Topics in Classical and Modern Analysis: In Memory of Yingkang Hu (Applied and Numerical Harmonic Analysis)

Different aspects of harmonic analysis, complex analysis, sampling theory, approximation theory and related topics are covered in this volume. The topics included are Fourier analysis, Padè approximation, dynamical systems and difference operators, splines, Christoffel functions, best approximation, discrepancy theory and Jackson-type theorems of approximation. The articles of this collection were originated from the International Conference in Approximation Theory, held in Savannah, GA in 2017, and organized by the editors of this volume. ANHA Series Preface Preface Contents Part I Yingkang Remembering Professor Yingkang Hu Remembrances On Some Properties of Moduli of Smoothness with Jacobi Weights 1 Introduction 2 The Main Result 3 The Polynomials of Best Approximation 4 Further Properties of the Moduli References Part II Approximation Theory, Harmonic and Complex Analysis, Splines and Classical Fourier Theory Special Difference Operators and the Constants in the Classical Jackson-Type Theorems 1 Introduction 2 Whitney's Constants 3 Bohr–Favard Inequality and Best Constants 4 Bohr–Favard Difference Inequality 5 Integral Approximation of the Characteristic Function 6 Neumann Series 7 Operators W2k 8 Jackson–Stechkin Inequality for W2k 9 Bernstein–Nikolskii–Stechkin Inequality 10 Approximation by Algebraic Polynomials 11 Open Problems References Comparison Theorems for Completely and Multiply Monotone Functions and Their Applications 1 Introduction 2 Definition of Considered Functional Classes 3 Main Results 4 Kolmogorov's Problem 4.1 Statement of the Problem 4.2 Auxiliary Results 4.3 Connections Between Stated Problems and Considered Classes 4.4 Some Properties of X-Perfect Splines 4.5 Solution to Kolmogorov's Problem 5 Some Other Applications 5.1 On the Smoothest Hermite–Birkhoff Interpolation 5.2 On Sharp Estimates for Intermediate Moments 5.3 On Extremal Distribution Functions References Concerning Exponential Bases on Multi-Rectangles of Rd 1 Introduction 2 Preliminaries 2.1 Bases and Frames 2.2 Exponential Bases on L2(Q0) 2.3 Stability of Riesz Bases 2.4 Scaling 2.5 Three Useful Lemmas 3 Proof of Theorem 1.3 4 Corollaries and Examples 4.1 A Stability Theorem 4.2 Two Cubes in Rd 4.3 Spectral Domains in Rd 4.4 Extracting Riesz Bases from Frames 5 Estimating the Frame Constants Appendix: Proof of Theorem 1.1 Proof of Theorem 1.1 References Hankel Transforms of General Monotone Functions 1 Introduction 2 Bessel Functions 3 Abel–Olivier Test for GM Functions and Sequences 4 Proofs References Univalence of a Certain Quartic Function 1 In Memoriam 2 Introduction 3 Proof 3.1 Decomposition 3.2 Injectivity 3.3 Main Result 3.4 Remark References Finding, Stabilizing, and Verifying Cycles of Nonlinear Dynamical Systems 1 Introduction 2 Closed Loop Systems 2.1 Characteristic Polynomials 2.2 Geometric Stability Criteria 2.3 Optimization Problem 2.3.1 Case γ=0 2.3.2 Case γ=0 2.4 Coefficients 2.4.1 Construction of the Polynomials q( z ) 2.4.2 Construction of the Polynomials p(z ) 3 Numerical Simulations 3.1 Hénon Map, n=1, ..., 1200 3.2 Elhadj–Sprott Map 3.3 Ikeda Map 3.4 Lozi Map 3.5 Holmes Cubic Map 3.6 Numerical Difficulties 4 Conclusion References Finding Orbits of Functions Using Suffridge Polynomials 1 Introduction and Statement of Main Results 2 Example 3 Conclusions and Further Directions of Research References The Sharp Remez-Type Inequality for Even Trigonometric Polynomials on the Period 1 Introduction 2 New Results 3 Lemmas 4 Proof of Theorem 2.1 References The Lebesgue Constants of Fourier Partial Sums 1 Introduction 2 Lebesgue Constants Generated by the Homothety of a Fixed Set 2.1 Cubic Partial Sums 2.2 Spherical Partial Sums 2.3 Hyperbolic Partial Sums 3 Polyhedral Partial Sums 3.1 General Estimates 3.2 Intermediate Growth 3.3 Asymptotics 4 Partial Increasing of Lebesgue Constants References Liouville–Weyl Derivatives of Double Trigonometric Series 1 Introduction 1.1 The One-Dimensional Case 1.2 The Two-Dimensional Case 2 Auxiliary Results 3 Proof of Theorem 1 References Inequalities in Approximation Theory Involving Fractional Smoothness in Lp, 0