Deals with the theory of function spaces and with the developments related to the numerous applications of the theory of function spaces to some neighbouring areas such as numerics, signal processing and fractal analysis. This book discusses typical building blocks as (non-smooth) atoms, quarks, wavelet bases and wavelet frames
InhaltsverzeichnisI Decompositions of Functions.1 Introduction, heuristics, and preliminaries.2 Spaces on ? n: the regular case.3 Spaces on ? n: the general case.4 An application: the Fubini property.5 Spaces on domains: localization and Hardy inequalities.6 Spaces on domains. decompositions.7 Spaces on manifolds.8 Taylor expansions of distributions.9 Traces on sets, related function spaces and their decompositions.II Sharp Inequalities.10 Introduction: Outline of methods and results.11 Classical inequalities.12 Envelopes.13 The critical case.14 The super-critical case.15 The sub-critical case.16 Hardy inequalities.17 Complements.III Fractal Elliptic Operators.18 Introduction.19 Spectral theory for the fractal Laplacian.20 The fractal Dirichlet problem.21 Spectral theory on manifolds.22 Isotropic fractals and related function spaces.23 Isotropic fractal drums.IV Truncations and Semi-linear Equations.24 Introduction.25 Truncations.26 The Q-operator.27 Semi-linear equations; the Q-method.References.Symbois.
Focusing on the interplay between fractal geometry and Fourier analysis, this book explores the spectral properties of fractal (pseudo)differential operators. It highlights recent advancements in function space theory that connect to fractal methods, particularly in spectral problems related to degenerate pseudodifferential operators. The text provides insights into entropy numbers and eigenvalue distribution, presenting new techniques not widely covered in existing literature. Additionally, it serves as a continuation of previous works in the field, enriching the understanding of fractal analysis.
Continuing the exploration of function spaces, this volume serves as both a supplement to the "Theory of Function Spaces" trilogy and a companion to the textbook by D.D. Haroske and the author on distributions, Sobolev spaces, and elliptic equations. It enhances the foundational concepts established in the earlier works, offering deeper insights and advanced discussions relevant to the field, making it a valuable resource for scholars and students alike.
InhaltsverzeichnisZahlen und Räume.Konvergenz und Stetigkeit.Differential- und Integralrechnung im R 1 (Grundbegriffe).Gewöhnliche Differentialgleichungen (Existenz- und Unitätssätze).Elementare Funktionen und Potenzreihen.Banachräume.Integralrechnung im R 1 (Fortsetzung).Differentialrechnung im R n.Integralrechnung im R n.Gewöhnliche Differentialgleichungen (Lösungsmethoden).Variationsrechnung.Prinzipien der klassischen Mechanik.Maßtheorie.Integrationstheorie.Funktionentheorie.Prinzipien der Hydrodynamik ebener Strömungen.Elemente der Geometrie.Orthogonalreihen.Partielle Differentialgleichungen.Operatoren in Banachräumen.Operatoren in Hilberträumen.Distributionen.Partielle Differentialgleichungen und Distributionen.Grundbegriffe der klassischen Feldtheorie.Prinzipien der speziellen Relativitätstheorie und der Elektrodynamik.Selbstadjungierte Operatoren im Hilbertraum.Differentialoperatoren und orthogonale Funktionen.Prinzipien der Quantenmechanik.Geometrie auf Mannigfaltigkeiten I (Tensoren).Allgemeine Relativitätstheorie I (Grundgleichungen).Allgemeine Relativitätstheorie II (Singularitäten, schwarze Löcher, Kosmologie).Geometrie auf Mannigfaltigkeiten II (Formen).Die Wellengleichung in gekrümmten Raum-Zeiten.Singularitätentheorie.Katastrophen: Theorie und Anwendung.