Harmonic approximation has recently matured into a coherent research area with extensive applications. This is the first book to give a systematic account of these developments, beginning with classical results concerning uniform approximation on compact sets, and progressing through fusion techniques to deal with approximation on unbounded sets. The author draws inspiration from holomorphic results such as the well-known theorems of Runge and Mergelyan. The final two chapters deal with wide ranging and surprising applications to the Dirichlet problem, maximum principle, Radon transform and the construction of pathological harmonic functions. This book is aimed at graduate students and researchers who have some knowledge of subharmonic functions (potential theory), or an interest in holomorphic approximation.
Harmonic approximation has recently matured into a coherent research area with extensive applications. This is the first book to give a systematic account of these developments, beginning with classical results concerning uniform approximation on compact sets, and progressing through fusion techniques to deal with approximation on unbounded sets. The author draws inspiration from holomorphic results such as the well-known theorems of Runge and Mergelyan. The final two chapters deal with wide ranging and surprising applications to the Dirichlet problem, maximum principle, Radon transform and the construction of pathological harmonic functions. This book is aimed at graduate students and researchers who have some knowledge of subharmonic functions (potential theory), or an interest in holomorphic approximation.
1. Review of thin sets; 2. Approximation on compact sets; 3. Fusion of harmonic functions; 4. Approximation on relatively closed sets; 5. Carleman approximation; 6. Tangential approximation at infinity; 7. Subharmonic extension and approximation; 8. The Dirichlet problem with non-compact boundary; 9. Further applications.
This book is aimed at graduate students and researchers who have some knowledge of subharmonic functions, or an interest in holomorphic approximation.
"This monograph should make the main results and techniques of harmonic approximation, much of which has been developed in the last 20 years, more familiar to a wider circle of mathematicians." P. Lappan, Mathematical Reviews
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