傅立叶级数和球面调和函数的几何应用
2000-6
世界图书出版公司
H.Groemer
329
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In 1901 Adolf Hurwitz published a short note showing that Fourier series can be used to prove the isoperimetric inequality for domains in the Euclidean plane,and in a subsequent article he showed how spherical harmonics can be utilized to prove an analogous inequality for three-dimensional convex bodies. A few years later Hermann Minkowski used spherical harmonics to prove an interesting characterization of (three-dimensional) convex bodies of constant width. The work of Hurwitz and Minkowski has convincingly shown that a study of this interplay of analysis and geometry, in particular of Fourier series and spherical harmonics on the one hand, and the theory of convex bodies on the other hand, can lead to interesting geometric results. Since then many articles have appeared that explored the possibilities of such methods.
Preface1 Analytic Preparations 1.1 Inner Product, Norm, and Orthogonality of Functions 1.2 The Gradient and Beltrami Operator 1.3 Spherical Integration and Orthogonal Transformations2 Geometric Preparations 2.1 Basic Features of Convex Sets 2.2 Support Functions 2.3 Metrics for Sets of Convex Bodies 2.4 Mixed Volumes and Mean Projection Measures 2.5 Inequalities 2.6 Difference Bodies, Projection Bodies, Steiner Point, and Centroid3 Fourier Series and Spherical Harmonics 3.1 From Fourier Series to Spherical Harmonics 3.2 Orthogonality, Completeness, and Series Expansions 3.3 Legendre Polynomials 3.4 Some Integral Transformations and the Funk-Hecke Theorem 3.5 Zonal Harmonics and Associated Legendre Functions 3.6 Estimates and Uniform Convergence4 Geometric Applications of Fourier Series 4.1 A Proof of Hurwitz of the Isoperimetric Inequality 4.2 The Fourier Expansion of the Support Function 4.3 The Isoperimetric and Related Inequalities 4.4 Wirtinger''s Inequality 4.5 Rotors and Tangential Polygons 4.6 Other Geometric Applications of Fourier Series5 Geometric Applications of Spherical Harmonics 5.1 The Harmonic Expansion of the Support Function 5.2 Inequalities for Mean Projection Measures and Mixed Volumes 5.3 The Isoperimetric Inequality 5.4 Wirtinger''s Inequality for Functions on the Sphere 5.5 Projections of Convex Bodies 5.6 Intersections of Convex Bodies with Planes or Half-Spaces 5.7 Rotors in Polytopes 5.8 Other Geometric Applications of Spherical HarmonicsReferencesList of SymbolsAuthor IndexSubject Index
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