Part VI — The Analysis Behind the Transforms

Booklet: Abridged Math Foundations for Signal Processing, Machine Learning, and Artificial Intelligence, Lessons 31–43 (PDF).

Sequences, series, and interchange of limits; metric spaces, continuity, and compactness; differentiation and the calculus theorems — the mean value theorem, L’Hôpital, Taylor — with the \(n\)-dimensional upgrade; the Riemann integral and the fundamental theorem of calculus; Lebesgue measure and integration and the swap theorems; Banach and Hilbert spaces, \(L^2\), and orthonormal bases; complex analysis — analytic functions, power series, and contour integrals, then zeros, poles, residues, and ROCs; the Fourier, Laplace, and \(z\)-transforms as operators, with the property tables proved; distributions and \(\delta\), Heisenberg’s inequality, and the Hilbert transform, the analytic signal, and causality; dual spaces, Riesz representation, and functional derivatives; numerical linear algebra — conditioning, stability, factorizations, and iteration; transforms of distributions — moment generating and characteristic functions, the Chernoff bound, and the central limit theorem with proof.

60 exercises across Lessons 31–43 — 30 [Hand], 30 [Proof].

Exercises

Worked sets are linked below as they are completed.

  • Lesson 31 — Limits Done Right: Sequences, Series, and When Swapping Is Legal — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 32 — Metric Spaces, Continuity, and Compactness — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 33 — Differentiation: The Calculus Theorems, Done Right — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 34 — The Riemann Integral and the Fundamental Theorem — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 35 — Integration Done Right: Lebesgue, Almost Everywhere, and the Swap Theorems — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 36 — Signal Space: Banach and Hilbert Spaces, \(L^2\), and Orthonormal Bases — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 37 — Complex Analysis I: Analytic Functions, Power Series, and Contour Integrals — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 38 — Complex Analysis II: Zeros, Poles, Residues, and Why ROCs Work — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 39 — The Fourier, Laplace, and \(z\)-Transforms as Operators: The Property Toolbox, Proved — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 40 — Distributions: Making \(\delta\) Honest — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 41 — Dual Spaces, Riesz Representation, and Functional Derivatives — 4 exercises (2 [Hand], 2 [Proof])
  • Lesson 42 — Numerical Linear Algebra: Conditioning, Stability, Factorizations, and Iteration — 12 exercises (6 [Hand], 6 [Proof])
  • Lesson 43 — Transforms of Distributions: Moment Generating Functions, Characteristic Functions, and the Central Limit Theorem — 4 exercises (2 [Hand], 2 [Proof])