William O. Bray Caslav V. Stanojevic

Editors

Analysis of Divergence Control and Management of Divergent Processes

Birkhäuser Boston • Basel • Berlin

Contents

Preface xv

Contributors xvii

Overview 1 W. 0. Bray, C. V. Stanojevic 1

I Convergence and Summability 11

1 Tauberian theorems for generalized Abelian summability methods C. V. Stanojevic, I. Canak, V. B. Stanojevic 13 1.1 Introduction 13 1.2 A General Summability Method 16 1.3 Generalized Abel's Summability Methods 20

2 Series summability of complete biorthogonal sequences W. H. Ruckle 27 2.1 Introduction 27 2.2 Preliminaries 28

2.2.1 Biorthogonal Sequences 28 2.2.2 Sequence Spaces 29 2.2.3 The Beta-Phi Topology on a Sequence Space . . . . 30 2.2.4 Biorthogonal Sequences and Sequence Spaces . . . . 31 2.2.5 Multiplier Algebras, Sums, and Sum Spaces 31 2.2.6 Convergence Properties of Sequence Spaces 32

2.3 Sums and Sum Spaces 32 2.3.1 Sums 32 2.3.2 Sum Spaces 33

2.4 Inclusion Theorems 37

3 Growth of Cesäro means of double Vilenkin-Fourier series of unbounded type W.R. Wade 41 3.1 Introduction 41 3.2 Fundamental concepts and notation 42 3.3 The Vilenkin-Fejer kernel 43

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3.4 The main results 45

4 A Substitute for summability in wavelet expansions G. G. Walter, X. Shen 51 4.1 Introduction 51 4.2 Background 52 4.3 Summability for Wavelets With Compact Support 53 4.4 The Properties Of The Summability Function 56

4.4.1 The rate of decrease of the filter coefhcients 56 4.4.2 The calculation of the positive estimation f^(t) . . . 59

5 Expansions in series of Legendre functions E. R. Love, M. N. Hunter 65 5.1 Introduction 65 5.2 Preliminaries and known results 66

5.2.1 Christoffel Summation Formula 66 5.2.2 Stieltjes's Inequality 66 5.2.3 Riemann-Lebesgue-type Theorem 67 5.2.4 Singular Integrals 67

5.3 Neumann's Integral and consequences 68 5.4 Hunter's Identities 71

6 Endpoint convergence of Legendre series M. A. Pinsky 79 6.1 Statement of results 79 6.2 Asymptotic estimates 80 6.3 Convergence at the endpoints 83

6.3.1 Convergence at x = 1 83 6.3.2 Convergence at x — — 1 84

7 Inversion of the horocycle transförm on real hyperbolic Spaces via a wavelet-like transförm W. 0. Bray, B. Rubin 87 7.1 Introduction 87 7.2 Preliminaries 88

7.2.1 Algebraic and geometric notions 88 7.2.2 The horocycle transförm and its dual 90 7.2.3 Approximate identities on H 93

7.3 Inversion of the Horocycle Transform 95

8 Fourier-Bessel expansions with general boundary condi-tions M. A. Pinsky 107 8.1 Introduction 107 8.2 Statement of Results 108

Contents ix

8.3 Proofs 109 8.4 Identifying the limit 113

8.4.1 An Abelian lemma 114

II Singular Integrals and Multipliers 117

9 Convolution Calderön-Zygmund singular integral Operators with rough kerneis L. Grafakos, A. Stefanov 119 9.1 Introduction 119 9.2 L? boundedness 121 9.3 LP boundedness, 1 < p < oo 123 9.4 The L1 theory 126 9.5 Another H1 condition in dimension 2 133 9.6 Maximal functions and maximal singular integrals 134

10 Haar multipliers, paraproducts, and weighted inequalities N. H. Katz, M. C. Pereyra 145 10.1 Introduction 145 10.2 Preliminaries 148

10.2.1 Dyadic intervals and Haar basis 148 10.2.2 Weights 149

10.3 Weight lemma and decaying stopping times 151 10.4 LP Lemmas for decaying stopping times 157

10.4.1 LP Plancherei Lemma 157 10.4.2 LP version of Cotlar's Lemma 160

10.5 Boundedness of Tt 161 10.5.1 Boundedness of Tu 161 10.5.2 Some corollaries 165

10.6 Haar multipliers and weighted inequalities 166

11 Multipliers and Square functions for Hp Spaces over Vilenkin groups J. E. Daly, K. L. Phillips 171 11.1 Introduction 171 11.2 Historical Comments 172 11.3 Multipliers for W (0 <p <1) 174 11.4 Square Function Characterization of Hp 181

