Some problems in bivariate interpolation and approximation

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A THESIS ON SOME PROBLEM IN

BIVARIATE INTERPOLATION AND APPROXIMATION

By

Jayarajan

Department of Mathematics Indian Institute of Technology

New Delhi

Submitted to the Indian Institute of Technology, New Delhi for the award of the Degree of Doctor of Philosophy

in Mathematics 1974

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'CERTIFICATE

This is to certify that the thesis entitled "On Some Problem in Bivariate Interpolation and Approximation"

which is being submitted by Mr. N. Jayarajan for the award of Doctor of Philosophy in Mathematics to the Indian

Institute of Technology, New

Delhi, is

a record of bonafide research work. He has worked for the last three years

under my guidance and supervision.

The thesis has reached the standard fulfilling the requainents of the regulations relating to the degree.

The results obtained in this thesis have not been submitted to any other University or Institute for the award of any degree

or

diploma,

(I4M. Cliawla) Assistant Professor Department of Mathematics • Indian Institute of Technology Hauz Khasi New Delhi-110029

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ACKNOWLEDGEMENTS

I express my sincere gratitude to Dr. M.,M„. thawla, Assistant Professor, Department of Mathematics, Indian Institute of Technology, Delhi under whose supervision and guidance I have done this work. His valuable suggests tions and constant encouragement have given me great

inspiration in carrying out this work.

I am very grateful to Professor M.K. Jain for his

suggestions and encouragement. I thank sincerely Professor K.-R. Parthasarathy, Head of the Department of Mathematics,

Delhi for providing all the facilities in the department in connection with this work and Professor

kW.- Singh for his keen interest in the progress of nay work.

I am thankful to the authorities of the Indian Institute of Technology, Delhi for providing me with all the necessary facilities in carrying out this work.

I acknowledge with thanks all the assistance rendered to me by the staff of the Computer Centre,

I also wish to express nay gratitude to Professor K.,M.!

Bahauddin, Principal, Regional. Engineering College, Calicut for arranging study leave from the college and for the keen interest he has shown in the progress of Iry work.'

My thanks are due to rry colleagues Mr. T.R. Ramakrishnan and Miss Veena Kaul for their co—operation, to Mr. Verma for helping me in the correction of the manuscript and to Mr. Dev Raj Joshi for typing the manuscript with great care.

1-41r9-' Jayarajan)

Department of Mathematics Indian Institute of Technology Hauz Khas, New Delhi,44110029;

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Page SYNOPSIS

CHAPTER 1 BIVARIATE CARDINAL INTERPO-

LATION AND CUBATURE FORMULAS 1 31

1.1 Introduction 1

1.2 Bivariate Cardinal Interpo- lation Series and its

Convergence 5

43 Cubature Formulas for co co

du ftect u)dx 15 oo moo

44 Cubature Formulas for 00 00

xsu)

du if dx u 24

0.#

References 31

CHAPTER II A GENERALIZATION OF HERIVITE'S INTERPOLATION

^_

FORMULA • N TWO VARIABL

^ES

2.1 Introduction 32

2.2 The Interpolation Formula 35 2.3 Illustration 45 2.4 The Error of Interpolation 47

References 54

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CHAPTER III OPTIMAL APPROXIMATIONS OVER HILBERT SPACES OF BIVARIATE

FUNCTIONS 55 — 106

3,1 Introduction 55

3, 2 Optimal Approximations with

Polynomial Precision 58 3•'3 Determination and Characteriza-

tion of the Optimal Rule 4 rn t n)*

3,4 Computation of the Optimal y n)*

Rules 70

MN

61 3..5 Er ror Bounds ' for the Optimal ( nit

Rules Qui n).

3,6 The Hilbert S paces Hid( C x Cr r and LAC xC ) , ri r2

3.7 Optimal Rules with Total Poly- nomial Precision

3. 8 Det ermlnation of the Approximate Rules Q

^(d)

3,.,9 computation of the Optimal ( d)*

Rules QN

3,10 Error Bounds for the Optimal Rules Q(Nd)*

References

72 80 89

92 99 1 ⁰⁰

105

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