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Axler S., Bourdon p., Ramey W. — Harmonic function theory
Axler S., Bourdon p., Ramey W. — Harmonic function theory

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Название: Harmonic function theory

Авторы: Axler S., Bourdon p., Ramey W.

Аннотация:

The authors have taken unusual care to motivate concepts and simplify proofs in this book about harmonic functions in Euclidean space. Readers with a background in real and complex analysis at the beginning graduate level will feel comfortable with the material presented here. Topics include basic properties of harmonic functions, Poisson integrals, the Kelvin transform, harmonic polynomials, spherical harmonics, harmonic Hardy spaces, harmonic Bergman spaces, the decomposition theorem, Laurent expansions, isolated singularities, and the Dirichlet problem.
This new edition contains a completely rewritten chapter on harmonic polynomials and spherical harmonics, as well as new material on Bocher's Theorem, norms for harmonic Hardy spaces, the Dirichlet problem for the half space, and the relationship between the Laplacian and the Kelvin transform. in addition, the authors have included new exercises and have made numerous minor improvements throughout the text.
The authors have developed a software package, available electronically without charge, that uses results from this book to calculate many of the expressions that arise in harmonic function theory. For example, the Poisson integral of any polynomial can be computed exactly.


Язык: en

Рубрика: Математика/Анализ/Продвинутый анализ/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 270

