頻譜Nevanlinna-Pick(SNP)插值??是為?提供強健控制器設計之「μ合成??」確?的?學??而發展，其目的在求從單位圓盤到單位頻球的解析(矩陣)函?，並滿足特定的函?值插值條件。?用矩陣的特徵多項式的性質，可將這?問題轉會成在對稱雙盤上的NP插值問題?求解。所謂的頻譜Caratheodory-Fejer(SCF)插值問題?是在SNP的插值條件中加入導?的要求而得。因此在求解SCF問題，必須解決在對稱雙盤上的CF插值問題才?。本研究計畫之目的?是探討這?問題的解之存在性與解的算法，著重在如何將已有的?點對稱雙盤上的NP插值問題的解法擴大為可求解對稱雙盤上的CF插值問題。延續以往計畫研究成果為基礎，結合SNP插值??，探討2x2矩陣SCF問題的解所形成之空間稱為插值體，刻畫出插值體條件以推導?確?的判斷方法。第一?以探討2階導?問題對應的插值體與解，第二?研究的將推廣到n階導?問題對應的插值體與解。透過本計畫的研究會對對稱雙盤的幾何性質有?深入的明瞭，並完全解決SCF插值問題。 The aim of this two years project is to find the Caratheodory-Fejer matricial interpolating function which is analytic from the open unit disc into the open spectral unit ball such that satisfies certain interpolation conditions on its values and its derivatives. It is obvious that this problem is called the spectral Caratheodory-Fejer (SCF) problem. Our approach is to transfer the SCF problem into a classical Caratheodory-Fejer (CF) problem such that a more efficient condition based on the given interpolation data is obtained for the existence of the SCF solution. Based on the result of our previous project together with the known SNP theory, direct solvability condition of 2x2 SCF problems can be characterized as an interpolation body. The properties of the interpolation body corresponding to the SCF problem up to second derivatives is analyzed in the first year of this two-year project. Furthermore, the construction of interpolating function is also considered. Extension to higher derivatives problem will be studies in the consecutive year.
