# Varun ..University of Lucknow · Department of Mathematics and Astronomy

Varun ..

M.Sc. NET-JRF GATE

## About

15

Publications

3,228

Reads

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6

Citations

Citations since 2017

Introduction

**Skills and Expertise**

Additional affiliations

January 2017 - present

Position

- PhD Student

Description

- I am research scholar at University of Lucknow. I am working on different types of interpolation techniques to analyse and approximate analytic and continous function in a unit disk via interpolatory polynomial. I am also interested in the applications of interpolation on the unit circle and numerically justify the works done in this direction.

## Publications

Publications (15)

The paper is devoted to the study of a Pal type (0;1) interpolation problem on the unit circle considering two disjoint sets of nodes. The nodal points are obtained by projecting vertically the zeros of the Jacobi polynomial P _n^{(α,β)}(x) and its derivative P _n^{(α,β)'}(x) , together with ±1 onto the unit circle. The Lagrange data are prescribed...

This research article aims to staunchly study the approximation using Lagrange interpolation on the unit circle. Nodal system constitutes the vertically projected zeros of Jacobi polynomial onto the unit circle with boundary points at ±1. Moreover, convergence is obtained by considering analytic functions on a suitable domain accompanied by some nu...

The paper is devoted to the study of a Pál type (0; 1) interpolation problem on the unit circle considering two disjoint sets of nodes. The nodal points are obtained by projecting vertically the zeros of the Jacobi polynomial P_n^(α, β)(x) and its derivative P_n^(α, β)' (x), together with ±1 onto the unit circle. The Lagrange data are prescribed on...

The aim of this paper is to study the approximation of functions using a higher order Hermite-Fejer interpolation process on the unit circle. The system of nodes is composed of vertically projected zeros of Jacobi polynomials onto the unit circle with boundary points at $ \pm1 $. Values of the polynomial and its first four derivatives are fixed by...

In this paper we have constructed a non-interpolatory spline on the unit circle. The rate of convergence and the error in approximation corresponding to the complex valued function has been considered.

In this presentation, authors brought into consideration the set of non-uniformly distributed nodes on the unit circle to investigate a Pal-type (1;0) interpolation problem in account with Hermite-Fej´er boundary condition. These nodes are obtained by projecting vertically the zeros of Jacobi polynomial as well as zeros of its derivative onto the u...

In this research manuscript, authors brought into consideration the set of non-uniformly distributed nodes on the unit circle to investigate a Pál-type (1;0) interpolation problem in account with Hermite-Fejér boundary condition. These nodes are obtained by projecting vertically the zeros of Jacobi polynomial as well as zeros of its derivative onto...

This paper deals with the Hermite-Fej´er interpolation problem on the
unit circle with the nodal system containing the vertically projected zeros of Jacobi’s polynomial with boundary points on the unit circle. We worked upon three
nodal structures throughout this paper and obtained rate of convergence for each
case. Moreover, we did a comparison of...

This paper deals with the Hermite-Fejér interpolation problem on the unit circle with the nodal system containing the vertically projected zeros of Jacobi's polynomial with boundary points on the unit circle. We worked upon three nodal structures throughout this paper and obtained rate of convergence for each case. Moreover, we did a comparison of...

In this research manuscript, authors brought into consideration the set of non-uniformly distributed nodes on the unit circle to investigate a Pal-type (1;0) interpolation problem in account with Hermite-Fejer boundary condition. These nodes are obtained by projecting vertically the zeros of Jacobi polynomial as well as zeros of its derivative onto...

In this paper we have constructed a non-interpolatory spline on the unit circle. The rate of convergence and the error in approximation corresponding to the complex valued function has been considered.

The aim of this paper is to study an interpolation problem ,which is an intermediate problem between Lagrange and Hermite. We consider this problem on the nodes obtained by projecting vertically the zeroes of the (1-x^2 )P_n (x) onto the unit circle, where P_n (x) stands for n^th Legendre polynomial. We prove the regularity of the problem, give exp...

The aim of this paper is to study an interpolation problem ,which is an intermediate problem between Lagrange and Hermite. We consider this problem on the nodes obtained by projecting vertically the zeroes of the (1-x^2 )P_n (x) onto the unit circle, where P_n (x) stands for n^th Legendre polynomial. We prove the regularity of the problem, give exp...

The aim of this paper is to study a Lagrange-Hermite interpolation on the nodes, which are obtained by projecting vertically the zeroes of the (1-x^2 )P_n (x) on the unit circle, where P_n (x) stands for n^th Legendre polynomial. We prove the regularity of the problem, give explicit forms and establish a convergence theorem for the same.

## Questions

Questions (5)

I wanted to know who went out of box and thought of choosing the interpolation nodes other than the real interval. Also, please attach the research article in this context.

I am aware of the facts that every totally bounded metric space is separable and a metric space is compact iff it is totally bounded and complete but I wanted to know, is every totally bounded metric space is locally compact or not. If not, then give an example of a metric space that is totally bounded but not locally compact.

Follow this question on the given link

It is cited in various research papers as

**Fejér, L.: Über Interpolation. Gött. Nachr, 66–91 (1916).**

Creating

**interpolating polynomial on the unit circle**by projecting the zeros of certain polynomials vertically on the unit circle as well as**reading their convergence behaviour**for the functions analytic inside the unit circle can how be seen for the**application part in the real life.**Where such type of interpolation can be applied to view a much wider perspective to such type of research problems.

Approximation theory of interpolation is of foundational importance in numerical analysis especially for various scientific computing problems.

Considerable amount of literature got accumulated on Lagrange, Hermite, Lacunary and Pal-type interpolation in past few years. Working out the interpolation on the real line has seen a numerical justification by many researchers, but

*on the complex plane, particularly I would say the unit disk hasn't seen much of the justification done numerically by the use of different programming platforms.*I would request other researchers who are part of this discussion to help me find out some useful papers in such direction.

I am also currently working through programming platform

**to view out numerical aspects of my research works.***MATHEMATICA*Hope to see you guys with some good results in future discussion.

## Projects

Project (1)