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Convener(s) |
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| Name: | Prof. A. Swaminathan | Prof. B. Praba | Dr. A. Sathish Kumar | Dr. S. Yugesh |
| Mailing Address: | Professor Deparment of Mathematics IIT Roorkee, Roorkee 247667 |
Professor & Head Department of Mathematics SSN College of Engineering, Kalavakkam , Tamil Nadu. 603 110 |
Assistant Professor Deparment of Mathematics IIT Madras, Chennai 600 036 |
Assistant Professor Department of Mathematics SSN College of Engineering, Kalavakkam ,Tamil Nadu, 603 110 |
| Email: | a.swaminathan at ma.iitr.ac.in | prabab at ssn.edu.in | sathishkumar at iitm.ac.in | yugeshs at ssn.edu.in |
Real Analysis forms the foundation of modern mathematics and plays a central role in various branches of analysis, with significant applications in approximation theory and learning theory. A solid understanding of real analysis is essential not only for exploring the structure and properties of functions and spaces, but also for developing rigorous methods in approximation and data-driven learning.
The instructional school aims to provide participants with a guided tour of the core concepts in Real Analysis and their connections to Approximation Theory and Learning Theory. It will also introduce advanced tools and applications such as best approximation, Padé approximation and real analytic functions, Shannon’s sampling theorem, convergence of sampling operators, the universal approximation theorem, reproducing kernel Hilbert spaces, and convergence analysis of regularized learning algorithms.
The topics are chosen to give participants exposure to both classical and modern perspectives in analysis and its applications, thereby strengthening their background for further research in mathematics, applied analysis, and data science.
Dates:
Venue:
Venue Address:
Sri Sivasubramaniya Nadar College of Engineering
Rajiv Gandhi Salai(OMR), Kalavakkam – 603 110, Tamil Nadu.
Venue State:
Venue City:
PIN:
Syllabus:
|
Name of the Speaker with affiliation |
No. of Lectures |
Detailed Syllabus |
|
Dr. B. Sriram |
6 |
Review of the real number system, Infimum and Supremum, Euclidean spaces and Metric spaces, Open and closed sets, Limit points, Bolzano-Weierstrass theorem, liminf and limsup of a sequence. |
|
Dr. Surjit Kumar |
6 |
Compactness, Characterization of compact sets, Heine Borel theorem, Cauchy sequence and Complete metric space |
|
Dr. G. Arunkumar |
6 |
Limits of functions, Continuous functions, Types of discontinuities, Uniform continuity, Differentiable functions and Mean value theorem, Convergence of sequences and series of numbers. |
|
Prof. A. Swaminathan |
6 |
Sequences and series of functions, Weierstrass M-test, Uniform convergence and its relation to continuity, Differentiation, and integration, Continued fractions, Fundamental results, Convergence, Best approximation and applications, Pade approximation and Real analytic functions. |
|
Dr. A. Sathish Kumar |
6 |
Approximation of real valued functions, Bernstein operators and its properties, Weierstass approximation theorem, Korovkin theorem, Shannon sampling theorem, Convergence of sampling operators and Universal approximation theorem. |
|
Prof. S. Sivananthan |
6 |
Reproducing kernel Hilbert space, Gaussian kernel and their RKHSs, Mercer’s theorem, Probabilistic inequalities, Tikhonov-type regularization, Representer theorem and Convergence analysis of regularized learning algorithm. |
References:
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L. Lorentzen and H. Waadeland, Continued Fractions, Volume 1: Convergence Theory, Atlantis Studies in Mathematics, 2nd edition, World Scientific, 2008.
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H. N. Mhaskar and D. V. Pai, Fundamentals of Approximation Theory, Narosa Publishing House, 2007.
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W. Rudin, Principles of Mathematical Analysis, Mcgraw-Hill, 1976.
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G. F. Simmons, Topology and Modern Analysis, Kreiger, 2003.
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D. F. Walnut, An Introduction to Wavelet Analysis, Birkhauser Boston, MA, 2003.
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D-X. Zhou and F. Cucker, Learning Theory: An Approximation Theory Viewpoint, Cambridge University Press, 2007.
Name of the tutors:
|
S. No. |
Name |
Affiliation |
|
1 |
Dr. S. Yugesh |
Assistant Professor, Department of Mathematics, |
|
2 |
Dr. M. Sundarakannan |
Assistant Professor, Department of Mathematics, |
|
3 |
Mr. Nitin Bartwal |
Research Scholar, Department of Mathematics, |
|
4 |
Ms. Puja Sonawane |
Research Scholar, Department of Mathematics, |
|
5 |
Mr. Arpan Kumar Dey |
Research Scholar, Department of Mathematics, |
|
6 |
Mr. M. Surya |
Research Scholar, Department of Mathematics, |
Time Table:
|
Day |
Date |
Lecture 1 (9.30–11.00) |
Tea (11.00 to |
Lecturer 2 (11.30–01.00) |
Lunch (1.00 |
Tutorial 1 (2.30-3.30) |
Tea (3.30 to |
Tutorial 2 (4.00-500) |
Snacks (5.00 |
|
|
|
|
|
||||||
|
(Speaker’s name) |
(Speaker’s name) |
(Speaker’s name) |
(Tutor’s name) |
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|
Mon |
01.06.26 |
BS |
SK |
BS, NB, AKD |
SK, NB, AKD |
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|
Tue |
02.06.26 |
BS |
SK |
BS, NB, AKD |
SK, NB, AKD |
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|
Wed |
03.06.26 |
BS |
SK |
MSK, MS, PS |
MSK, MS, PS |
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|
Thu |
04.06.26 |
BS |
SK |
MSK, MS, PS |
MSK, MS, PS |
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|
Fri |
05.06.26 |
GA |
AS |
GA, SY, MSK |
MSK, SY, PS |
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|
Sat |
06.06.26 |
GA |
AS |
GA, SY, MSK |
AS, SY, MSK |
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|
Sunday |
|||||||||
|
Mon |
08.06.26 |
AS |
Tea (11.00 |
GA |
Lunch (1.00 |
AS, NB, AKD |
Tea (3.30 to |
PS, NB, AKD |
Snacks (5.00
|
|
Tue |
09.06.26 |
AS |
GA |
SY, MS, PS |
SY, MS, PS |
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|
Wed |
10.06.26 |
ASK |
SS |
NB, PS, MS |
SS, MS, AKD |
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|
Thu |
11.06.26 |
ASK |
SS |
SY, NB, AKD |
SS, NB, AKD |
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|
Fri |
12.06.26 |
SS |
ASK |
PS, NB, AKD |
ASK, NB, AKD |
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|
Sat |
13.06.26 |
SS |
ASK |
ASK, SY, MS |
SY, MS, PS |
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Full forms for the abbreviations of speakers and tutors:
AS: Prof. A. Swaminathan
SS: Prof. S. Sivananthan
BS: Dr. B. Sriram
SK: Dr. Surjit Kumar
GA: Dr. G. Arunkumar
ASK: Dr. A. Sathish Kumar
SY: Dr. S. Yugesh
MSK: Dr. M. Sundarakannan
NB: Mr. Nitin Bartwal
PS: Ms. Puja Sonawane
AKD: Mr. Arpan Kumar Dey
Selected Applicants:
How to Reach:
TBA