臺灣博碩士論文加值系統

對稱雙盤定義為根在單位圓盤內的一元二次方程式之兩個係數所形成的集合，本文發
展matlab 圖形介面來展示對稱雙盤以及定義在其上之頻譜插值函數的圖形。由於對稱雙盤
是C^2上的集合，分別在固定其中一個實部或虛部的情況下，展現在三度空間的投影圖形，並透
過圖形介面的滑尺功能，以觀察當固定值改變時，圖形間的變化情形。同時，文中也展示對稱
雙盤對應圓盤r 半徑改變時，對於圖形的影響。此外，本文中亦介紹定義在對稱雙盤上頻譜
插值函數的兩種計算方式以及它們的圖形；同時，應用此插值函數計算頻譜Nevanlinna-Pick
插值的解。

關鍵字：單位圓盤、對稱雙盤、一元二次方程式、matlab、圖形介面、頻譜插值問題

The symmetrrized bidisc is defined as the set of two coefficients of a quadratic equation
with its roots located inside the unit disc. In this thesis, a matlab-based GUI is
developed to the graphs of the symmetrized bidisc and associated spectral interpolating
functions. Since the symmetrized bidisc belongs to C^2, its 3D projection is plotted as the
real or imaginary part of one variable is fixed. By the way, the graph of the symmetrized
bidisc is also shown when the radius of the root's location changes. Furthermorre, two
kinds of approaches are used to construct the spectral interoplating function defined on
the symmetrized bidisc are introduced and their graphs are depicted as well. Once the
interpolating function is computed, we demo how to construct the interpolation function
to solve the two-by-two spectral Nevanlinna-Pick problem.

Keywords: unit disc, symmetrized bidisc, quadratic equation, matlab, GUI, spectral Nevanlinna-
Pick interpolation problemn

圖目錄
摘要
Abstract
符號說明
第一章前言
第二章數學預備知識
2.1 預備知識
2.2 Nevanlinna-Pick插值問題
2.3 Spectral Nevanlinna-Pick插值問題
2.4 Γ2上的一些性質
第三章對稱雙盤之圖形展示
3.1 設計操作介面
3.2 程式編輯
3.3 Γ2圖形展示
3.4 延伸討論
第四章頻譜插值函數及其圖形
4.1 第一種頻譜NP 插值問題的解
4.2 第二種頻譜NP 插值問題的解
第五章結論與展望
參考文獻

[1] J. Agler and N.J. Young, The two-point spectral Nevalinna-Pick problem, Integral Equations Operator
Theory 37 (2000) 375-385.
[2] J. Alger 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 Nevenlinna-Pick problem, Transactions of the American
Mathematical Society 356(2) (2003) 573-585.
[4] J. Alger, F.B. Yeh, and N.J. Young, Realization of functions into the symmetrised bidisc, Operator
Theory : Advances and Applications, Birkhauser 143 (2003) 1-37.
[5] T. Constantinescu. Schur parameters, Factorization and Dilation Problems. Birkhauser Verlag, Basel,1996.
[6] S. Dineen, The Schwarz Lemma, Oxford University Press, 1989.
[7] H. N. Huang, S. A. M. Marcantognini and N. J. Young, The spectral Caratheodory-Fejer Problem,
Integral Equations and Operator Theory, 56, Number 2, pp. 229-256(28), 2005
[8] C.M. Lin, Realization of Spectral Nevanlinna-Pick Interpolation on Symmetrized Bidisc, Master Thesis,
Department of Mathematics, Tunghai University, Taiwan, July, 2003.
[9] C.T. Lin, Schwarz Lemma On Symmetrized Bidisc, Master Thesis, Department of Mathematics, Tunghai
University, Taiwan, July, 2001.
[10] T.D. Lin, Spectral Nevalinna-Pick Interpolation On Symmetrized Bidisc, Master Thesis, Department of
Mathematics, Tunghai University, Taiwan, July 2001.
[11] K. Zhou, J.C. Doyle, and K.Glover, Robust and Optimal Control, Prentice-Hall, New York, 1996.
[12] http://blinkdagger.com/matlab/matlab-gui-tutorial-slider/
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