions,we will study hyperbolic operators, hyperbolic Helmholtz operators,Dunkl-Laplace operators,and Umbral-Helmholtz operators.The classical Almansi type decomposition theorems deal with the Laplace operator and its iterated operators.Our approaches are no longer restricted to classical complex-valued functions,but instead we study Clifford-valued functions. It should be noted that classical Clifford analysis is mainly concerned with the Clifford algebra Cl_0,n, while our theory is developed within the Clifford algebra Cl_p,q·(see Ch 2 and 4)As an application of finite Almansi type decompositions,we fully solve the Riquier problem of hyperbolic operators in the unit ball.Moreover using the Almansi type decomposition of Dunkl operators,we derive the growth rate estimate of poly-Dunkl harmonic functions,hence proving the Liouville theorem for poly-Dunkl harmonic functions.(see Ch 4 and 5) The infinite Almansi type decompositions are in series form.We extend the decompositions from polyharmonic functions to analytic functions.We derive the series form representation of analytic functions in star-like domains.The components of the sum are wave-harmonic functions,which are the generalization of spherical harmonics of square integrable functions in the unit ball.(see Ch 4)The infinite Almansi type decompositions in series form are given by normalized systems of hyperbolic operators.This needs an in-depth study of normalized system. We derive the normalized systems of wave and Dunkl-Laplace operators.Classically, operators commute in the theory of normalized system.The operators we consider in Clifford analysis are non-commuting.Thus the study of normalized systems has to be extended to the non-commuting case.As applications,we give non-trivial formal solutions of Helmholtz equations and study the Riquier problem of the wave operators.(see Ch 3)Using Almansi decomposition theorems,we intend to study polymonogenic function theory in Clifford analysis,for example the Berezin transform theory,which has important applications in physics.The classical Berezin transform theory focuses mainly on holomorphic or harmonic functions on the unit ball.Our primary result gives asymptotic properties of Berezin transform of monogenic functions.Our approach uses the M(o｜¨)bius transforms to handle complicated Bergman kernel functions in Clifford analysis.(see Ch 6)Almansi type decompositions have broad application foreground,due to their important roles in Clifford analysis,Dunkl analysis and Umbral analysis.