Analysis of Divergence : Control and Management of Divergent Processes

The 7th International Workshop in Analysis and its Applications (IWAA) was held at the University of Maine, June 1-6, 1997 and featured approxi­ mately 60 mathematicians. The principal theme of the workshop shares the title of this volume and the latter is a direct outgrowth of the workshop. IWAA was founded in 1984 by Professor Caslav V. Stanojevic. The first meeting was held in the resort complex Kupuri, Yugoslavia, June 1-10, 1986, with two pilot meetings preceding. The Organization Committee to­ gether with the Advisory Committee (R. P. Boas, R. R. Goldberg, J. P. Kahne) set forward the format and content of future meetings. A certain number of papers were presented that later appeared individually in such journals as the Proceedings of the AMS, Bulletin of the AMS, Mathematis­ chen Annalen, and the Journal of Mathematical Analysis and its Applica­ tions. The second meeting took place June 1-10, 1987, at the same location. At the plenary session of this meeting it was decided that future meetings should have a principal theme. The theme for the third meeting (June 1- 10, 1989, Kupuri) was Karamata's Regular Variation. The principal theme for the fourth meeting (June 1-10, 1990, Kupuri) was Inner Product and Convexity Structures in Analysis, Mathematical Physics, and Economics. The fifth meeting was to have had the theme, Analysis and Foundations, organized in cooperation with Professor A. Blass (June 1-10, 1991, Kupuri).

Overview.- I Convergence and Summability.- 1 Tauberian theorems for generalized Abelian summability methods.- 1.1 Introduction.- 1.2 A General Summability Method.- 1.3 Generalized Abel’s Summability Methods.- 2 Series summability of complete biorthogonal sequences.- 2.1 Introduction.- 2.2 Preliminaries.- 2.2.1 Biorthogonal Sequences.- 2.2.2 Sequence Spaces.- 2.2.3 The Beta-Phi Topology on a Sequence Space.- 2.2.4 Biorthogonal Sequences and Sequence Spaces.- 2.2.5 Multiplier Algebras, Sums, and Sum Spaces.- 2.2.6 Convergence Properties of Sequence Spaces.- 2.3 Sums and Sum Spaces.- 2.3.1 Sums.- 2.3.2 Sum Spaces.- 2.4 Inclusion Theorems.- 3 Growth of Cesàro means of double Vilenkin-Fourier series of unbounded type.- 3.1 Introduction.- 3.2 Fundamental concepts and notation.- 3.3 The Vilenkin-Fejér kernel.- 3.4 The main results.- 4 A substitute for summability in wavelet expansions.- 4.1 Introduction.- 4.2 Background.- 4.3 Summability for Wavelets With Compact Support.- 4.4 The Properties Of The Summability Function.- 4.4.1 The rate of decrease of the filter coefficients.- 4.4.2 The calculation of the positive estimation $$f^r_m(t)$$.- 5 Expansions in series of Legendre functions.- 5.1 Introduction.- 5.2 Preliminaries and known results.- 5.2.1 Christoffel Summation Formula.- 5.2.2 Stieltjes’s Inequality.- 5.2.3 Riemann-Lebesgue-type Theorem.- 5.2.4 Singular Integrals.- 5.3 Neumann’s Integral and consequences.- 5.4 Hunter’s Identities.- 6 Endpoint convergence of Legendre series.- 6.1 Statement of results.- 6.2 Asymptotic estimates.- 6.3 Convergence at the endpoints.- 6.3.1 Convergence at x = 1.- 6.3.2 Convergence at x = -1.- 7 Inversion of the horocycle transform on real hyperbolic spaces via a wavelet-like transform.- 7.1 Introduction.- 7.2 Preliminaries.- 7.2.1 Algebraic and geometric notions.- 7.2.2 The horocycle transform and its dual.- 7.2.3 Approximate identities on ?.- 7.3 Inversion of the Horocycle Transform.- 8 Fourier-Bessel expansions with general boundary conditions.- 8.1 Introduction.- 8.2 Statement of Results.- 8.3 Proofs.- 8.4 Identifying the limit.- 8.4.1 An Abelian lemma.- II Singular Integrals and Multipliers.- 9 Convolution Calderón-Zygmund singular integral operators with rough kernels.- 9.1 Introduction.- 9.2 L2 boundedness.- 9.3 Lp boundedness, 1 < p < ?.- 9.4 The L1 theory.- 9.5 Another H1 condition in dimension 2.- 9.6 Maximal functions and maximal singular integrals.