Preface	6
	Contents	9
	Constructive Elements and Approaches in Approximation Theory	13
	Introduction to Approximation Theory	13
	Basic Notions	13
	Algebraic and Trigonometric Polynomials	16
	Best Approximation by Polynomials	19
	Chebyshev Polynomials	21
	Basic Properties	21
	Differential Equation	22
	Zeros and Extremal Points	23
	Chebyshev Polynomials in the Complex Plane	24
	Some Other Relations	25
	Orthogonality	26
	Chebyshev Extremal Problems	26
	The Extremal Problem in the Uniform Norm	26
	The Extremal Problem in L1-norm	28
	Chebyshev Alternation Theorem	29
	Some Classical Special Cases	31
	Numerical Methods	32
	Basic Facts on Trigonometric Approximation	36
	Trigonometric Kernels	36
	Fourier Series and Sums	42
	Moduli of Smoothness, Best Approximation and Besov Spaces	44
	Chebyshev Systems and Interpolation	50
	Chebyshev Systems and Spaces	50
	Algebraic Lagrange Interpolation	51
	Trigonometric Interpolation	52
	Riesz Interpolation Formula	56
	A General Interpolation Problem	58
	Interpolation by Algebraic Polynomials	60
	Representations and Computation of Interpolation Polynomials	60
	Interpolation Array and Lagrange Operators	63
	Interpolation Error for Some Classes of Functions	66
	The Error in the Class of Continuous-Differentiable Functions	66
	The Error in the Class of Analytic Functions	67
	Uniform Convergence in the Class of Analytic Functions	68
	Bernstein's Example of Pointwise Divergence	73
	Lebesgue Function and Some Estimates for the Lebesgue Constant	75
	Equidistant Nodes	76
	Chebyshev Nodes	77
	Algorithm for Finding Optimal Nodes	80
	Orthogonal Polynomials and Weighted Polynomial Approximation	86
	Orthogonal Systems and Polynomials	86
	Inner Product Space and Orthogonal Systems	86
	Fourier Expansion and Best Approximation	88
	Examples of Orthogonal Systems	90
	Trigonometric System	90
	Chebyshev Polynomials	90
	Orthogonal Polynomials on the Unit Circle	91
	Orthogonal Polynomials on the Unit Disk	91
	Orthogonal Polynomials on the Ellipse	91
	Malmquist-Takenaka System of Rational Functions	92
	Polynomials Orthogonal on the Radial Rays	92
	Müntz Orthogonal Polynomials	93
	Müntz Orthogonal Polynomials of the Second Kind	95
	Generalized Exponential Polynomials	96
	Discrete Chebyshev Polynomials	96
	Formal Orthogonal Polynomials with Respect to a Moment Functional	97
	Basic Facts on Orthogonal Polynomials and Extremal Problems	100
	Zeros of Orthogonal Polynomials	104
	Orthogonal Polynomials on the Real Line	106
	Basic Properties	106
	Three-Term Recurrence Relations	107
	Christoffel's Formulae	109
	Zeros	110
	Some Special Weights	112
	Asymptotic Properties of Orthogonal Polynomials	114
	Bernstein-Szego Identities	119
	The Fokas-Its-Kitaev (Riemann-Hilbert) Identity	120
	Rakhmanov's Identity	122
	Associated Polynomials and Christoffel Numbers	122
	Associated Polynomials	122
	Stieltjes Transform of the Measure and Christoffel Numbers	125
	Markov's Moment Problem	127
	Functions of the Second Kind and Stieltjes Polynomials	128
	Classical Orthogonal Polynomials	132
	Definition of the Classical Orthogonal Polynomials	132
	General Properties of the Classical Orthogonal Polynomials	135
	Generating Function	139
	Jacobi Polynomials	142
	Special Cases	144
	Zeros	146
	Inequalities and Asymptotics	147
	Christoffel Function and Christoffel Numbers	150
	Generalized Laguerre Polynomials	151
	Zeros	152
	Inequalities	153
	Christoffel Function and Christoffel Numbers	155
	Hermite Polynomials	156
	Nonclassical Orthogonal Polynomials	157
	Semi-classical Orthogonal Polynomials	157
	Generalized Gegenbauer Polynomials	158
	Generalized Jacobi Polynomials	159
	Sonin-Markov Orthogonal Polynomials	163
	Freud Orthogonal Polynomials	165
