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Koosis P. The Logarithmic Integral (Vol. 1)
Koosis P.  The Logarithmic Integral (Vol. 1)









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: The Logarithmic Integral (Vol. 1)

: Koosis P.

:

The theme of this unique 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.This, the first of two volumes, is self-contained, but more importantly, 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 He has brought together here many results, some unpublished, some new, and some available only in inaccessible journals.


: en

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ed2k: ed2k stats

: 1997

: 606

: 02.04.2005

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$w^{\ast}$ convergence      41
Akhiezers description of entire functions, arising in weighted approximation      160 174
Akhiezers theorems about weighted polynomial approximation      158ff 424 523
Akhiezers theorems on weighted approximation by sums of imaginary exponentials      174 424 432 445
Approximation index $M_p(A)$, Beurlings      293
Approximation index M(A), Beurlings      275
Approximation, weighted      145ff 385 424
Benedicks, M.      434ff
Benedicks lemma on harmonic measure for slit regions bounded by a circle      400
Benedicks theorem on existence of a Phragmen Lindelof function      418 431
Benedicks theorem on harmonic measure for slit regions      404
Bernstein approximation problem      146ff
Bernstein intervals associated with a set of points on (0, oo)      454ff
Bernsteins lemma      102
Bernsteins theorem on weighted polynomial approximation      169
Beurling quasianalyticity      275ff
Beurling quasianalyticity for $L_p$, functions      292ff
Beurling Dynkin theorem on the Legendre transform      333
Beurling, A. And Malliavin, P.      550 568
Beurlings approximation indices      see under Approximation index
Beurlings gap theorem      237 305
Beurlings identity for certain bilinear forms      484
Beurlings theorem about Fourier Stieltjes transforms vanishing on a set of positive measure      268
Beurlings theorem about his quasi-analyticity      276
Beurlings theorem on his Lp quasi-analyticity      293
Boundary values, non-tangential      10 43ff 265 269 286ff
Canonical product      21
Carlemans criterion for quasianalyticity      80
Carlemans criterion for quasianalyticity, its necessity      89
Carlemans inequality      96
Carlesons lemma on linear forms      392 398
Carlesons theorem on harmonic measure for slit regions      394 404 430
Cartan Gorny theorem      104
Cauchy principal value, definition of      533
Cauchy transform, planar      320ff
Class $\mathcal{C}_I({M_n})$ of infinitely differentiate functions      79
Class $\mathcal{C}_I({M_n})$ of infinitely differentiate functions, its quasianalyticity      80
Convex logarithmic regularization ${M_n}$ of a sequence ${M_n}$      83ff 92ff 104ff 130 226
de Branges lemma      187
de Branges theorem      192
Density, of a measurable sequence      178
Dirichlet integral      479 500 510ff
Dirichlet problem      251 360 387 388
Discussion about      198ff
Dynkins extension theorem      339 359 373
Energy of a measure on $(0,\infty)$      479ff 549ff 562 568
Energy of a measure on $(0,\infty)$, bilinear form associated thereto      482 487 494ff 508 512ff 551 552 553 563 566
Energy of a measure on $(0,\infty)$, formulas for      479 485 497 512
Energy of a measure on $(0,\infty)$, positivity of      482 493
Entire functions of exponential type      15ff
Entire functions of exponential type as majorants on subsets of R      555ff 562 564 568
Entire functions of exponential type, arising in weighted approximation      160 174 218 219 525
Entire functions of exponential type, expansions      203ff 205
Exponential type, entire functions of      see Entire functions of exponential type expansions
Extension of domain, principle of      259 289 368 372 529 531
Extension of positive linear functionals      111ff 116
Extreme point of a convex $w^{\ast}$ compact set of measures      186ff
Fejer and Riesz, lemma of      281
Function of exponential type, entire      15ff (see also under Entire functions)
Function T{r) used in study of quasianalyticity      80ff
Gap theorem, Beurlings      237
Gauss quadrature formula      134 137ff
Green potential      479 551 552 553 560 562 563 566
Green, George, homage to      419-22
Greens function      400ff 406 407 410 418ff 439 479 526ff 547ff 550
Greens function, estimates for in slit regions      401 439 442 548
Greens function, symmetry of      401 415 418ff 530
Hadamard factorization for entire functions of exponential type      16 19 22 54 56 70 201 556 561
Hall of mirrors argument      157 158 184 208 375 523
Hall, T., his theorem on weighted polynomial approximation      169
Hankel matrix      117
Harmonic conjugate      46 59 61
Harmonic conjugate, existence a.e. Of      47 532 537
Harmonic estimation, statement of theorem on it      256
Harmonic functions, positive, representations for, in half plane      41
Harmonic functions, positive, representations for, in unit disk      39
Harmonic measure      251ff
Harmonic measure in curvilinear strips, use of estimate for      355
Harmonic measure in slit regions      385 389ff 394 403 404 430 437 443 444 446 522 525ff 530 541 545ff 554 562 565
Harmonic measure, approximate identity property of      253 261
Harmonic measure, boundary behaviour of      261ff 265
Harmonic measure, definition of      255
Harmonic measure, Volbergs theorem on      349 353 362 364 366
Harnacks inequality      254 372 410 430
Hilbert transform      47 61 62 63 65 532 534 538ff
Jensens formula      2 4 7 21 76 163 291 559
Kargaevs example on Beurlings gap theorem      305ff 315
