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本篇論文當中，吾人推導出gamma(Γ)2上同構的另一種證法，並將Caratheodory-Fejer問題化成傳統Nevanlinna-Pick問題來求解。進而將此方法運用來求二階導數 頻譜之Caratheodory-Fejer問題的解存在之必要條件，以及建立解的求法。

In this thesis, we first prove the automorphism of gamma(Γ)2. The Caratheodory-Fejer problem is transformed into the Nevanlinna-Pick problem for finding its solution. Afterward, this idea is applied to derive a necessary condition for the existence of solutions of the spectral Caratheodory-Fejer (SCF) problem with two derivatives as well as the technique to construct the solution of SCF problem.

中文摘要 i
Abstract ii
符號 v

1 前言 　　　　　1
1.1 Mu-設計理論．．．．．．．．．．．．．．．．．．．．． 1
1.2 研究動機．．．．．．．．．．．．．．．．．．．．．．． 3

2 數學預備知識 5
2.1 預備知識．．．．．．．．．．．．．．．．．．．．．． 5
2.2 Nevanlinna-Pick插值理論．．．．．．．．．．．．．． 7
2.3 Spectral Nevanlinna-Pick問題．．．．．．．．．．． 8
2.4 gamma(Γ)2上的一些性質．．．．．．．．．．．．．．．．12

3 主要結果 　　　　13
3.1 上的同構．．．．．．．．．．．．．．．．．．．．．．．13
3.2 Caratheordory-Fejer問題．．．．．．．．．．．．．． 20
3.3 對稱雙盤上之Caratheordory-Fejer問題．．．．．．．． 29
3.4 Spectral Caratheordory-Fejer問題．．．．．．．．．．32

4 結論與展望 38

參考文獻 　　　　　　　　　39

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[2] J. Agler and N. J. Young, A Schwarz lemma for the symmetrized bidisc, Bull. London Math. Soc. 33 (2001), 175-186.

[3] J. Agler and N. J. Young, The two-by-two spectral Nevanlinna-Pick Problem, Transactions of the American Mathematical Society 356(2) (2004), 573-538.

[4] M. Jarnicki and P. Pflug, On automorphisms of the symmetrized bidisc,Arch, Math. 83 (2004),264-266.

[5] H. N. Huang, S. A. M. Marcantognini and N. J. Young, Chain Rule for Higher Derivatives, The Mathematical Intelligiencer 28(2) (2006), 1-9.

[6] H. N. Huang, S. A. M. Marcantognini and N. J. Young, The spectral Caratheodory-Fejer Problem, Integral Equations and Operator Theory (2005).
(Published in On-line First Form)

[7] C. T. Lin, Schwarz Lemma On Symmetrized Bidise, Master Thesis, Deparment of Mathematics, Tunghai University, Taiwan, July (2001).

[8] T. D. Lin, Spectral Nevanlinna-Pick Interpolation On Symmetrized Bidise, Master Thesis, Deparment of Mathematics, Tunghai University, Taiwan, July (2001).

[9] C. M. Lin, Realization of Spectral Nevanlinna-Pick Interpolation on Symmetrized Bidise, Master Thesis, Deparment of Mathematics, Tunghai University, Taiwan, July (2003).