Orthonormal Systems and Banach Space Geometry describes the
interplay between orthonormal expansions and Banach space geometry.
Using harmonic analysis as a starting platform, classical
inequalities and special functions are used to study orthonormal
systems leading to an understanding of the advantages of systems
consisting of characters on compact Abelian groups. Probabilistic
concepts such as random variables and martingales are employed and
Ramsey's theorem is used to study the theory of super-reflexivity.
The text yields a detailed insight into concepts including type and
co-type of Banach spaces, B-convexity, super-reflexivity, the
vector-valued Fourier transform, the vector-valued Hilbert
transform and the unconditionality property for martingale
differences (UMD). A long list of unsolved problems is included as
a starting point for research. This book should be accessible to
graduate students and researchers with some basic knowledge of
Banach space theory, real analysis, probability and algebra.
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