Halmos, Measure Theory, Springer-Verlag, New York, 1974. Zorich, Mathematical Analysis, 4th edn, Springer, Berlin Heidelberg, 2004. Conway, Functions of One Complex Variable I, 2nd edn, Springer, Berlin Heidelberg, 1978. Li, Privarov’s problem about the uniqueness of holomorphic functions, Pure Appl. Rudin, Functional Analysis, 2nd edn, McGraw-Hill, New York, 1973. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, 2008. cstc2019jcyj-msxmX0390).Ĭonflict of interest: The authors state no conflict of interest. GJJ180944 and GJJ190963) and Chongqing Natural Science Foundation Project (No. This work was supported in part by Science and Technology Project of Jiangxi Provincial Department of Education (Nos. Of course, analytical function f ( z ) can also be expressed by the function values on a curve, that is the following famous Cauchy integral formula (see ). In fact, in addition to the power series representation, the analytical function f ( z ) on C has many other important characteristics, and it even could be expressed by G derivative only according to the function values on a sequence of points with at least one aggregation point (see ), which is a profound supplement to the uniqueness theorem of analytic functions. Is there any other equivalent definition of the matrix functions corresponding to analytic functions on the complex field C? This is a natural problem because the analytic functions have many different equivalent definitions and representations. Fortunately, on the basis of annihilation polynomial of matrix and Lagrange-Sylvester theorem, we can give the polynomial matrix function equivalent to f ( A ), which is convenient to calculate the value of f ( A ) (see ). For example, even for matrix A in matrix ring C n × n, not to mention the element in general Banach algebra on C, it is difficult to calculate the value of f ( A ). If f ( z ) is an analytic function on complex field C, then it can be expanded by power series f ( z ) = ∑ k = 0 k = ∞ z k, and naturally, the f ( A ) is defined as f ( A ) ≔ ∑ k = 0 k = ∞ A k (see ).Īlthough the aforementioned definition of f ( A ) is intuitive, it has some inconveniences. Whether matrix rings or more general Banach algebras, there is always a good scheme to the definition of f ( A ) for analytical function f ( z ) on complex field C. In fact, not only matrix ring C n × n but also the general Banach algebra on complex field C has the problem about reasonable definition of function f ( A ). However, for more general function f ( z ), the meaning of the corresponding matrix function f ( A ) is not so obvious, and how to define it reasonably is an important problem. If f ( z ) is a polynomial function on C and A is a n-order matrix in C n × n, then the meaning of matrix function f : A ∈ C n × n → f ( A ) ∈ C n × n, abbreviated as f ( A ), is obvious, that is a polynomial whose variable is the matrix A (see ). Let C denote the complex field and C n × n denote the matrix ring on C.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |