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Compact operators on Hilbert space
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
In Exercise 8.5.11 we saw how to “evaluate a function at an operator”, that is, how to make sense of g(A) when g is an analytic function on a disc Dr(0), with r > ║A║. The term functional calculus refers to the procedure of evaluating a function g (from within a specified class of functions) at an operator A (from within a specified class of operators), to form a new operator g(A). This notion also goes under the names function calculus, operational calculus, or function of an operator. The purpose of this section is to introduce the functional calculus for compact normal operators.
Representation Theory and Operational Calculus for SU(2) and SO(3)
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
In this chapter we presented a concrete overview of the representation theory of SO(3) and SU(2) together with harmonic analysis on these groups. Irreducible unitary representation matrices were given in terms of several parameterizations. It was shown how differential operators acting on functions of these groups can be defined in a concrete way, and how the matrix elements of IURs behave under these operators. The results of this chapter are used in Chapter 10 to define the Fourier transform and operational calculus for SE(3), and in Chapters 16 and 17 in the context of applications.
Application of the Laplace transform in problems of studying the dynamic properties of a material system and in engineering technologies
Published in Mangey Ram, J. Paulo Davim, Advanced Mathematical Techniques in Engineering Sciences, 2018
Lubov Mironova, Leonid Kondratenko
The first case. The function f(t) is the original. This function takes the unit value t > 0 and is zero for all t < 0. This function is often called either a single step function, or a single jump function or simply a single jump, respectively, and various notations are used. For example, a unit function is widely used of the mathematical apparatus of theories of automatic control, signal processing and other technical applications. In the operational calculus, this function is called the Heaviside unit function and has the form
Survey on algebraic numerical differentiation: historical developments, parametrization, examples, and applications
Published in International Journal of Systems Science, 2022
Amine Othmane, Lothar Kiltz, Joachim Rudolph
Consider the convergent Taylor series expansion of x at t = 0 given as For the estimation of the n-th order derivative of x, the truncated Taylor series expansion of x with is considered. It satisfies the differential equation which using operational calculus (Laplace transform or Mikusinski's operational calculus detailed in Doetsch (1974), Mikusinski (1983) respectively) reads where is the operational counterpart of . The sought derivatives are linearly identifiable parameters of (5) (compare with the results in Fliess & Sira-Ramírez, 2003a).
Teaching transfer functions without the Laplace transform
Published in European Journal of Engineering Education, 2022
Imad Abou-Hayt, Bettina Dahl, Camilla Østerberg Rump
In 1809, Laplace extended his z-transform to find solutions to linear differential equations, giving the world the Laplace transform as we know it today (Struik 2012). However, the transform was not given a true physical meaning until the English mathematician and electrical engineer Oliver Heaviside (1850–1925) invented operational calculus, which is a new method using the operator D notation that allowed him to transform difficult differential equations into simple algebraic equations.
On a linear input–output approach for the control of nonlinear flat systems
Published in International Journal of Control, 2018
H. Sira-Ramírez, E. W. Zurita-Bustamante, E. Hernández-Flores, M. A. Aguilar-Orduña
The transfer function representation, in the operational calculus notation, of the proposed controller readily follows from Equation (15) The proposed controller is thus a proper classical compensation network of the proportional-plus-iterated tracking error integrals type. This type of controller has also been derived, in a very natural way, from the module theoretic approach to linear control. They received the name of GPI controllers.