The theme of this unique work, the logarithmic integral, is found throughout much of twentieth century analysis. It is a thread connecting many apparently separate parts of the subject, and so is a natural point at which to begin a serious study of real and complex analysis. The author's aim is to show how, from simple ideas, one can build up an investigation that explains and clarifies many different, seemingly unrelated problems; to show, in effect, how mathematics grows.
Author(s): Koosis P.
Year: 1992
Language: English
Pages: 574
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
