The Logarithmic Integral: Volume 2 (Cambridge Studies in Advanced Mathematics, Series Number 21)

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. 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 Cambridge University Press Cover 1 Title 4 Copyright 5 Contents 8 Foreword to volume II, with an example for the end of volume I 12 Errata for volume I 26 IX Jensen's Formula Again 28 A Polya's gap theorem 28 B Scholium. A converse to Pblya's gap theorem 35 1 Special case. E measurable and of density D > 0 35 Problem 29 36 2 General case; Y. not measurable. Beginning of Fuchs' construction 40 3 Bringing in the gamma function 47 Problem 30 49 4 Formation of the group products R;(z) 51 5 Behaviour of (1/x) log I (x - 2)/(x + 2)1 56 6 Behaviour of (1/x)logIR;(x)I outside the interval [Xi,YY] 58 7 Behaviour of (1/x)logIRj(x)I inside [Xi, YY] 61 8 Formation of Fuchs' function F(z). Discussion 70 9 Converse of Pblya's gap theorem in general case 79 C A Jensen formula involving confocal ellipses instead of circles 84 D A condition for completeness of a collection of imaginary exponentials on a finite interval 89 Problem 31 91 1 Application of the formula from ? 92 2 Beurling and Malliavin's effective density DA 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 114 1 Introduction to extremal length and to its use in estimating harmonic measure 115 Problem 32 128 Problem 33 135 Problem 34 136 2 Real zeros of functions f (z) of exponential type with (log+ I f(x)1/(1 + x2))dx < oo 137 F Scholium. Extension of results in ?.1. Pfluger's theorem and Tsuji's inequality 153 1 Logarithmic capacity and the conductor potential 154 Problem 35 158 2 A conformal mapping. Pfluger's theorem 159 3 Application to the estimation of harmonic measure. Tsuji's inequality 167 Problem 36 173 Problem 37 184 X Why we want to have multiplier theorems 185 A Meaning of term `multiplier theorem' in this book 185 Problem 38 186 1 The weight is even and increasing on the positive real axis 186 2 Statement of the Beurling-Malliavin multiplier theorem 191 B Completeness of sets of exponentials on finite intervals 192 1 The Hadamard product over E 196 2 The little multiplier theorem 200 3 Determination of the completeness radius for real and complex sequences A 216 Problem 39 222 C The multiplier theorem for weights with uniformly continuous logarithms 222 1 The multiplier theorem 222 2 A theorem of Beurling 229 Problem 40 235 D Poisson integrals of certain functions having given weighted quadratic norms 236 E Hilbert transforms of certain functions having given weighted quadratic norms 252 1 HP spaces for people who don't want to really learn about them 253 Problem 41 261 Problem 42 275 2 Statement of the problem, and simple reductions of it 276 3 Application of HP space theory; use of duality 287 4 Solution of our problem in terms of multipliers 299 Problem 43 306 F Relation of material in preceding ?to the geometry of unit sphere in L./HO 309 Problem 44 319 Problem 45 320 Problem 46 322 Problem 47 323 XI Multiplier theorems 325 A Some rudimentary potential theory 325 1 Superharmonic functions; their basic properties 325 2 The Riesz representation of superharmonic functions 338 Problem 48 354 Problem 49 355 3 A maximum principle for pure logarithmic potentials. 356 Problem 50 361 Problem 51 366 B Relation of the existence of multipliers to the finitness of a superharmonic majorant 368 1 Discussion of a certain regularity condition on weights 368 Problem 52 388 Problem 53 389 2 The smallest superharmonic majorant 390 Problem 54 396 Problem 55 397 Problem 56 398 3 How 931F gives us a multiplier if it is finite 401 Problem 57 410 C Theorems of Beurling and Malliavin 416 1 Use of the domains from ? of Chapter VIII 418 2 Weight is the modulus of an entire function of exponential type 422 Problem 58 432 3 A quantitative version of the preceding result 434 Problem 59 439 Problem 60 440 4 Still more about the energy. Description of the Hilbert space used in Chapter VIII, ?.5 445 Problem 61 470 Problem 62 471 5 Even weights W with II log W(x)/x IIE < ao 473 Problem 63 478 Problem 64 479 D Search for the presumed essential condition 479 1 Example. Uniform Lip I condition on log log W(x) not sufficient 481 2 Discussion 494 Problem 65 496 3 Comparison of energies 499 Problem 66 510 Problem 67 511 Problem 68 514 4 Example. The finite energy condition not necessary 514 5 Further discussion and a conjecture 529 E A necessary and sufficient condition for weights meeting the local regularity requirement 538 1 Five lemmas 539 2 Proof of the conjecture from ?.5 551 Problem 69 585 Problem 70 588 Problem 71 592 Bibliography for volume II 593 Index 599 0521102545,9780521102544,9780521309073,0521309077 The theme of this work, the logarithmic integral, lies athwart 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. Professor Koosis' aim is to show how, from simple ideas, one can build up an investigation which explains and clarifies many different, seemingly unrelated problems; to show, in effect, how mathematics grows. The presentation is straightforward, so that by following the theme, Professor Koosis has produced a work that can be read as a whole. He has brought together here many results, some unpublished, some new, and some available only in inaccessible journals Publisher Description (unedited publisher data) 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 The logarithmic integral lies athwart much of twentieth century analysis and connects many apparently separate parts of the subject. Koosis' 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. He has brought together here many results, some new and unpublished, making this a key reference for graduate, students and researchers. The derivations of the two main results in this chapter - Polya's gap theorem and a lower bound for the completeness radius of a set of imaginary exponentials - are both based on the same simple idea: application of Jensen's formula with a circle of varying radius and moving centre.

a Unique Work Giving A Straightforward Presentation Of The Logarithmic Integral, A Theme Which Lies Athwart Much Of Twentieth-century Analysis.