Alphabeta MathOrthonormal Bases, Parseval and Fourier Series — Examples
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Functional Analysis
- Bounded Linear Operators and Quotient Spaces
- Complex Norms and Realification
- Finite Dimensional Normed Spaces and Riesz Lemma
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Orthonormal Bases, Parseval and Fourier Series
- The Analytic Hahn Banach Theorem
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Areas of Elementary Plane Figures
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Convergence: Nets and Filters
- Countability Axioms and Cardinal Functions
- Function Space Topologies and the Exponential Law
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Separation Axioms: the Hierarchy
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Example
The standard basis of ℓ²(ℕ)
Example
Legendre polynomials from Gram–Schmidt
Example
The Haar orthonormal basis of L²((0,1))
Example
The Fourier series of a sawtooth and the Basel sum
Example
The Fourier series of a square wave and the odd reciprocal-square sum
Sources
- Theo Bühler and Dietmar Salamon, Functional Analysis — Example 2.65, p.87
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, p.52
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, p.50, example after Theorem 2.3
- Theo Bühler and Dietmar Salamon, Functional Analysis — Exercise 2.63, p.87
- Vladimir Dobrushkin, Eigenfunction Expansion, APMA0360 course tutorial — Example 5, dyadic functions and span discussion
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, p.50
- Henri P. Gavin, Dynamic Periodic Response to Periodic Forcing, System Identification course notes, Fall 2013 — §4.2, p.7
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, pp.64–66
- Henri P. Gavin, Dynamic Periodic Response to Periodic Forcing, System Identification course notes, Fall 2013 — §4.1, p.6