- 10 Haar multipliers, paraproducts, and weighted inequalities.- 10.1 Introduction.- 10.2 Preliminaries.- 10.2.1 Dyadic intervals and Haar basis.- 10.2.2 Weights.- 10.3 Weight lemma and decaying stopping times.- 10.4 Lp Lemmas for decaying stopping times.- 10.4.1 Lp Plancherel Lemma.- 10.4.2 Lp version of Cotlar’s Lemma.- 10.5 Boundedness of $$ T_\omega ^t $$.- 10.5.1 Boundedness of T?.- 10.5.2 Some corollaries.- 10.6 Haar multipliers and weighted inequalities.- 11 Multipliers and square functions for Hp spaces over Vilenkin groups.- 11.1 Introduction.- 11.2 Historical Comments.- 11.3 Multipliers for Hp (0 for oscillatory Fourier transforms.- 14.1 Introduction.- 14.2 Lp(L?)-estimates.- 14.3 L2(L2)-estimates.- 15 Optimal spaces for the S’-convolution with the Marcel Riesz kernels and the N-dimensional Hilbert kernel.- 15.1 Introduction.- 15.2 Definitions and notation.- 15.2.1 Function and distribution spaces.- 15.2.2 The S’-convolution.- 15.2.3 Partition of unity on $$\mathbb{R}^n$$.- 15.3 Optimal space for the S’-convolution with the vector Riesz kernel.- 15.4 Optimal space for the S’-convolution with $$p\nu \frac{1}{x_1} \otimes \cdots \otimes p\nu \frac{1}{x_n}$$.- 15.5 Necessary condition for the S’-convolvability with a single Riesz kernel.- III Integral Operators and Functional Analysis.- 16 Asymptotic expansions and linear wavelet packets on certain hypergroups.- 16.1 Introduction.- 16.2 The Chébli-Trimèche hypergroups $$(\mathbb{R}_+,*_A)$$.- 16.3 The dual of the hypergroups $$(\mathbb{R}_+,*_A)$$.- 16.4 Asymptotic expansions and integral representations of Mehler and Schläfli type.- 16.4.1 The asymptotic expansions.- 16.4.2 Integral representations of Mehler and Schläfli type.- 16.5 Harmonic analysis and maximal ideal spaces of some algebras.- 16.5.1 Harmonic analysis.- 16.5.2 The maximal ideal spaces of the algebras and $$ L^1 (m_A )\;\text{and}\;M_b (\text{R}_ + ) $$.- 16.6 Continuous linear wavelet transform and its discretization.- 16.6.1 Linear wavelets on $$(\mathbb{R}_+,*_A)$$.- 16.6.2 Linear wavelet packet on $$(\mathbb{R}_+,*_A)$$.- 16.6.3 Scale discrete L-scaling function on $$(\mathbb{R}_+,*_A)$$.- 17 Hardy-type inequalities for a new class of integral operators.- 17.1 Introduction.- 17.2 Starshaped regions.- 17.3 Prom regions to kernels.- 18 Regularly bounded functions and Hardy’s inequality.- 18.1 Introduction.- 18.2 Definition and Uniform Boundedness.- 18.3 The global bounds.- 18.4 The representation theorem.- 18.5 The multiplicative class.- 18.6 Abelian Theorems.- 18.7 Hardy’s Inequality.- 19 Extremal problems in generalized Sobolev classes.- 19.1 Introduction.- 19.1.1 General problem of sharp inequalities for intermediate derivatives.- 19.1.2 Functional classes $$W^rH^\omega(\mathbb{I})$$.- 19.1.3 The Kolmogorov problem in $$W^rH^\omega(\mathbb{I})$$.- 19.2 Maximization of integral functional over H?[a, b].- 19.2.1 Simple kernels ?(·) and their rearrangements $$ \Re (\Psi \text{;} \cdot ) $$.- 19.2.2 The Korneichuk lemma.- 19.2.3 Extremal functions of functionals over H?[a, b].- 19.2.4 Structural properties of extremal functions $$ x_{\omega ,\psi } $$.- 19.3 Kolmogorov problem for intermediate derivatives.- 19.3.1 Differentiation formulae for f(m)(0), 0 ? m < r.- 19.3.2 Differentiation formula for f(r)(0).- 19.3.3 Sufficient conditions of extremality.- 19.3.4 Extremaiity conditions in the form of an operator equation.- 19.3.5 Sharp additive inequalities for intermediate derivatives.- 19.3.6 Kolmogorov problem in Hölder classes.- 19.4 Kolmogorov problem in $$W^1H^\omega(\mathbb{R}_+)$$ and $$W^1H^\omega(\mathbb{R}_+)$$.- 19.4.1 Preliminary remarks.- 19.4.2 Maximization of the norm $$\Vert f \Vert L_{\infty}(\mathbb{R}_+)$$.- 19.4.3 Extremal functions in Hölder classes $${{\rm H}^\omega }[a,b]$$.- 19.4.4 Maximization of the norm $$\Vert f^\prime \Vert L_{\infty}(\mathbb{R}_+)$$.