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The first chapter lists the basic results of orthogonal
polynomials, Jacobi, Laguerre, and Hermite polynomials, and
collects some frequently used theorems and formulas. As a base and
useful tool, the representation and quantitative theory of Hermite
interpolation is the subject of Chapter 2. The theory of power
orthogonal polynomials begins in Chapter 3: existence, uniqueness,
Characterisations, properties of zeros, and continuity with respect
to the measure and the indices are all considered. Chapter 4 deals
with Gaussian quadrature formulas and their convergence. Chapter 5
is devoted to the theory of Christo (R)el type functions, which are
related to Gaussian quadrature formulas and is one of the important
contents of power orthogonal polynomials. The explicit
representation of power orthogonal polynomials is an interesting
problem and is discussed in Chapter 6. Chapter 7 is a detailed
treatment of zeros in power orthogonal polynomials. Chapter 8 is
devoted to bounds and inequalities of power orthogonal polynomials.
In Chapters 9 and 10 we study asymptotics of general polynomials
and power orthogonal polynomials, respectively. In Chapter 11 we
discuss convergence of power orthogonal series, Lagrange and
Hermite interpolation, and two positive operators constructed by
power orthogonal polynomials. In Chapter 12 we investigate Gaussian
quadrature formulas for extended Chebyshev spaces. In Chapter 13 we
give construction methods for power orthogonal polynomials and
Gaussian quadrature formulas; we also provide numerical results and
numerical tables.
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