Vishnu Narayan Mishra_ICFA 2021

Biography
Prof. Vishnu Narayan Mishra Indira Gandhi National Tribal University, India
Title: Some approximation properties of a new class of Linear Operators
Abstract: Approximation theory is the branch of mathematics which studies the process of approximating general functions by simple functions such as polynomials, finite elements or Fourier series. Approximation processes arise in a very natural way in many problems dealing with the constructive approximation of functions as well as solutions to (partial) differential equations and integral equations. The study of such subject falls into an intensive research area, developing in different directions by many mathematicians. Several investigations have been devoted to the approximation properties of new sequences of operators, which might generalize or modify well-known ones, in order to get better results. Issues related to these studies are, for instance, shape preserving properties of the approximating operators, estimates of the rate of convergence, asymptotic formulae, saturation problems, approximation of semigroups of operators, asymptotic behavior, direct, and converse results. Several approximation processes have been successfully applied for example in Computer Aided Geometric Design, in the theory of artificial neural networks, and in evolution problems arising in population genetics, financial mathematics, and other fields. The goal of this talk is to attract researchers as well as scientists who are working in the recent advances in operator methods in approximation theory and related applications. Potential topics of this talk include but are not limited to the following: · Approximation by positive operators · Approximation by linear/nonlinear operators · Approximation by integral operators · Rate of convergence and moduli of smoothness · Simultaneous approximation · Approximation problems for semigroups of operators and evolution equations · Multidimensional problems · Abstract approximation theory · Quantum & Post Quantum Calculus in Approximation Theory The theory of summability arises from the process of summation of series and the significance of the concept of summability has been strikingly demonstrated in various contexts e. g. in Analytic Continuation, Quantum Mechanics, Probability Theory, Fourier Analysis, Approximation Theory and Fixed Point Theory. The methods of almost summability and statistical summability have become an active area of research in recent years. This short monograph is the first one to deal exclusively with the study of some summability methods and their interesting applications. We consider here some special regular matrix methods as well as non-matrix methods of summability. Broadly speaking, signals are treated as functions of one variable and images are represented by functions of two variables. Positive approximation processes play an important role in Approximation Theory and appear in a very natural way dealing with approximation of continuous functions, especially one, which requires further qualitative properties such as monotonicity, convexity and shape preservation and so on. Analysis of signals or time functions is of great importance, because it conveys information or attributes of some phenomenon. The engineers and scientists use properties of Fourier approximation for designing digital filters. In this talk, we discuss the basic tools of approximation theory & determine the error (degree) in approximation of a signal (function) by different types of positive linear operators in various Function spaces like as in Lp-spaces. During this talk, few applications of approximations of signals will also be highlighted. Approximation processes arise in a very natural way in many problems dealing with the constructive approximation of functions as well as solutions to (partial) differential equations and integral equations. The study of such subject falls into an intensive research area, developing in different directions by many mathematicians. Several investigations have been devoted to the approximation properties of new sequences of operators, which might generalize or modify well-known ones, in order to get better results. Issues related to these studies are, for instance, shape preserving properties of the approximating operators, estimates of the rate of convergence, asymptotic formulae, saturation problems, approximation of semigroups of operators, asymptotic behavior, direct, and converse results. Several approximation processes have been successfully applied for example in Computer Aided Geometric Design, in the theory of artificial neural networks, and in evolution problems arising in population genetics, financial mathematics, and other fields. The