On Some Identities And Inequalities For F_a-frame Sequences In L~2(R₊)

Frame theory includes two aspects:the frame theory in abstract spaces,the structured frame theory in function spaces.In the past more than thirty years,we have witnessed the rapid development and wide application of the frame theory.In particular,the research of wavelet and Gabor frames on L2(R)has achieved fruitful results.However,the analysis on the half space L2(R+)has been less reported.This is because the field R of real numbers is a group under addition,while R+ is not.It results in the fact that L2(R+)admits no shift invariant system,and thus admits no wavelet or Gabor frame.On the other hand,L2(R+)is not closed under the Fourier transform when identified as a closed linear subspace of L2(R)composed of all functions in L2(R)that vanish on the left half real line.As a result,the Fourier transform tool is not suitable for the analysis on L2(R+).Therefore,the analysis on L2(R+)is nontrivial.In practice,the time variable cannot be negative,and the space L2(R+)models the causal signal space.Observe that R+ is a group under multiplication although it is not a group under addition.In recent years,a multiplication-based function valued inner product on L2(R+)called Fa-inner product and its associated Fa-frames have interested some scholars.This thesis addresses identities and inequalities of Fa-frame sequences in L2(R+).The study of some identities and inequalities of frames and frame sequences in general Hilbert spaces is related to the fast algorithm of phase free reconstruction(also known as phase retrieval).Therefore,it has been concerned by some scholars,and its research has made some progress.In this thesis,we investigate identities and inequalities of Fa-frame sequences in the setting of Fa-frame theory.By studying Fa-frame properties,we obtain some new identities and inequalities of Fa-frame sequences in L2(R+),define the optimal upper and lower bounds of inequalities and give their estimates.This paper belongs to theoretical research,and its application needs further exploration.