Kolmogorovs theorem on the harmonic conjugate      62ff
Krein Milman theorem, its use      186 199
Kreins theorem on certain entire functions      205
Kroneckers lemma      119
Legendre transform $h(\xi)$ of an increasing function $M(\nu)$      323ff
Levinson (and Cartwright), theorem on distribution of zeros for functions with real zeros only      66
Levinson (and Cartwright), theorem on distribution of zeros, general form of      69
Levinson (and Cartwright), theorem on distribution of zeros, use of      175 178
Levinsons log log theorem      374ff 376 379ff
Levinsons theorem about Fourier Stieltjes transforms vanishing on an interval      248 347 361
Levinsons theorem on weighted approximation by sums of imaginary exponentials      243
Lindelofs theorem on conformal mapping      264
Lindelofs theorems about the zeros of entire functions of exponential type, statements      20 21
Log log theorem      see under Levinson
Lower polynomial regularization $W_{\ast}(x)$ of a weight W(x) its definition      158
Lower regularization $W_A(x)$ of a weight W(x) by entire functions of exponential type $\ieq A$      175 428
Lower regularization $W_A(x)$ of a weight W(x) by entire functions of exponential type $\ieq A$ for Lip 1 weights      236
Lower regularization $W_A(x)$ of a weight W(x) by entire functions of exponential type $\ieq A$ for weights increasing on $[0,\infty)$      242
Lower regularizations $W_{A,E}(x) of a weight W(x) corresponding to closed, unbounded sets $E\subseteq R$      428
Markov Riesz Pollard trick      139 155 171 182 190
Maximum principle, extended, its statement      23
Measurable sequence      178
Mergelians theorems about weighted, polynomial approximation      147ff
Mergelians theorems on weighted approximation by sums of imaginary exponentials      173 174 432
Moment problem      see under Riesz
Moment sequences, definition of      109
Moment sequences, determinacy of      109 126 128 129 131 141 143
Moment sequences, indeterminacy of      109 128 133 143
Moment sequences, Riesz characterization of      110
Moment sequences, same in terms of determinants      121
Newton polygon      83ff
Non-tangential limit      11
Paley and Wiener, their construction of certain entire functions      100
Paley and Wiener, theorem of      31
Paley and Wiener, theorem of, $L_1$ version of same      36
Phragmen Lindelof argument      25 405 406 553
Phragmen Lindelof function      25 386 406 407 418 431 441 525ff 541 555
Phragmen Lindelof theorems, fifth      29
Phragmen Lindelof theorems, first      23
Phragmen Lindelof theorems, fourth      28
Phragmen Lindelof theorems, second      25
Phragmen Lindelof theorems, third      27
Poisson kernel for half plane      38 42 384 534 536 539
Poisson kernel for rectangle      299
Poisson kernel for unit disk      7 8 10ff
Poisson kernel, pointwise approximate identity property of latter      10
Pollards theorem      164 433
Pollards theorem for weighted approximation by sums of imaginary exponentials      181 428
Polya maximum density for a positive increasing sequence      176ff
Polyas theorem      178
Quasianalytic classes $\mathcal{C}_I({M_n})$, their characterization      91
Quasianalyticity of a class $\mathcal{C}_I({M_n})$      80
Quasianalyticity of a class $\mathcal{C}_I({M_n})$, Carlemans criterion for it      80
Quasianalyticity of a class $\mathcal{C}_I({M_n})$, necessity of same      89
Quasianalyticity, Beurlings      275ff
Representations for positive harmonic functions      see under Harmonic functions
Riesz Fejer theorem      55 556
Riesz, F. and M.      259 276 286
Riesz criterion for existence of a solution to moment problem      110 121
Riesz criterion for indeterminacy of the moment problem      133
Simultaneous polynomial approximation, Volbergs theorem on      344 349
Slit regions (whose boundary consists of slits along real axis)      384 3861T 401 402 418 430 439 441 525ff 540ft 545 553 564 568
Spaces $\mathcal{C}_W(0)$ and $\mathcal{C}_W(0+)$      212
Spaces $\mathcal{C}_W(0)$ and $\mathcal{C}_W(0+)$, conditions on W for their equality      223 226
Spaces $\mathcal{C}_W(0)$ and $\mathcal{C}_W(0+)$, weights W for which they differ      229ff 244ff
Spaces $\mathcal{L}_p(\mathcal{D}_0)$      281ff
Spaces of functions used in studying weighted approximation, their definitions $\mathcal{C}_W(\mathbb{R})$      145
Spaces of functions used in studying weighted approximation, their definitions, $\mathcal{C}_W(0)$, $\mathcal{C}_W(A)$$\mathcal{C}_W(A+)$      211
Spaces of functions used in studying weighted approximation, their definitions, $\mathcal{C}_W(E)$, $\mathcal{C}_W(A,E)$, $\mathcal{C}_W(0,E)$      424
Spaces of functions used in studying weighted approximation, their definitions, $\mathcal{C}_W(\mathbb{Z})$, $\mathcal{C}_W(0,\mathbb{Z})$      522
Szegos theorem      7 291 292
Szegos theorem, extension of same by Krein      9
Two constants, theorem on      257
Volbergs theorems on harmonic measures      349 353 362 364 366
Volbergs theorems on simultaneous polynomial approximation      344 349
Volbergs theorems on the logarithmic integral      317ff 357
Weight      145ff
Weighted approximation      145ff 385 424
Weighted approximation by polynomials      147ff 169 247 433 445
Weighted approximation by sums of imaginary exponentials      171ff
Weighted approximation by sums of imaginary exponentials on closed unbounded subsets of R      428 444
Weighted approximation by sums of imaginary exponentials with a Lip 1 weight      236
Weighted approximation by sums of imaginary exponentials with a weight increasing on $[0,\infty)$      243 247 Mergelian)
Weighted approximation on $\mathbb{Z}$      447ff 523 Mergelian)
Well disposed, definition of term      452
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