goal of this talk is to attract researchers, engineers as well as scientists who are working in the recent advances in operator methods in approximation theory and related applications. The study of sequence spaces occupies a very prominent position in analysis. The convergence problems have always been of great interest. The theory of sequence spaces has widely used in several branches of mathematics such as the structural theory of topological vector spaces, law of large numbers and the theory of functions. It has a significant contribution in enveloping the classical summability theory via matrix transformations from the one sequence space to another sequence space. The study of sequence spaces came into existence by special results in the theory of summability. The concept of summability is the generalization of the concept of convergence. Summability is a theory of assigning the value to a series whose sequence of partial sums diverges. It is an extremely constructive area for the application of functional analysis. In 1890, Italian analyst Ernesto Cesàro was the first to deal with the sum of divergent series and defined Cesàro summation. Several alternative methods of assigning a value to an infinite series were invented by mathematicians; these are known as “summability methods”. Some of the most familiar methods of summability are those that are associated with the names of great mathematicians like Hölder summability, Abel summability, Borel summability, Nörlund summability, Riesz summability etc. References: 1. V.N. Mishra, K. Khatri, L.N. Mishra, Deepmala; Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators, Journal of Inequalities and Applications 2013, 2013:586. doi:10.1186/1029-242X-2013-586. 2. V.N. Mishra, H.H. Khan, K. Khatri, L.N. Mishra; Hypergeometric Representation for Baskakov-Durrmeyer-Stancu Type Operators, Bulletin of Mathematical Analysis and Applications, Volume 5 Issue 3 (2013), Pages 18-26. 3. V.N. Mishra, K. Khatri, L.N. Mishra; On Simultaneous Approximation for Baskakov-Durrmeyer-Stancu type operators, Journal of Ultra Scientist of Physical Sciences, Vol. 24, No. (3) A, 2012, pp. 567-577. 4. V.N. Mishra, K. Khatri, L.N. Mishra; Some approximation properties of q-Baskakov-Beta-Stancu type operators, Journal of Calculus of Variations, Volume 2013, Article ID 814824, 8 pages. http://dx.doi.org/10.1155/2013/814824 5. V.N. Mishra, K. Khatri, L.N. Mishra; Statistical approximation by Kantorovich type Discrete $q-$Beta operators, Advances in Difference Equations 2013, 2013:345, DOI: 10.1186/10.1186/1687-1847-2013-345. 6. V.N. Mishra, P. Sharma, L.N. Mishra; On statistical approximation properties of $q-$Baskakov-Sz\'{a}sz-Stancu operators, Journal of Egyptian Mathematical Society, Vol. 24, Issue 3, 2016, pp. 396-401. DOI: 10.1016/j.joems.2015.07.005. 7. A.R. Gairola, Deepmala, L.N. Mishra, Rate of Approximation by Finite Iterates of q-Durrmeyer Operators, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. (April–June 2016) 86(2):229–234 (2016). doi: 10.1007/s40010-016-0267-z 8. K.K. Singh, A.R. Gairola, Deepmala, Approximation theorems for $q-$ analouge of a linear positive operator by A. Lupas, Int. J. Anal. Appl. Vol. 12, No. 1, (2016), pp. 30-37. 9. A.R. Gairola, Deepmala, L.N. Mishra, On the $q-$derivatives of a certain linear positive operators, Iranian Journal of Science & Technology, Transactions A: Science, Vol. 42, No. 3, (2018), pp. 1409-1417. DOI 10.1007/s40995-017-0227-8. 10. R.B. Gandhi, Deepmala, V.N. Mishra, Local and global results for modified Sz\'{a}sz - Mirakjan operators, Math. Method. Appl. Sci., Vol. 40, Issue 7, (2017), pp. 2491-2504. DOI: 10.1002/mma.4171. 11. A. Kumar, Vandana, Approximation by genuine Lupa\c{s}-Beta-Stancu Operators, J. Appl. Math. Inf. Sci., Vol. 36, No. 1-2, (2018), 15-28. 12. A. Kumar, Vandana, Some approximation properties of generalized integral type operators, Tbilisi Mathematical Journal, 11(1) (2018), pp. 99–116. 13. A. Kumar, L.N. Mishra, Approximation by modified Jain-Baskakov-Stancu operators, Tbilisi Mathematical Journal, 10(2) (2017), 185–199. 14. V.N. Mishra, P. Patel, L.N. Mishra, The Integral type Modification of Jain Operators and its Approximation Properties, Numerical Functional Analysis and Optimization, (2018), DOI: 10.1080/01630563.2018.1477796. https://doi.org/10.1080/01630563.2018.1477796.
Biography: Associate Professor & Head Qualification : B.Sc.(Gold Medalist),M.Sc. (Double Gold Medalist), Ph.D. (I.I.T. Roorkee) Specialization : Linear Positive Operators, Approximation theory, Functional Analytic aspects (methods) in Summability, Fourier Approximation, Quantum Calculus, Asymptotic expansions, Fixed point theory and applications in dynamic programming, , Signal Analysis & Image processing.