ECE 735 Signal Synthesis and Recovery Techniques, Fall 2006

9/5     Tuesday.  Course overview.  For precise definition of field and
	vector space, see p. 1.  For simple vector spaces, see Section 1.3 and
	left-hand column of p. 4.  Def. of inner product space and Cauchy-Schwarz
	inequality p. 19.  Def. of norm top of p. 16.  Def. of metric see
	first paragraph of Section 4.

9/7	Section 1.4, part of Section 1.5: Span, linear independence,
	basis, dimension.  Worked Problems 2-4, 2-6.  Discussed Theorem 2.9.
	Introduced least upper bound axiom.
	Assigned HW#1: 1-1 (see proof of Lemma 1.2), 2-5,
	2-7, 2-9 DUE THU 9/14.

9/12	Defined supremum, infimum.  Proved Bolzano-Weierstrass Theorem.
	Started Section 4 Metric Spaces.  Skipped proofs of Theorems 4.13
	and 4.14.

9/14	Covered Def. 4.15 (boundary).  Covered Section 4.3 Sequences and
	Convergence in Metric Spaces through Theorem 4.26.  Gave different
	proofs of Theorems 4.19 and 4.20.  Download variation on new proofs.
	Assigned HW#2: 4-3, 4-4, 4-7, 4-8 4-10 DUE THU 9/21.
	NOTE: In Problem 4-8, the formula for the closed ball should
	read {x ∈ Q : |x| ≤ r }.

9/19	Discussed the completeness of Rd.  Discussed upper
	and lower bounds on Euclidean distance.  Theorem 4.28: Closed
	and bounded subsets of Rd are sequentially compact.
	Covered Section 4.4 Fixed Points and Contraction Mappings.

9/21	Covered Section 4.5 Continuous Functions.  Worked Problem 4-18.
	Started Section 4.6 Compact Sets.  Proved Finite Intersection
	Property Theorem 4.38.  Proved Lemma 4.40, Corollary 4.41, and
	Lemma 4.42.  Did not prove Lemma 4.43.  Defined uniform continuity.
	Assigned HW#3: 4-13, 4-14, 4-16(⇒ only), 4-17, 4-19 DUE THU 9/28.

9/26	Went over solutions to some problems in HW#2.  Proved Theorem 4.47.
	Started Section 5 Normed Vector Spaces.  Covered Section 5.1
	Projections - Introduction.  Gave lots of examples.

9/28	Solved Problem 5-2(b)(c).  Proved Lemma 5.4.  Discussed Corollary 5.5
	and Theorem 5.6.
	Assigned HW#4: 5-3, 5-5, 5-6, 5-7, 6-3, 6-5, 6-6, 6-10 DUE THU 10/5.

10/3	Covered Section 5.3 Projections onto Closed Finite-Dimensional Subsets.
	Proved Th. 5.8, 5.9, 5.10.  Started Section 6.  Discussed Problem 6-3,
	parallelogram law, polarization identity, orthogonal complement,
	Problem 6-11 and Remark following it.  Derived the Orthogonality
	Principle (Th. 6.7).  Discussed projections onto finite-dimensional
	subspaces in Section 6.1.

10/5	Discussed (6.7) and (6.8).  Covered Section 6.2 Projections onto
	Infinite-Dimensional Spaces, Section 6.3 The Projection Theorem.
	Derived Proposition 6.13 (inner-product characterization of best
	approximation from a convex set).
	Assigned HW#5: 6-11, 6-12, 6-13, 6-16, 6-17 DUE THU 10/12.

10/10	Proved Strict Separation Th. 6.14.  Covered Sections 7.1 and 7.2.

10/12	Started Section 7.3 Adjoint Operators.  Skip Lemma 7.19, Lemma 7.20,
	Problem 7-32, 7-33.  Briefly discussed Section 7.3.1 Ellipses.
	Quickly went over Section 7.3.2 Projections onto range A.
	Assigned HW#6: 6-23, 6-26, 6-29, 6-32, 6-33, 7-8 DUE THU 10/19.

