Part VII — The Analysis Behind the Transforms

Booklet: Abridged Math Foundations for Signals and Systems, Lessons 45–52 (PDF).

Sequences, series, and interchange of limits; Lebesgue measure and integration and the swap theorems; Banach and Hilbert spaces, \(L^2\), and orthonormal bases; complex analysis — poles, ROCs, and inversion integrals; distributions and \(\delta\); dual spaces, Riesz representation, and functional derivatives; numerical linear algebra — conditioning and stability.

28 exercises across Lessons 45–51 — 14 [Hand], 14 [Proof]. Lesson 52 is Capstone VII (the certification lab, no exercise block).

Exercises

Worked sets are linked below as they are completed.

• Lesson 45 — Limits Done Right: Sequences, Series, and When Swapping Is Legal — 4 exercises (2 [Hand], 2 [Proof])
• Lesson 46 — Integration Done Right: Lebesgue, Almost Everywhere, and the Swap Theorems — 4 exercises (2 [Hand], 2 [Proof])
• Lesson 47 — Signal Space: Banach and Hilbert Spaces, \(L^2\), and Orthonormal Bases — 4 exercises (2 [Hand], 2 [Proof])
• Lesson 48 — Complex Analysis: Why Poles, ROCs, and Inversion Integrals Work — 4 exercises (2 [Hand], 2 [Proof])
• Lesson 49 — Distributions: Making \(\delta\) Honest — 4 exercises (2 [Hand], 2 [Proof])
• Lesson 50 — Dual Spaces, Riesz Representation, and Functional Derivatives — 4 exercises (2 [Hand], 2 [Proof])
• Lesson 51 — Numerical Linear Algebra: Conditioning, Stability, and Structure — 4 exercises (2 [Hand], 2 [Proof])
• Lesson 52 — Capstone VII: The Certification Lab — project