  This paper investigates the mathematical nature of qualitative uncertainty
principle (QUP), which plays an important role in mathematics, physics and
engineering fields. Consider a 3-tuple (K, H1, H2) that K: H1 -> H2 is an
integral operator. Suppose a signal f in H1, {\Omega}1 and {\Omega}2 are
domains on which f, Kf define respectively. Does this signal f vanish if
|{\Sigma}(f)|<|{\Omega}1|and|{\Sigma}(Kf)|<|{\Omega}2|? The excesses and
deficiencies of integral kernel K({\omega}, t) are found to be greatly related
to this general formulation of QUP. The complete point theory of integral
kernel is so established to deal with the QUP. This theory addresses the
density and linear independence of integral kernels. Some algebraic and
geometric properties of complete points are presented. It is shown that the
satisfaction of QUP depends on the existence of some complete points. By
recognizing complete points of their corresponding integral kernels, the QUP
with Fourier transform, Wigner-Ville distribution, Gabor transform and wavelet
are studied. It is shown the QUP only holds for good behaved integral
operators. An investigation of full violation of QUP shows that L2 space is
large for high resolution harmonic analysis. And the invertible linear integral
transforms whose kernels are complete in L2 probably lead to the satisfaction
of QUP. It indicates the performance limitation of linear integral transforms
in harmonic analysis. Two possible ways bypassing uncertainty principle,
nonlinear method and sparse representation, are thus suggested. The notion of
operator family is developed and is applied to understand remarkable
performances of recent sparse representation.
