Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

Spectral discretization methods are well established methods for the computation of characteristic roots of time-delay systems. In this paper a method is presented for computing all characteristic roots in a given right half plane. In particular, a procedure for the automatic selection of the number of discretization points is described. This procedure is grounded in the connection between a spectral discretization and a rational approximation of exponential functions. First, a region that contains all desired characteristic roots is estimated. Second, the number of discretization points is selected in such a way that in this region the rational approximation of the exponential functions is accurate. Finally, the characteristic roots approximations, obtained from solving the discretized eigenvalue problem, are corrected up to the desired precision by a local method. The effectiveness and robustness of the procedure are illustrated with several examples and compared with DDE-BIFTOOL.

Computing all characteristic roots of delay differential equations in a given right half plane using a spectral method

In recent decades, increasingly intensive research attention has been given to dynamical systems containing delays and those affected by the after-effect phenomenon. Such research covers a wide range of human activities and the solutions of related engineering problems often require interdisciplinary cooperation. The knowledge of the spectrum of these so-called time-delay systems (TDSs) is very crucial for the analysis of their dynamical properties, especially stability, periodicity, and dumping effect. A great volume of mathematical methods and techniques to analyze the spectrum of the TDSs have been developed and further applied in the most recent times. Although a broad family of nonlinear, stochastic, sampled-data, time-variant or time-varying-delay systems has been considered, the study of the most fundamental continuous linear time-invariant (LTI) TDSs with fixed delays is still the dominant research direction with ever-increasing new results and novel applications. This paper is primarily aimed at a (systematic) literature overview of recent (mostly published between 2013 to 2017) advances regarding the spectrum analysis of the LTI-TDSs. Specifically, a total of 137 collected articles–which are most closely related to the research area–are eventually reviewed. There are two main objectives of this review paper: First, to provide the reader with a detailed literature survey on the selected recent results on the topic and Second, to suggest possible future research directions to be tackled by scientists and engineers in the field.

Spectrum Analysis of LTI Continuous-Time Systems With Constant Delays: A Literature Overview of Some Recent Results

A variety of block Krylov subspace methods have been successfully developed for linear systems and matrix equations. The application of block Krylov methods to compute matrix functions is, however, less established, despite the growing prevalence of matrix functions in scientific computing. Of particular importance is the evaluation of a matrix function on not just one but multiple vectors. The main contribution of this paper is a class of efficient block Krylov subspace methods tailored precisely to this task. With the full orthogonalization method (FOM) for linear systems forming the backbone of our theory, the resulting methods are referred to as B(FOM)2: block FOM for functions of matrices. Many other important results are obtained in the process of developing these new methods. Matrix-valued inner products are used to construct a general framework for block Krylov subspaces that encompasses already established results in the literature. Convergence bounds for B(FOM)2 are proven for Stieltjes functions applied to a class of matrices which are self-adjoint and positive definite with respect to the matrix-valued inner product. A detailed algorithm for B(FOM)2 with restarts is developed, whose efficiency is based on a recursive expression for the error, which is also used to update the solution. Numerical experiments demonstrate the power and versatility of this new class of methods for a variety of matrix-valued inner products, functions, and matrices.

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Block Krylov subspace methods for functions of matrices

In this paper, we analyze all possible situations of rank deficiency that cause breakdown in block conjugate gradient (BCG) solvers. A simple solution, breakdown-free block conjugate gradient (BFBCG), is designed to address the rank deficiency problem. The rationale of the BFBCG algorithm is to derive new forms of parameter matrices based on the potentially reduced search subspace to handle rank deficiency. Orthogonality properties and convergence of BFBCG in case of rank deficiency are justified accordingly with mathematical rigor. BFBCG yields faster convergence than restarting BCG when breakdown occurs. Numerical examples suffering from rank deficiency are provided to demonstrate the robustness of BFBCG.

Hua Dai

Papers

A new family of global methods for linear systems with multiple right-hand sides

Generalized global conjugate gradient squared algorithm

An inverse eigenvalue problem for Jacobi matrix

Stability analysis of time-delay systems using a contour integral method

An inverse problem for undamped gyroscopic systems