12 Spectra of pseudo-differential Operators in the Hörmander class J. Alvarez 187 12.1 Introduction 187 12.2 Preliminary results 189

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12.3 The ivp-spectrum of translation invariant pseudo-differential Operators 191

12.4 The set Ka 192 12.5 Applications 194

13 Scaling properties of infinitely flat curves and surfaces A. Iosevich 201 13.1 Introduction 201 13.2 Scaling 204

13.2.1 Simplification 206 13.2.2 Three dimensions 207

13.3 Orlicz norms of dilation Operators 208 13.4 Examples 210

14 Some LP(L°°)— and L2(L2)— estimates for oscillatory Fourier transforms B. Walther 213 14.1 Introduction 213 14.2 Lp(L°°)-estimates 215 14.3 L2(L2)-estimates 221

15 Optimal Spaces for the S'-convolution with the Marcel Riesz kerneis and the iV-dimensional Hubert kernel J. Alvarez, C. Carton-Lebrun 233 15.1 Introduction 233 15.2 Definitions and notation 234

15.2.1 Function and distribution Spaces 235 15.2.2 The «S'-convolution 236 15.2.3 Partition of unity on E" 236

15.3 Optimal space for the <S'-convolution with the vector Riesz kernel 237

15.4 Optimal space for the 5'-convolution with pv-£- ® • • • <S)pv~ 240 15.5 Necessary condition for the iS'-convolvability with a Single

Riesz kernel 244

III Integral Operators and Functional Analysis 249

16 Asymptotic expansions and linear wavelet packets on cer-tain hypergroups K. Trimeche 251 16.1 Introduction 251 16.2 The Chebli-Trimeche hypergroups (R + ,* A ) 252 16.3 The dual of the hypergroups (R, *A) 255

Contents xi

16.4 Asymptotic expansions and integral representations of Mehler and Schläfli type 258 16.4.1 The asymptotic expansions 258 16.4.2 Integral representations of Mehler and Schläfli type . 262

16.5 Harmonie analysis and maximal ideal Spaces of some algebras265 16.5.1 Harmonie analysis 265 16.5.2 The maximal ideal Spaces of the algebras LNJUA)

and M6(M+) 270 16.6 Continuous linear wavelet transform and its discretization . 272

16.6.1 Linear wavelets on (R + ,*^) 272 16.6.2 Linear wavelet packet on (R+,*A) 282 16.6.3 Scale discrete L-scaling funetion on (R+,*yi) . . . . 287

17 Hardy-type inequalities for a new class of integral Opera­tors G. Sinnamon 297 17.1 Introduction 297 17.2 Starshaped regions 298 17.3 Prom regions to kerneis 304

18 Regularly bounded funetions and Hardy's inequality T. Ostrogorski 309 18.1 Introduction 309 18.2 Definition and Uniform Boundedness 310 18.3 The global bounds 314 18.4 The representation theorem 315 18.5 The multiplicative class 317 18.6 Abelian Theorems 319 18.7 Hardy's Inequality 321

19 Extremal problems in generalized Sobolev classes S. K. Bagdasarov 327 19.1 Introduction 327

19.1.1 General problem of sharp inequalities for intermedi-ate derivatives 327

19.1.2 Punctional classes WrHu(J) 329 19.1.3 The Kolmogorov problem in WrHu(l) 330

19.2 Maximization of integral functionals over H^la, b] 330 19.2.1 Simple kerneis *(•) and their rearrangements 5R(*; •) 331 19.2.2 The Korneichuk lemma 332 19.2.3 Extremal funetions of functionals over H"[a, b] . . . 333 19.2.4 Structural properties of extremal funetions x^^ . . 337

19.3 Kolmogorov problem for intermediate derivatives 342 19.3.1 Differentiation formulae for / ( m ) (0 ) , 0<m<r . . . 343 19.3.2 Differentiation formula for/( r)(0) 343

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19.3.3 Sufficient conditions of extremality 344 19.3.4 Extremality conditions in the form of an Operator

equation 344 19.3.5 Sharp additive inequalities for intermediate derivatives345 19.3.6 Kolmogorov problem in Holder classes 347

19.4 Kolmogorov problem in W1HU'(R+) and W1HU(R) . . . . 347 19.4.1 Preliminary remarks 347 19.4.2 Maximization of the norm ||/||ioo(R+) 348 19.4.3 Extremal functions in Holder classes W1Ha(R+) . . 350 19.4.4 Maximization of the norm | | / ' | | L 0 0 ( R + ) 350 19.4.5 Maximization of the norm ll/Hioo(K) 352 19.4.6 Maximization of the norm | | / ' ^ ( R ) 353

