考慮古典的 Sturm-Liouville 系統： $$ left{
egin{array}{c}
-y'+q(x)y = la y
y(0) cos al + y'(0) sin al =0
y(1) cos e + y'(1) sin e =0
end{array} ight. ,
$$ 其中 $qin L^{1}(0,1)$，$al$、$ein [0,pi)$。

令$0<x_{1}^{(n)}<x_{2}^{(n)}<...<x_{n-1}^{(n)}<1$
為(0,1)區間第$n$個特徵函數的$n-1$個零點(節點)。
節點反演問題是討論如何利用已知的特徵函數的零點來決定原系統的參數
$(q,al,e)$。 這個問題首先由McLaughlin 提出研究，並給出了當$qin
C^1$時，$q(x)$的重構公式；1999年，Law-Shen-Yang改良了X. F. Yang
的結果並證明了同一公式，當$qin L^1$時，亦為點態收斂。

在這篇論文中，我們發現了Law-Shen-Yang在計算$s_{n}$和$l_{j}^{(n)}$的漸近公式時的錯誤，
我們重新計算並做修正
。同時，我們也給出Law-Shen-Yang重構公式的證明，藉以了解此錯誤並未影響結果。


在這篇論文的第二部分，我們將收斂性加強至$L^{1}$收斂。
我們所使用的證明方法與證明點態收斂的精神是一致的。

Consider the Sturm-Liouville system :
8 > > > > > < > > > > > :
&#8722; 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 &#8722; 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.

1 Introduction 2

2 On the Reconstruction Formula 7
2.1 Some asymptotic formulas . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Reconstruction formula . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Proof of asymptotic formulas for sn and x(n)i . . . . . . . . . . . . 11

3 L 1 Convergence of the Reconstruction Formula 18
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Proof of L 1 convergence . . . . . . . . . . . . . . . . . . . . . . . 20

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