Fourier Series and Orthogonal Polynomials Book

I.	Fourier Series
1.	Definition of Fourier series	1
2.	Orthogonality of sines and cosines	2
3.	Determination of the coefficients	3
4.	Series of cosines and series of sines	6
5.	Examples	8
6.	Magnitude of coefficients under special hypotheses	11
7.	Riemann's theorem on limit of general coefficient	14
8.	Evaluation of a sum of cosines	17
9.	Integral formula for partial sum of Fourier series	17
10.	Convergence at a point of continuity	18
11.	Uniform convergence under special hypotheses	21
12.	Convergence at a point of discontinuity	22
13.	Sufficiency of conditions relating to a restricted neighborhood	24
14.	Weierstrass's theorem on trigonometric approximation	25
15.	Least-square property	27
16.	Parseval's theorem	29
17.	Summation of series	31
18.	Fejer's theorem for a continuous function	32
19.	Proof of Weierstrass's theorem by means of de la Vallee Poussin's integral	35
20.	The Lebesgue constants	40
21.	Proof of uniform convergence by the method of Lebesgue	42
II.	Legendre Polynomials
1.	Preliminary orientation	45
2.	Definition of the Legendre polynomials by means of the generating function	45
3.	Recurrence formula	46
4.	Differential equation and related formulas	48
5.	Orthogonality	50
6.	Normalizing factor	51
7.	Expansion of an arbitrary function in series	53
8.	Christoffel's identity	54
9.	Solution of the differential equation	55
10.	Rodrigues's formula	57
11.	Integral representation	58
12.	Bounds of P[subscript n](x)	61
13.	Convergence at a point of continuity interior to the interval	63
14.	Convergence at a point of discontinuity interior to the interval	65
III.	Bessel Functions
1.	Preliminary orientation	69
2.	Definition of J[subscript 0](x)	69
3.	Orthogonality	71
4.	Integral representation of J[subscript 0](x)	74
5.	Zeros of J[subscript 0](x) and related functions	76
6.	Expansion of an arbitrary function in series	79
7.	Definition of J[subscript n](x)	80
8.	Orthogonality: developments in series	82
9.	Integral representation of J[subscript n](x)	84
10.	Recurrence formulas	85
11.	Zeros	87
12.	Asymptotic formula	87
13.	Orthogonal functions arising from linear boundary value problems	88
IV.	Boundary Value Problems
1.	Fourier series: Laplace's equation in an infinite strip	91
2.	Fourier series: Laplace's equation in a rectangle	95
3.	Fourier series: vibrating string	96
4.	Fourier series: damped vibrating string	100
5.	Polar coordinates in the plane	101
6.	Fourier series: Laplace's equation in a circle; Poisson's integral	103
7.	Transformation of Laplace's equation in three dimensions	105
8.	Legendre series: Leplace's equation in a sphere	107
9.	Bessel series: Laplace's equation in a cylinder	109
10.	Bessel series: circular drumhead	112
V.	Double Series; Laplace Series
1.	Boundary value problem in a cube; double Fourier series	115
2.	General spherical harmonics	118
3.	Laplace series	121
4.	Harmonic polynomials	126
5.	Rotation of axes	129
6.	Integral representation for group of terms in the Laplace series	132
7.	Completeness of the Laplace series	137
8.	Boundary value problem in a cylinder; series involving Bessel functions of positive order	138
VI.	The Pearson Frequency Functions
1.	The Pearson differential equation	142
2.	Quadratic denominator, real roots	142
3.	Quadratic denominator, complex roots	145
4.	Linear or constant denominator	146
5.	Finiteness of moments	147
VII.	Orthogonal Polynomials
1.	Weight function	149
2.	Schmidt's process	151
3.	Orthogonal polynomials corresponding to an arbitrary weight function	153
4.	Development of an arbitrary function in series	155
5.	Formula of recurrence	156
6.	Christoffel-Darboux identity	157
7.	Symmetry	158
8.	Zeros	159
9.	Least-square property	160
10.	Differential equation	161
VIII.	Jacobi Polynomials
1.	Derivative definition	166
2.	Orthogonality	167
3.	Leading coefficients	169
4.	Normalizing factor; series of Jacobi polynomials	171
5.	Recurrence formula	172
6.	Differential equation	173
IX.	Hermite Polynomials
1.	Derivative definition	176
2.	Orthogonality and normalizing factor	177
3.	Hermite and Gram-Charlier series	178
4.	Recurrence formulas; differential equation	179
5.	Generating function	181
6.	Wave equation of the linear oscillator	181
X.	Laguerre Polynomials
1.	Derivative definition	184
2.	Orthogonality; normalizing factor; Laguerre series	184
3.	Differential equation and recurrence formulas	186
4.	Generating function	187
5.	Wave equation of the hydrogen atom	188
XI.	Convergence
1.	Scope of the discussion	191
2.	Magnitude of the coefficients; first hypothesis	192
3.	Convergence; first hypothesis	194
4.	Magnitude of the coefficients; second hypothesis	197
5.	Convergence; second hypothesis	199
6.	Special Jacobi polynomials	200
7.	Multiplication or division of the weight function by a polynomial	201
8.	Korous's theorem on bounds of orthonormal polynomials	205
	Exercises	209
	Bibliography	229
	Index	231