	Mhaskar-Rakhmanov-Saff Number	165
	Basic Properties of Freud Polynomials	166
	Strong Asymptotics	168
	Orthogonal Polynomials with Respect to Abel, Lindelöf, and Logistic Weights	170
	Strong Non-classical Orthogonal Polynomials	170
	Numerical Construction of Orthogonal Polynomials	171
	Modified Chebyshev Algorithm	171
	Discretized Stieltjes-Gautschi Procedure	173
	Weighted Polynomial Approximation	177
	Weighted Functional Spaces, Moduli of Smoothness and K-functionals	177
	Weighted Best Polynomial Approximation on [-1,1]	181
	Weighted Approximation on the Semi-axis	185
	Weighted K-functionals and Moduli of Smoothness	186
	Weighted Best Polynomial Approximation	187
	Weighted Besov Type Spaces	188
	Weighted Approximation on the Real Line	189
	Weighted Polynomial Approximation of Functions Having Isolated Interior Singularities	193
	Trigonometric Approximation	204
	Approximating Properties of Operators	204
	Approximation by Fourier Sums	204
	Approximation by Fejér and de la Vallée Poussin Means	206
	Discrete Operators	208
	A Quadrature Formula	208
	Discrete Versions of Fourier and de la Vallée Poussin Sums	213
	Marcinkiewicz Inequalities	216
	Uniform Approximation	221
	Lagrange Interpolation Error in Lp	223
	Some Estimates of the Interpolation Errors in L1-Sobolev Spaces	232
	The Weighted Case	235
	Algebraic Interpolation in Uniform Norm	245
	Introduction and Preliminaries	245
	Interpolation at Zeros of Orthogonal Polynomials	245
	Some Auxiliary Results	249
	Optimal Systems of Nodes	258
	Optimal Systems of Knots on [-1,1]	258
	Interpolation at Jacobi Abscissas	258
	Interpolation at the ``Practical Abscissas''	259
	Additional Nodes Method with Jacobi Zeros	262
	Other ``Optimal'' Interpolation Processes	274
	Interpolation with Associated Polynomials	274
	Interpolation at Stieltjes Zeros	276
	Extended Interpolation	276
	Some Simultaneous Interpolation Processes	278
	Weighted Interpolation	281
	Weighted Interpolation at Jacobi Zeros	281
	Lagrange Interpolation in Sobolev Spaces	286
	Interpolation at Laguerre Zeros	288
	Interpolation at Hermite Zeros	297
	Interpolation of Functions with Internal Isolated Singularities	302
	Interpolation Processes on Bounded Intervals	305
	Interpolation Processes on Unbounded Intervals	316
	Numerical Examples	319
	Applications	329
	Quadrature Formulae	329
	Introduction	329
	Some Remarks on Newton-Cotes Rules with Jacobi Weights	332
	Gauss-Christoffel Quadrature Rules	334
	Gauss-Christoffel Quadratures for the Classical Weights	334
	Computation of Gauss-Christoffel Quadratures	335
	Gauss-Radau and Gauss-Lobatto Quadrature Rules	338
	Gauss-Radau Quadrature Formula	339
	Gauss-Lobatto Quadrature Formula	340
	Error Estimates of Gaussian Rules for Some Classes of Functions	342
	Error Estimates for Analytic Functions	344
	Error Estimates for Some Classes of Continuous Functions	347
	Error Estimates for Gauss-Laguerre Formula	351
	Error Estimates for Freud-Gaussian Rules	353
	Product Integration Rules	355
	Integration of Periodic Functions on the Real Line with Rational Weight	360
	Integral Equations	372
	Some Basic Facts	372
	Fredholm Integral Equations of the Second Kind	379
	Locally Smooth Kernels	380
	Numerical Examples	386
	Weakly Singular Kernels	389
	Nyström Method	392
	Moment-Preserving Approximation	395
	The Standard L2-Approximation	395
	Generalization	397
	The Constrained L2-Polynomial Approximation	398
	Moment-Preserving Spline Approximation	399
	Approximation on [0,+)	399
	Approximation on a Compact Interval	405
	Summation of Slowly Convergent Series	407
	Laplace Transform Method	408
	Contour Integration Over a Rectangle	411
	Remarks on Some Slowly Convergent Power Series	421
	References	424
	Index	445