The Logarithmic Integral 2

Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 8
Foreword to volume II, with an example for the end of volume I......Page 12
Errata for volume I......Page 26
A Polya's gap theorem......Page 28
1 Special case. E measurable and of density D > 0......Page 35
Problem 29......Page 36
2 General case; Y. not measurable. Beginning of Fuchs' construction......Page 40
3 Bringing in the gamma function......Page 47
Problem 30......Page 49
4 Formation of the group products R;(z)......Page 51
5 Behaviour of (1/x) log I (x - 2)/(x + 2)1......Page 56
6 Behaviour of (1/x)logIR;(x)I outside the interval [Xi,YY]......Page 58
7 Behaviour of (1/x)logIRj(x)I inside [Xi, YY]......Page 61
8 Formation of Fuchs' function F(z). Discussion......Page 70
9 Converse of Pblya's gap theorem in general case......Page 79
C A Jensen formula involving confocal ellipses instead of circles......Page 84
D A condition for completeness of a collection of imaginary exponentials on a finite interval......Page 89
Problem 31......Page 91
1 Application of the formula from ?......Page 92
2 Beurling and Malliavin's effective density DA......Page 97
E Extension of the results in ? to the zero distribution of entire functions f (z) of exponential type with f?. (log` (f(x)I/(1 +x2))dx convergent......Page 114
1 Introduction to extremal length and to its use in estimating harmonic measure......Page 115
Problem 32......Page 128
Problem 33......Page 135
Problem 34......Page 136
2 Real zeros of functions f (z) of exponential type with (log+ I f(x)1/(1 + x2))dx < oo......Page 137
F Scholium. Extension of results in ?.1. Pfluger's theorem and Tsuji's inequality......Page 153
1 Logarithmic capacity and the conductor potential......Page 154
Problem 35......Page 158
2 A conformal mapping. Pfluger's theorem......Page 159
3 Application to the estimation of harmonic measure. Tsuji's inequality......Page 167
Problem 36......Page 173
Problem 37......Page 184
A Meaning of term `multiplier theorem' in this book......Page 185
1 The weight is even and increasing on the positive real axis......Page 186
2 Statement of the Beurling-Malliavin multiplier theorem......Page 191
B Completeness of sets of exponentials on finite intervals......Page 192
1 The Hadamard product over E......Page 196
2 The little multiplier theorem......Page 200
3 Determination of the completeness radius for real and complex sequences A......Page 216
1 The multiplier theorem......Page 222
2 A theorem of Beurling......Page 229
Problem 40......Page 235
D Poisson integrals of certain functions having given weighted quadratic norms......Page 236
E Hilbert transforms of certain functions having given weighted quadratic norms......Page 252
1 HP spaces for people who don't want to really learn about them......Page 253
Problem 41......Page 261
Problem 42......Page 275
2 Statement of the problem, and simple reductions of it......Page 276
3 Application of HP space theory; use of duality......Page 287
4 Solution of our problem in terms of multipliers......Page 299
Problem 43......Page 306
F Relation of material in preceding ?to the geometry of unit sphere in L./HO......Page 309
Problem 44......Page 319
Problem 45......Page 320
Problem 46......Page 322
Problem 47......Page 323
1 Superharmonic functions; their basic properties......Page 325
2 The Riesz representation of superharmonic functions......Page 338
Problem 48......Page 354
Problem 49......Page 355
3 A maximum principle for pure logarithmic potentials.......Page 356
Problem 50......Page 361
Problem 51......Page 366
1 Discussion of a certain regularity condition on weights......Page 368
Problem 52......Page 388
Problem 53......Page 389
2 The smallest superharmonic majorant......Page 390
Problem 54......Page 396
Problem 55......Page 397
Problem 56......Page 398
3 How 931F gives us a multiplier if it is finite......Page 401
Problem 57......Page 410
C Theorems of Beurling and Malliavin......Page 416
1 Use of the domains from ? of Chapter VIII......Page 418
2 Weight is the modulus of an entire function of exponential type......Page 422
Problem 58......Page 432
3 A quantitative version of the preceding result......Page 434
Problem 59......Page 439
Problem 60......Page 440
4 Still more about the energy. Description of the Hilbert space used in Chapter VIII, ?.5......Page 445
Problem 61......Page 470
Problem 62......Page 471
5 Even weights W with II log W(x)/x IIE < ao......Page 473
Problem 63......Page 478
D Search for the presumed essential condition......Page 479
1 Example. Uniform Lip I condition on log log W(x) not sufficient......Page 481
2 Discussion......Page 494
Problem 65......Page 496
3 Comparison of energies......Page 499
Problem 66......Page 510
Problem 67......Page 511
4 Example. The finite energy condition not necessary......Page 514
5 Further discussion and a conjecture......Page 529
E A necessary and sufficient condition for weights meeting the local regularity requirement......Page 538
1 Five lemmas......Page 539
2 Proof of the conjecture from ?.5......Page 551
Problem 69......Page 585
Problem 70......Page 588
Problem 71......Page 592
Bibliography for volume II......Page 593
Index......Page 599