20 On angularly perturbed Laplace equations in the unit ball and their distributional boundary values P. R. Massopust 359 20.1 Introduction 359 20.2 Notation and Preliminaries 360 20.3 Bounded Solutions on B n + 2 362 20.4 Distributional Boundary Values 372 20.5 Generalities 376

21 Nonresonant semilinear equations and applications to bound­ary value problems P. S. Milojevic 379 21.1 Introduction 379 21.2 Semi-abstract nonresonance problems 380 21.3 Strong solvability of elliptic BVP's 390 21.4 Time periodic Solutions of BVP's for nonlinear parabolic and

hyperbolic equations 394 21.4.1 Nonlinear parabolic equations 394 21.4.2 Applications to the heat equation 396 21.4.3 Nonlinear hyperbolic equtions 398 21.4.4 Applications to the telegraph equation 400 21.4.5 Application to the beam equation with damping. . 401

22 A topological and fimctional analytic approach to Statistical convergence J. Connor 403 22.1 Introduction: 403 22.2 The support set of a measure 406 22.3 Invariants of Statistical convergence 407 22.4 Summability theorems 408

Contents xiii

IV Asymptotics and Applications 415

23 Optimal control of divergent control Systems D. A. Carlson 417 23.1 Introduction and History 417 23.2 Basic modeis and hypotheses 420 23.3 Existence of optimal Solutions 423

23.3.1 Existence of overtaking optimal Solutions without dis-counting 423

23.3.2 Existence of overtaking optimal Solutions with dis-counting 427

23.4 The associated uncoupled optimal control problems 429 23.4.1 The undiscounted case 429 23.4.2 The discounted case 430

23.5 Optimal Solutions of the explicitly State constrained optimal control problem 431 23.5.1 The undiscounted case 435

23.6 Conclusions 437

24 Surfaces minimizing integrals of divergent integrands H. R. Parks 441 24.1 Introduction 441 24.2 Surfaces and Integrands 443 24.3 Overtaking Minimizers 446 24.4 A Radially Symmetrie Example 450 24.5 Hypotheses for Regularity 452 24.6 Barriers 453 24.7 A Result in Differential Geometry 455 24.8 Bounding the Curvature 458

25 Sparse exponential sums with low sidelobes G. Benke 463 25.1 Introduction 463 25.2 Generalized Rudin-Shapiro Polynomials 465 25.3 Exponential Sums with Low Sidelobes 469

26 Spline type summability for multivariate sampling W. R. Madych 475 26.1 Introduction 475

26.1.1 Sampling theory 475 26.1.2 Splines and sampling theory 478 26.1.3 Contents, notation, and acknowledgements 480

26.2 Regulär sampling of multivariate funetions and their recov­ery via splines 481 26.2.1 Band limited funetions and polyharmonic splines . . 481

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26.2.2 The Spaces L2'k(Mn) and l2'k(ZZn) and the varia-tional properties of polyharmonic splines 485

26.2.3 The Paley-Wiener space PW% 488 26.2.4 Convergence of m-harmonic splines as m —> oo . . . 492

26.3 Generalizations, related methods, and computational issues 495 26.3.1 Generalizations 495 26.3.2 Multivariate analogues of the Paley-Wiener Theorem

and the sampling theorem 499 26.3.3 Box splines 501 26.3.4 Computing polyharmonic splines 503

27 ß-Splines and orthonormal sets in Paley-Wiener space A. I. Zayed 513 27.1 Introduction 513 27.2 Preliminaries: 514 27.3 Sampling and Orthonormal Functions 516 27.4 ß-splines and Orthonormal Sets in the Paley-Wiener Space 519

28 Norms of powers and a central limit theorem B. Baishanski 523 28.1 Introduction 523 28.2 The Five Parameters 524 28.3 Boundedness 525

28.3.1 Power Series 525 28.3.2 Trigonometrie Series 527

28.4 Asymptotic Behavior 527 28.5 Asymptotic Series 529 28.6 Changing the Question 532 28.7 Behavior of Scaled ^ for Large n 533 28.8 Another Kind of Central Limit Theorems 538

29 Quasiasymptotics at zero and nonlinear problems in a frame-work of Colombeau generalized funetions S. Pilipovic, M. Stojanovic 545 29.1 Introduction 545 29.2 Algebra of generalized funetions 547 29.3 ^/-quasiasymptotics at zero 550 29.4 Application of Q quasiasymptotics to generalized Solutions . 552

29.4.1 System of nonlinear Volterra integral equations with non-Lipschitz nonlinearity 553

29.4.2 Semilinear hyperbolic System 558 29.4.3 Nonlinear wave equation 560 29.4.4 Euler-Lagrange equation 562 29.4.5 Goursat problem 562