Wavelet decomposition techniques have grown over the last two decades into a powerful tool
in signal analysis. Similarly, spline functions have enjoyed a sustained high popularity in the
approximation of data.
In this thesis, we study the cardinal B-spline wavelet construction procedure based on quasiinterpolation
and local linear projection, before specialising to the cubic B-spline on a bounded
interval.
First, we present some fundamental results on cardinal B-splines, which are piecewise polynomials
with uniformly spaced breakpoints at the dyadic points Z/2r, for r ∈ Z. We start our wavelet
decomposition method with a quasi-interpolation operator Qm,r mapping, for every integer r,
real-valued functions on R into Sr
m where Sr
m is the space of cardinal splines of order m, such
that the polynomial reproduction property Qm,rp = p, p ∈ m−1, r ∈ Z is satisfied. We then
give the explicit construction of Qm,r.
We next introduce, in Chapter 3, a local linear projection operator sequence {Pm,r : r ∈ Z}, with
Pm,r : Sr+1
m → Sr
m , r ∈ Z, in terms of a Laurent polynomial m solution of minimally length
which satisfies a certain Bezout identity based on the refinement mask symbol Am, which we
give explicitly.
With such a linear projection operator sequence, we define, in Chapter 4, the error space sequence
Wr
m = {f − Pm,rf : f ∈ Sr+1
m }. We then show by solving a certain Bezout identity that there
exists a finitely supported function m ∈ S1
m such that, for every r ∈ Z, the integer shift
sequence { m(2 · −j)} spans the linear space Wr
m . According to our definition, we then call
m the mth order cardinal B-spline wavelet. The wavelet decomposition algorithm based on the
quasi-interpolation operator Qm,r, the local linear projection operator Pm,r, and the wavelet m,
is then based on finite sequences, and is shown to possess, for a given signal f, the essential
property of yielding relatively small wavelet coefficients in regions where the support interval of
m(2r · −j) overlaps with a Cm-smooth region of f.
Finally, in Chapter 5, we explicitly construct minimally supported cubic B-spline wavelets on a
bounded interval [0, n]. We also develop a corresponding explicit decomposition algorithm for a
signal f on a bounded interval.
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Throughout Chapters 2 to 5, numerical examples are provided to graphically illustrate the theoretical
results.