Добавлена в каталог: 02.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Annular region      20
Approximate identity      13 144
Arzela — Ascoli theorem      36
Baire’s theorem      42
Balls internally tangent to B      190
Barrier      227
Barrier function      227
Barrier problem      228
Basis of $\mathcal{H}_m(\mathbb{R}^n)$, $\mathcal{H}_m(S)$      92
Bergman projection      175
Bergman space      171
Bergman, Stefan      171
Bloch space      43 189
Bocher’s Theorem      50 57 197 199
Boundary data      15
Bounded harmonic function      31
Bounded harmonic function on B      40 119
Canonical projection of $\mathcal{P}_m(\mathbb{R}^n)$ onto $\mathcal{H}_m(\mathbb{R}^n)$      77
Cauchy, Augustin-Louis      34
Cauchy’s estimates      33
Computer package      80 90 93 106 125 180 218 221 247
Cone      232
Conformal map      60
Conjugate index      112
Convex region      230
Covering lemma      133
Decomposition theorem      195
Decomposition theorem for holomorphic functions      206
Degree      23 80
Dilate      2
Direct sum      81
Dirichlet problem      12 223
Dirichlet problem for annular regions      215
Dirichlet problem for annular regions (n = 2)      221
Dirichlet problem for convex regions      230
Dirichlet problem for H      146
Dirichlet problem for smooth regions      230
Dirichlet, Johann Peter Gustav Lejeune      13
Divergence theorem      4
Dual space      115
Equicontinuity      117
Essential singularity      211
Essential singularity (n = 2)      219
Essential singularity at $\infty$      220
Exterior Dirichlet problem      66
Exterior Poisson integral      66
Exterior Poisson kernel      66
External ball condition      229
External cone condition      232
External segment condition      237
Extremal function      124
Extreme point      140
Fatou theorem      128 160
Finitely connected      203
Fourier series      82 97
Fundamental pole      211
Fundamental pole (n = 2)      219
Fundamental pole at $\infty$      220
Fundamental solution of the Laplacian      193
Gamma function      245
Generalized annular Dirichlet problem      221
Generalized Dirichlet problem      106
Green’s identity      4
Hardy — Littlewood maximal function      131
Hardy, G. H.      118
Harmonic      1
Harmonic at $\infty$      63
Harmonic Bergman space      171
Harmonic Bloch space      43 189
Harmonic conjugate      203
Harmonic functions, limits of      16 49
Harmonic Hardy space      118 151
Harmonic measure      237
Harmonic motion      25
Harmonics      25
Harnack’s inequality      48
Harnack’s Inequality for B      47 56
Harnack’s principle      49
Holomorphic at $\infty$      71
Homogeneous expansion      24 99
Homogeneous harmonic polynomial      24 75
Homogeneous part      75
Homogeneous polynomial      23 75
Hopf Lemma      28
inversion      60
Inversion map      153
Isolated singularity      32 210
Isolated singularity at $\infty$      61
Isolated singularity of positive harmonic function      50
Isolated zero      6
Kelvin transform      59 61 155
Laplace’s equation      1
Laplacian      1
Laurent series      193 209
Law of Cosines      130
Lebesgue decomposition      136
Lebesgue Differentiation Theorem      165
Lebesgue point      141
Limits along rays      39
Liouville’s theorem      31
Liouville’s Theorem for positive harmonic functions      45 56 198
Local denning function      230
Local Fatou Theorem      161
Locally connected      235
Locally integrable      18
Logarithmic conjugation theorem      203
Mathematica      247
Maximum principle      7 36
Maximum principle for subharmonic functions      224
Maximum principle, local      23
Mean-value property      5
Mean-value property, converse of      17
Mean-value property, volume version      6
Minimum principle      7
Morera’s theorem      213
Multi-index      15
Neumann problem      108
Nonextendability of harmonic functions      233
Nontangential approach region      38 128
Nontangential limit      38 128 160
Nontangential maximal function      129
Nontangentially bounded      161
Normal family      3 5
North pole      103
One-point compactification      59
One-radius theorem      28
Open mapping      27
Open mapping theorem      181
Operator norm      115
Order of a pole      211
Order of a pole (n = 2)      219
Orthogonal transformation      3 95
Orthonormal basis of $b^2(B_2)$      189
Parallel orthogonal to $\eta$      100
Perron family      226
Perron function      226
Picard’s Theorem      212
Point evaluation      172
Point of density      165
Poisson integral      12
Poisson integral for annular region      217
Poisson integral for H      146
Poisson integral of a measure      111
Poisson integral of a polynomial      74 89 98 105
Poisson kernel      9 12 99 122 157
Poisson kernel for annular region      215
Poisson kernel for H      144 157 185
Poisson kernel, expansion into zonal harmonics      99
Poisson modification      225
Poisson’s equation      193
Polar coordinates, integration in      6
Polar coordinates, Laplacian in      26
Pole      211
Pole (n = 2)      219
Positive harmonic function      45
Positive harmonic function on $\mathbb{R}^2\{0\}$      46
Positive harmonic function on $\mathbb{R}^n\{0\}$      54
Positive harmonic function on B      55 119
Positive harmonic function on H      156
Power series      19
Principal part      211
Principal part (n = 2)      219
Product rule for Laplacian      13
Punctured ball      218
Radial derivative      141
Radial function      27 52 57 101
Radial harmonic function      52 57
Radial limit      185
Radial maximal function      129
Real analytic      19
Reflection about a hyperplane      67
Reflection about a sphere      68
Removable sets      200
Removable singularity      32 188 211
Removable singularity at $\infty$      61 220
Reproducing kernel      172
Reproducing kernel of B      176
Reproducing kernel of H      185
Residue      213
Residue (n = 2)      221
Residue theorem      213
Residue theorem (n = 2)      221
Riemann — Lebesgue lemma      183
Riesz representation theorem      111
Rotation      4
Schwarz lemma      123
Schwarz Lemma for $h^2$      141
Schwarz Lemma for $\nabla u$      125
Schwarz reflection principle      67
Schwarz, Hermann Amandus      123
Separable normed linear space      117
Simply connnected      203
Singular measure      136
Singularity at $\infty$      220
Slice integration      241
Smooth boundary      230
Software for harmonic functions      80 90 93 106 125 180 218 221 247
Spherical average      50
Spherical cap      131
Spherical coordinates, Laplacian in      26
Spherical harmonic      25 80
Spherical harmonics via differentiation      85
Stone — Weierstrass theorem      81 106 216 217
Subharmonic      224
Submean-value property      224
Support      192
Surface area of S      240
Symmetry about a hyperplane      67
Symmetry about a sphere      68
Symmetry lemma      10
Total variation norm      111
translate      2
Uniform boundedness principle      138
Uniformly continuous function      147 167
Uniformly integrable      139
Upper half-space      143
Volume of B      239
weak$^{\ast}$ convergence      115 149
Zonal harmonic      173
Zonal harmonic expansion of Poisson kernel      99
Zonal harmonic with pole $\eta$      94
Zonal harmonic, formula for      104
Zonal harmonic, geometric characterization      100
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