- 19.4.5 Maximization of the norm $$\Vert f \Vert L_{\infty}(\mathbb{R}_+)$$.- 19.4.6 Maximization of the norm $$\Vert f^\prime \Vert L_{\infty}(\mathbb{R}_+)$$.- 20 On angularly perturbed Laplace equations in the unit ball and their distributional boundary values.- 20.1 Introduction.- 20.2 Notation and Preliminaries.- 20.3 Bounded Solutions on Bn+2.- 20.4 Distributional Boundary Values.- 20.5 Generalities.- 21 Nonresonant semilinear equations and applications to boundary value problems.- 21.1 Introduction.- 21.2 Semi-abstract nonresonance problems.- 21.3 Strong solvability of elliptic BVP’s.- 21.4 Time periodic solutions of BVP’s for nonlinear parabolic and hyperbolic equations.- 21.4.1 Nonlinear parabolic equations.- 21.4.2 Applications to the heat equation.- 21.4.3 Nonlinear hyperbolic equtions.- 21.4.4 Applications to the telegraph equation.- 21.4.5 Application to the beam equation with damping.- 22 A topological and functional analytic approach to statistical convergence.- 22.1 Introduction.- 22.2 The support set of a measure.- 22.3 Invariants of statistical convergence.- 22.4 Summability theorems.- IV Asymptotics and Applications.- 23 Optimal control of divergent control systems.- 23.1 Introduction and History.- 23.2 Basic models and hypotheses.- 23.3 Existence of optimal solutions.- 23.3.1 Existence of overtaking optimal solutions without discounting.- 23.3.2 Existence of overtaking optimal solutions with discounting.- 23.4 The associated uncoupled optimal control problems.- 23.4.1 The undiscounted case.- 23.4.2 The discounted case.- 23.5 Optimal solutions of the explicitly state constrained optimal control problem.- 23.5.1 The undiscounted case.- 23.6 Conclusions.- 24 Surfaces minimizing integrals of divergent integrands.- 24.1 Introduction.- 24.2 Surfaces and Integrands.- 24.3 Overtaking Minimizers.- 24.4 A Radially Symmetric Example.- 24.5 Hypotheses for Regularity.- 24.6 Barriers.- 24.7 A Result in Differential Geometry.- 24.8 Bounding the Curvature.- 25 Sparse exponential sums with low sidelobes.- 25.1 Introduction.- 25.2 Generalized Rudin-Shapiro Polynomials.- 25.3 Exponential Sums with Low Sidelobes.- 26 Spline type summability for multivariate sampling.- 26.1 Introduction.- 26.1.1 Sampling theory.- 26.1.2 Splines and sampling theory.- 26.1.3 Contents, notation, and acknowledgements.- 26.2 Regular sampling of multivariate functions and their recovery via splines.- 26.2.1 Band limited functions and polyharmonic splines.- 26.2.2 The spaces $$L^{2,k}(\mathbb{R}^n)$$ and $$L^{2,k}(\mathbb{Z}^n)$$ and the variational properties of polyharmonic splines.- 26.2.3 The Paley-Wiener space $$PW^k_\pi$$.- 26.2.4 Convergence of m-harmonic splines as m ? ?.- 26.3 Generalizations, related methods, and computational issues.- 26.3.1 Generalizations.- 26.3.2 Multivariate analogues of the Paley-Wiener Theorem and the sampling theorem.- 26.3.3 Box splines.- 26.3.4 Computing polyharmonic splines.- 27 B-Splines and orthonormal sets in Paley-Wiener space.- 27.1 Introduction.- 27.2 Preliminaries.- 27.3 Sampling and Orthonormal Functions.- 27.4 B-splines and Orthonormal Sets in the Paley-Wiener Space.- 28 Norms of powers and a central limit theorem.- 28.1 Introduction.- 28.2 The Five Parameters.- 28.3 Boundedness.- 28.3.1 Power Series.- 28.3.2 Trigonometric Series.- 28.4 Asymptotic Behavior.- 28.5 Asymptotic Series.- 28.6 Changing the Question.- 28.7 Behavior of Scaled ?(n) for Large n.- 28.8 Another Kind of Central Limit Theorems.- 29 Quasiasymptotics at zero and nonlinear problems in a framework of Colombeau generalized functions.- 29.1 Introduction.- 29.2 Algebra of generalized functions.- 29.3 G-quasiasymptotics at zero.- 29.4 Application of G quasiasymptotics to generalized solutions.- 29.4.1 System of nonlinear Volterra integral equations with non-Lipschitz nonlinearity.- 29.4.2 Semilinear hyperbolic system.- 29.4.3 Nonlinear wave equation.- 29.4.4 Euler-Lagrange equation.- 29.4.5 Goursat problem.