| Summary: | 碩士 === 國立中山大學 === 應用數學系研究所 === 89 === Consider the Sturm-Liouville system :
8 > > > > > < > > > > > :
− y00 + q(x)y = y
y(0) cos + y0(0) sin = 0
y(1) cos + y0(1) sin = 0
,
where q 2 L 1 (0, 1) and , 2 [0, ˇ).
Let 0 < x(n)1 < x(n)2 < ... < x(n)n − 1 < 1 be the nodal points of n-th eigenfunction
in (0,1). The inverse nodal problem involves the determination of the parameters
(q, , ) in the system by the knowledge of the nodal points . This problem was
first proposed and studied by McLaughlin. Hald-McLaughlin gave a reconstruc-
tion formula of q(x) when q 2 C 1 . In 1999, Law-Shen-Yang improved a result
of X. F. Yang to show that the same formula converges to q pointwisely for a.e.
x 2 (0, 1), when q 2 L 1 .
We found that there are some mistakes in the proof of the asymptotic formulas
for sn and l(n)j in Law-Shen-Yang’s paper. So, in this thesis, we correct the
mistakes and prove the reconstruction formula for q 2 L 1 again. Fortunately, the
mistakes do not affect this result.Furthermore, we show that this reconstruction formula converges to q in
L 1 (0, 1) . Our method is similar to that in the proof of pointwise convergence.
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