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Geometry of the Unit Sphere in Polynomial Spaces

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This brief presents a global perspective on the geometry of spaces of polynomials. Its particular focus is on polynomial spaces of dimension 3, providing, in that case, a graphical representation of the unit ball. Also, the extreme points in the unit ball of several polynomial spaces are characterized. Finally, a number of applications to obtain sharp classical polynomial inequalities are presented.

The study performed is the first ever complete account on the geometry of the unit ball of polynomial spaces. Nowadays there are hundreds of research papers on this topic and our work gathers the state of the art of the main and/or relevant results up to now. This book is intended for a broad audience, including undergraduate and graduate students, junior and senior researchers and it also serves as a source book for consultation. In addition to that, we made this work visually attractive by including in it over 50 original figures in order to help in the understanding of all the results and techniques included in the book.

Author(s): Jesús Ferrer, Domingo García, Manuel Maestre, Gustavo A. Muñoz, Daniel L. Rodríguez, Juan B. Seoane
Series: SpringerBriefs in Mathematics
Publisher: Springer
Year: 2023

Language: English
Pages: 139
City: Cham

Contents
1 Introduction
2 Polynomials of Degree n
2.1 On the Real Line
2.1.1 Polynomials Bounded by a Majorant
2.2 On the Complex Plane
3 Spaces of Trinomials
3.1 On the Real Line with the Supremum Norm
3.1.1 The Geometry of Bm,n,∞ for Odd Numbers m,n
3.1.2 The Geometry of Bm,n,∞ for m Odd and n Even
3.1.3 The Geometry of Bm,n,∞ for m Even and n Odd
3.1.4 The Geometry of Bm,n,∞ for Even Numbers m,n
3.2 On the Real Line with the Lp-Norm
3.3 On the Real Plane
3.3.1 The Geometry of B2n,n,∞,2 for n Odd
3.3.2 The Geometry of B2n,n,∞,2 for n Even
3.3.3 The Geometry of Bm,n,∞,2 for m Odd
3.4 On the Complex Plane
4 Polynomials on Non-Balanced Convex Bodies
4.1 The Simplex
4.1.1 Polynomials of Degree at Most 2
4.2 The Unit Square
4.3 Circular Sectors
4.3.1 The Geometry of BD(β) When β≥π
4.3.2 The Geometry of BD(β) When β=π4,π2,3π4
4.3.3 The General Case of BD(β)
5 Sequence Banach Spaces
5.1 The Space 12
5.2 The Space ∞2
5.3 The Space p2 when 1 5.4 The Space p2 when 2 5.5 The Space c0
5.6 The Space 1
5.7 The Space p when p>2
6 Polynomials with the Hexagonal and Octagonal Norms
6.1 Octagonal Norm
6.2 Hexagonal Norm
7 Hilbert Spaces
7.1 The Real and Complex Case for 2-Homogeneous Polynomials
7.2 Polynomials of Degree n
8 Banach Spaces
8.1 Integral and Nuclear Polynomials
8.2 Orthogonally Additive Polynomials
9 Applications
9.1 Bernstein-Markov Type Inequalities
9.2 Polarization Constants
9.3 Unconditional Constants
9.4 Bohnenblust–Hille and Hardy–Littlewood Constants
9.4.1 On the Complex Case
9.4.2 On the Real Case
References