- Course Instructor: Aditya Tatu, FB-1, Room 1206, Ext: 540

Course credits: 3-0-0-3

Schedule: Mon,Wed: 09:30 - 10:30, Thu: 08:30 - 09:30.

Venue: CEP 205

## Weekly outline

### 26 July - 1 August

- Lecture 1: Introduction
- Reading material:

Chapter 2 from:

Evolution of Applied Harmonic Analysis: Models of the Real World

- Elena Prestini. - Brief lecture notes
- Lecture 2: Sets and Relations
- Brief lecture notes
- Assignment 1: Sets and relations

Due date: August 4, 2010 (Wednesday)

- Lecture 3: Equivalence relations and Metric spaces
- Brief lecture notes

- What is the size of a set?

Difference between finite, countably infinite and uncountably infinite: Cantor's diagonal argument.

### 2 August - 8 August

- Lecture 4: Metric space and Continuous functions
- Brief lecture notes

- Lecture 5: Metric spaces and Continuous functions between general metric spaces
- Brief lecture notes

- Assignment 2: Metric spaces.

Due date: 11 August 2010 (Wednesday)

- Lecture 6: Continuity and Convergence
- Brief Lecture Notes

### 9 August - 15 August

- Lecture 7: Convergence in a metric space
- Brief lecture notes

- Extra reading material:

Following book uses the concept of convergence in the analysis of numerical schemes to solve PDEs. One should use a scheme that is convergent. Convergence in the field of numerical analysis is explained in Section 2.6. The necessary background is built in Chapter 1 and part of Chapter 2 before Section 2.6.

- Chapter 1 and 2 from: Numerical Solutions of PDE: An introduction, by Morton and Mayers

- Interesting book:

Proofs and Refutations: The logic of mathematical discovery.

Imre Lakatos

Cambridge University Press. - Note:

One copy of the following book has been put into the Course Reserve section of the RC, while the 5 remaining copies will soon be put in the Short Loan section:

Introductory functional analysis with applications. New York. John Wiley and Sons

Erwin, Kreyszig

- Lecture 8: Convergence of a sequence of functions

- Brief lecture notes

- Lecture 9: Example of a Complete space

- Brief lecture notes

### 16 August - 22 August

- Lecture 10: Vector spaces
- Brief lecture notes

- Lecture 11 & 12: Subspaces and Basis
- Brief lecture notes

### 23 August - 29 August

- Lecture 13: Normed vector spaces
- Brief lecture notes

- Lecture 14-15: Completion of NVS, L
^{p}spaces. - Brief lecture notes

- Assignment 3: Normed Vector Spaces

Due date: Monday, 13th of September

Solutions to Assignment 2:

### 30 August - 5 September

- Reference for Lebesgue Integral and L
^{p }spaces:

The Lebesgue- Stieltjes Integral: A Practical Introduction

by M.Carter and B.van Brunt

Other books on Lebesgue integrals, real analysis, measure theory and functional analysis may also cover the relevant topics.

- Exam week

### 6 September - 12 September

- Lecture 16: l
^{p}, L^{\infty}and l^{\infty}spaces - Brief lecture notes

- Lecture 17: Linear Systems
- Brief lecture notes

- Lecture 18: Continuity and Boundedness of a Linear operator
- Brief lecture notes

- Interesting paper: How to Read a Paper by S. Keshav

### 13 September - 19 September

- Lecture 19: Operator norm
- Brief lecture notes

- Lecture 20 & 21: Solved examples on convergence in metric and NVS.
- These notes contain a minor correction in the alternative proof for Q.2 and the complete proof of Q.3

### 20 September - 26 September

- Solutions of Assignment 3

Lecture 22: Convergence of sequence of functions

- Brief lecture notes

- Interesting applications using the concept of convergence:
- Suggested by: Pratik Shah

- Lecture 23: Inner Product Spaces
- Brief lecture notes

- Lecture 24: Hilbert spaces
- Brief lecture notes

### 27 September - 3 October

- Assignment 4: Optional submission, will not be graded.

Lecture 25: Gram-Schmidt Orthonormalization, Orthonormal basis, Best Approximation theorem- Brief lecture notes

- Lecture 26: Parseval's relation and theorem as special cases of Pythagoras theorem
- Brief lecture notes

- Lecture 27: Convex optimization
- Brief lecture notes

### 4 October - 10 October

- Term paper/project: Splines

Submission deadline: 15 November 2010

- Reference papers:

Solutions to Assignment 4

Lecture notes for Linear Algebra portion of the course will not be put up.

Please directly refer to Linear algebra and its applications by Gilbert Strang.- Lecture 28: Linear equations: Row and Column picture. Interpreting the solution of linear equations

- Lecture 29: Solution of linear equations using Gaussian Elimination

- Lecture 30: Gauss Jordan method of finding inverse of a matrix

### 11 October - 17 October

- In-sem Question paper

Lecture 31: In-sem solutions

Assignment 5: No submissions. Solutions to be discussed in Tutorial class on Friday if required.

### 18 October - 24 October

- Lecture 32: Rectangular system of linear equations, Non- Singular, Inconsistent and Under-determined system of linear equations.

- Lecture 33: Rank of a matrix, Row space, Column space, Nullspace and Left nullspace with their dimensions.

- Lecture 34: Fundamental theorem of Linear algebra, Inverses and Moore- Penrose Pseudo-inverse of a matrix.

### 25 October - 31 October

- Lecture 35: Least square filters - Using Calculus

Lecture 36: Least square filter design - Using projection

- Lecture 37: QR decomposition

- Linear-Phase FIR Filter Design By Least Squares

- by Ivan Selesnick - I expect the Communication specialization students to help others (incl. me) understand these notes.

Spline project report LATEX file:

### 1 November - 7 November

- Diwali break

### 8 November - 14 November

- Lecture 38: Eigenvalues and Eigenvectors with applications to solutions of Diff. eqns.

- Lecture 39: Diagnolization of a matrix, Eigenvalues related to invertibility and Eigenvectors related to diagnolization, Eigenfunctions of LTI systems.
- Submitting the spline project:

Submit single zip file. Only one file per group.

The main directory should contain a readme.txt file explaining the contents of all files (in one line each). The readme.txt file should also contain the names and Id's of the students in the group.

The main directory should contain a pdf report of your project (max 2 pages) and your matlab codes in a sub-directory.

The main matlab file should be named spline2dclosed.m

Lecture 40: Similarity transformations

### 15 November - 21 November

- Lecture 41: Differentiability, differential and the derivative

### 22 November - 28 November

- Final exam question paper:

Solutions to exam questions:

Your answer sheets will be available on Thursday 2nd December 2010 in CEP 205 from 11 am to 11:30 am. Please pass this message to all your colleagues.

### 29 November - 5 December