10/17	Section 7.4 Solving Linear Equations.  Introduced Gaussian Quadrature
	and distributed handout.  Recommend reading the proof of Theorem 9;
	use the Gram-Schmidt formula (6).

10/19	Discussed approximation of functions, use of Matlab functions polyfit
	and polyval.  Distributed handout on Piecewise Polynomial Functions
	and discussed through Section 2.1.
	Assigned HW#7: 7-24, 7-25, 7-26, 7-34, 7-35, 7-41 DUE THU 10/26.
	See additional notes.

10/24	Introduced the B-splines in Section 2.2 of the handout.  Showed how
	to use splinetool, export splines, use fnder and fnint for splines.
	Started discussion of Section 8: Spectral Theory.

10/26
	Exam 1 Review Problems: 2-8, 6-31, 6-34, 6-35, 7-3, 7-4, 7-6,
	7-39, 7-43, 7-44, 7-45, 7-46.  Solutions.

10/31	Exam 1.  Example showing linear independence.  Example showing
	how the trace problem arises.

11/2	Went over Exam 1.  Proved Lemmas 8.2 and 8.3.
	Assigned HW#8: Matlab problem on handout and 8-1, 8-2, 8-3,
	8-9, 8-10, 8-11 DUE THU 11/9.

11/7	Proved the Spectral Theorem.  Covered matrix example, sqrare-root
	of an operator application, Fredholm eq. of the 2nd kind example.
	Stated Theorem 8.5 giving sufficient conditions for an operator's
	eigenfunctions to form a complete orthonormal set.

11/9	Discussed HW#9, how to efficiently implement quadrature in Matlab.
	Proved Theorems 8.5, 8.6, and the SVD.
	Assigned HW#9: Matlab problem on handout and 8-12, 8-14, 8-17
	DUE THU 11/16.  legendrequad01.m

11/14	Additional material on approximation of functions.  See writeup.
	Matlab scripts discussed: lincmb.m, smpthm.m, aprxscript.m, stscript.m
	Discussed Fredholm equations of the 2nd kind and regularization.

11/16	Briefly discussed function approximation and related it to the SVD
	and pseudoinverse.  Started Section 9.  Covered Sections 9.1, 9.2,
	and 9.4.  We are skipping Section 9.3.
	Assigned HW#10: 8-18, 9-1, 9-2, 9-3, 9-5, 9-6, 9-8 DUE THU 11/30.

11/21	Covered Section 9.5 Kuhn-Tucker Sufficiency Conditions, Section 9.6
	More on Gateaux derivatives.  Skip Section 9.7.  Section 9.8 only
	the part after the proof of Lemma 9.6 to the end of the section.
	Section 9.9 The Frechet Derivative.

11/23	THANKSGIVING -- NO CLASS

11/28	Proved Theorem 9.30.  Skipped Section 9.10.  Covered Section 9.11
	The Chain Rule and the Product Rule.

11/30	Section 9.12 Applications.  Section 9.13 Functions Concentrated in
	Time and Frequency - thru (9.41) (prolate spheroidal wave functions).
	Assigned HW#11: 9-12, 9-14, 9-15, 9-16, 9-17 DUE THU 12/7.

12/5	Almost finished Section 9.13.

12/7	Worked on minimizing the out-of-band energy of time-limited signals.
	Distrbuted teaching evaluations.

12/12	Gave a precise definition of the prolate spheroidal wave functions.
	Discussed sampled version of integral equation for \psi_k.  Derived
	the sampling theorem and showed both pointwise and norm convergence.

12/14	Discussed Hilbert-Schmidt operators.
	Review Problems: 7-8, 7-16, 7-28, 7-42, 8-6, 8-7, 9-22, 9-23.
	Also 8-20 FROM TODAY's HANDOUT.

12/18	MONDAY, Final Exam, 12:25 pm in 3534 EH.