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Current and historical research methods in approximation theory are
presented in this book beginning with the 1800s and following the
evolution of approximation theory via the refinement and extension
of classical methods and ending with recent techniques and
methodologies. Graduate students, postdocs, and researchers in
mathematics, specifically those working in the theory of functions,
approximation theory, geometric function theory, and optimization
will find new insights as well as a guide to advanced topics. The
chapters in this book are grouped into four themes; the first,
polynomials (Chapters 1 -8), includes inequalities for polynomials
and rational functions, orthogonal polynomials, and location of
zeros. The second, inequalities and extremal problems are discussed
in Chapters 9 -13. The third, approximation of functions, involves
the approximants being polynomials, rational functions, and other
types of functions and are covered in Chapters 14 -19. The last
theme, quadrature, cubature and applications, comprises the final
three chapters and includes an article coauthored by Rahman. This
volume serves as a memorial volume to commemorate the distinguished
career of Qazi Ibadur Rahman (1934-2013) of the Universite de
Montreal. Rahman was considered by his peers as one of the
prominent experts in analytic theory of polynomials and entire
functions. The novelty of his work lies in his profound abilities
and skills in applying techniques from other areas of mathematics,
such as optimization theory and variational principles, to obtain
final answers to countless open problems.
Paul Butzer, who is considered the academic father and grandfather
of many prominent mathematicians, has established one of the best
schools in approximation and sampling theory in the world. He is
one of the leading figures in approximation, sampling theory, and
harmonic analysis. Although on April 15, 2013, Paul Butzer turned
85 years old, remarkably, he is still an active research
mathematician. In celebration of Paul Butzer's 85th birthday, New
Perspectives on Approximation and Sampling Theory is a collection
of invited chapters on approximation, sampling, and harmonic
analysis written by students, friends, colleagues, and prominent
active mathematicians. Topics covered include approximation methods
using wavelets, multi-scale analysis, frames, and special
functions. New Perspectives on Approximation and Sampling Theory
requires basic knowledge of mathematical analysis, but efforts were
made to keep the exposition clear and the chapters self-contained.
This volume will appeal to researchers and graduate students in
mathematics, applied mathematics and engineering, in particular,
engineers working in signal and image processing.
Current and historical research methods in approximation theory are
presented in this book beginning with the 1800s and following the
evolution of approximation theory via the refinement and extension
of classical methods and ending with recent techniques and
methodologies. Graduate students, postdocs, and researchers in
mathematics, specifically those working in the theory of functions,
approximation theory, geometric function theory, and optimization
will find new insights as well as a guide to advanced topics. The
chapters in this book are grouped into four themes; the first,
polynomials (Chapters 1 -8), includes inequalities for polynomials
and rational functions, orthogonal polynomials, and location of
zeros. The second, inequalities and extremal problems are discussed
in Chapters 9 -13. The third, approximation of functions, involves
the approximants being polynomials, rational functions, and other
types of functions and are covered in Chapters 14 -19. The last
theme, quadrature, cubature and applications, comprises the final
three chapters and includes an article coauthored by Rahman. This
volume serves as a memorial volume to commemorate the distinguished
career of Qazi Ibadur Rahman (1934-2013) of the Universite de
Montreal. Rahman was considered by his peers as one of the
prominent experts in analytic theory of polynomials and entire
functions. The novelty of his work lies in his profound abilities
and skills in applying techniques from other areas of mathematics,
such as optimization theory and variational principles, to obtain
final answers to countless open problems.
Presents easy to understand proofs of some of the most difficult results about polynomials demonstrated by means of applications.
Paul Butzer, who is considered the academic father and grandfather
of many prominent mathematicians, has established one of the best
schools in approximation and sampling theory in the world. He is
one of the leading figures in approximation, sampling theory, and
harmonic analysis. Although on April 15, 2013, Paul Butzer turned
85 years old, remarkably, he is still an active research
mathematician. In celebration of Paul Butzer's 85th birthday, New
Perspectives on Approximation and Sampling Theory is a collection
of invited chapters on approximation, sampling, and harmonic
analysis written by students, friends, colleagues, and prominent
active mathematicians. Topics covered include approximation methods
using wavelets, multi-scale analysis, frames, and special
functions. New Perspectives on Approximation and Sampling Theory
requires basic knowledge of mathematical analysis, but efforts were
made to keep the exposition clear and the chapters self-contained.
This volume will appeal to researchers and graduate students in
mathematics, applied mathematics and engineering, in particular,
engineers working in signal and image processing.
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