Boundary properties and applications of the differentiated Poisson integral for different domains

Title Page......Page 4
Contents......Page 6
Preface......Page 8
1.1 Notation, Definitions and the Well-Known Statements......Page 12
1.2 Auxiliary Statements......Page 16
1.3 The Boundary Properties of Derivatives of the Poisson Integral for a Half-Plane......Page 19
1.4 The Dirichlet Problem for a Half-Plane......Page 29
2.1 Notation and Definitions......Page 32
2.2 Auxiliary Statements......Page 33
2.3 Boundary Properties of Derivatives of the Poisson Integral for a Circle......Page 36
2.4 The Dirichlet Problem for a Circle......Page 45
3.1 Notation, Definitions and Statement of Some Well-Known Facts......Page 48
3.2 The Poisson Integral for a Ball......Page 56
3.3 A Generalized Laplace Operator on the Unit Sphere Sk−1......Page 58
3.4 Boundary Properties of the Integral [omitted]......Page 67
3.5 The Boundary Properties of the Integral [omitted]......Page 78
3.6 The Dirichlet Problem for a Ball......Page 85
3.7 Representation by the Laplace Series of an Arbitrary Measurable Function Defined on the Unit Sphere S2......Page 94
4.1 Generalized Partial First Order Derivatives of a Function of Several Variables......Page 98
4.2 The Boundary Properties of First Order Partial Derivatives of the Poisson Integral for a Half-Space RR[sup(k+1)][sub(+)]  (k > 1)......Page 100
4.3 Generalized Partial Second Order Derivatives for a Function of Several Variables......Page 113
4.4 The Boundary Properties of Partial Second Order Derivatives of the Poisson Integral for a Half-Space RR[sup(k+1)][sub(+)]  (k > 1)......Page 115
4.5 The Dirichlet Problem for a Half-Space R[sup(3)][sub(+)]......Page 130
4.6 Generalized Partial Derivatives of Arbitrary Order......Page 132
4.7 The Boundary Properties of Arbitrary Order Partial Derivatives of the Poisson Integral for a Half-Space RR[sup(k+1)][sub(+)]......Page 133
4.8 Generalized Mixed Derivatives and Differentials of Arbitrary Order......Page 138
4.9 The Boundary Properties of Mixed Derivatives and Differentials of Arbitrary Order of the Poisson Integral for the Half-Space RR[sup(k+1)][sub(+)]  (k > 1)......Page 140
4.10 The Generalized Laplace Operator in R[sup(k)] (k ≥ 2)......Page 147
4.11 The Boundary Properties of the Integral [omitted]......Page 150
5.1 Notation and Definitions......Page 156
5.2 The Boundary Properties of Arbitrary Poisson Integral for a Bicylinder......Page 158
5.3 Representation of a Function of Two Variables by a Trigonometric Series in the Case of Spherical Convergence......Page 169
5.4 On One Method of Summation of Double Fourier Series......Page 170
5.5 Representation of a Function of Two Variables by a Double Trigonometric Series in the Case of Ptingsheim’s Convergence......Page 173
Bibliography......Page 180
Index......Page 190