The logarithmic integral 1

Cover......Page 1
Title......Page 4
Copyright......Page 5
Notice......Page 8
Contents......Page 10
Preface......Page 16
Introduction......Page 18
I Jensen's formula......Page 20
Problem 1......Page 24
A The theorem......Page 26
B The pointwise approximate identity property of the Poisson kernel......Page 29
Problem 2......Page 32
A Hadamard factorization......Page 34
B Characterization of the set of zeros of an entire function of exponential type. Lindelof's theorems......Page 38
Problem 3......Page 41
C Phragmbn-Lindelof theorems......Page 42
D The Paley-Wiener theorem......Page 49
E Introduction to the condition......Page 56
1. The representation......Page 58
2. Digression on the a.e. existence of boundary values......Page 62
1 Functions without zeros in 3z > 0......Page 66
2 Convergence of f?.(log -If(x)I/(1+x2))dx......Page 68
3 Taking the zeros in 3z > 0 into account. Use of Blaschke products......Page 71
H Levinson's theorem on the density of the zeros ^g......Page 77
1 Kolmogorov's theorem on the harmonic conjugate......Page 78
2 Functions with only real zeros......Page 84
3 The zeros not necessarily real......Page 88
Problem 5......Page 95
1 Definition of the classes ^7({Mn})......Page 97
2 The function T(r). Carleman's criterion......Page 99
1 Definition of the sequence {Mj. Its relation to {Mn} and T(r)......Page 102
2 Necessity of Carleman's criterion and the characterization of quasianalytic classes......Page 108
C Scholium. Direct establishment of the equivalence between the three conditions......Page 111
Problem 6......Page 115
D The Paley-Wiener construction of entire functions of small exponential type decreasing fairly rapidly along the real axis......Page 116
E Theorem of Cartan and Gorny on equality of ''({M}) and W an algebra......Page 121
Problem 7......Page 122
V The moment problem on the real line......Page 128
A Characterization of moment sequences. Method based on extension of positive linear functionals......Page 129
B Scholium. Determinantal criterion for to be a moment sequence......Page 135
1 Carleman's sufficient condition......Page 145
2 A necessary condition......Page 147
Problem 8......Page 150
1 The criterion with Riesz' function R(z)......Page 151
2 Derivation of the results in ? from the above one......Page 161
VI Weighted approximation on the real line......Page 164
1 Criterion in terms of finiteness of ((z)......Page 166
2 A computation......Page 169
3 Criterion in terms of J?. (log Sl;(t)/(1 + t2) )dt......Page 172
1 Criterion in terms of f "'. (log W (x)/(1 + x2))dx......Page 177
2 Description of II II w limits of polynomials when log W (t) dt < 1+t2 00......Page 179
3 Strengthened version of Akhiezer's criterion. Pollard's theorem......Page 182
C Mergelian's criterion really more general in scope than Akhiezer's. Example......Page 184
D Some partial results involving the weight W explicitly.......Page 188
1 Equivalence with weighted approximation by certain entire function of exponential type. The collection 8.......Page 190
2 The functions c14(z) and WA(z). Analogues of Mergelian's and Akhiezer's theorems......Page 192
3 Scholium. P61ya's maximum density......Page 194
4 The analogue of Pollard's theorem......Page 199
F L. de Branges' description of extremal unit measures orthogonal to the ei2"/ W(x), - A < 2 < A, when 'A is not dense in %w(18)......Page 203
1 Three lemmas......Page 206
2 De Branges' theorem......Page 209
3 Discussion of the theorem......Page 217
4 Scholium. Krein's functions......Page 222
Problem 10......Page 228
G Weighted approximation with LP norms......Page 229
H Comparison of weighted approximation by polynomials and by functions in 8A......Page 230
1 Characterization of the functions in cw(A +)......Page 231
2 Sufficient conditions for equality of 'w(0) and 'w(0 +)......Page 238
3 Example of a weight W with 1w(0) ''w(0 +) # W w(1d)......Page 248
VII How small can the Fourier transform of a rapidly decreasing non-zero function be?......Page 252
1 Some shop math......Page 253
2 Beurling's gap theorem......Page 255
Problem 11......Page 257
3 Weights which increase along the positive real axis......Page 258
4 Example on the comparison of weighted approximation by polynomials and that by exponential sums......Page 262
5 Levinson's theorem......Page 266
B The Fourier transform vanishes on a set of positive measure Beurling's theorems......Page 269
1 What is harmonic measure?......Page 270
2 Beurling's improvement of Levinson's theorem......Page 284
3 Beurling's study of quasianalyticity......Page 294
4 The spaces 5p(-90), especially .91(-90)......Page 299
5 Beurling's quasianalyticity theorem for LP approximation by functions in 9'p(90)......Page 311
C Kargaev's example......Page 324
1 Two lemmas......Page 325
2 The example......Page 331
D Volberg's work......Page 335
Problem 12......Page 337
1 The planar Cauchy transform......Page 338
Problem 13......Page 341
2 The function M(v) and its Legendre transform......Page 342
Problem 14(a)......Page 346
Problem 14(c)......Page 355
3 Dynkin's extension of F(e') to { Iz 15 1) with control on I FZ{z)1......Page 357
4 Material about weighted planar approximation by polynomials......Page 362
5 Volberg's theorem on harmonic measures......Page 367
6 Volberg's theorem on the logarithmic integral......Page 375
7 Scholium. Levinson's log log theorem......Page 393
VIII Persistence of the form dx/(1 + x2)......Page 403
A The set E has positive lower uniform density......Page 405
1 Harmonic measure for -9......Page 406
2 Green's function and a Phragmen-Lindelof function for -9......Page 419
Problem 16......Page 423
Problem 17(a)......Page 430
Problem 17(b)......Page 432
3 Weighted approximation on the sets E......Page 443
Problem 18......Page 451
4 What happens when the set E is sparse......Page 453
Problem 19......Page 462
B The set E reduces to the integers......Page 464
Problem 20......Page 465
1 Using certain sums as upper bounds for integrals corresponding to them......Page 466
2 Construction of certain intervals containing the zeros of p(x)......Page 473
3 Replacement of the distribution n(t) by a continuous one......Page 487
4 Some formulas......Page 492
5 The energy integral......Page 497
Problem 22......Page 503
6 A lower estimate for 1.11 o log I 1- (x2/t2)Idp(t)(dx/x2)......Page 506
7 Effect of taking x to be constant on each of the intervals Jk......Page 511
8 An auxiliary harmonic function......Page 514
Problem 23......Page 516
9 Lower estimate for f n f o log 11- (x2/t2)Idp(t)(dx/x2)......Page 525
10 Return to polynomials......Page 535
Problem 24......Page 537
11 Weighted polynomial approximation on 7L......Page 541
C Harmonic estimation in slit regions......Page 544
1 Some relations between Green's function and harmonic measure for our domains .9......Page 545
2 An estimate for harmonic measure......Page 559
Problem 26......Page 564
3 The energy integral again......Page 567
4 Harmonic estimation in 9......Page 572
5 When majorant is the logarithm of an entire function of exponential type......Page 574
Problem 27......Page 580
Problem 28......Page 587
1 Brennan's improvement, for M(v)/v1/2 monotone increasing......Page 589
2 Discussion......Page 593
3 Extension to functions F(ei,) in L1(-rr,n)......Page 601
4 Lemma about harmonic functions......Page 609
Bibliography for volume 1......Page 615
Index......Page 619
Contents of volume II......Page 622