Fourier Series and Orthogonal Polynomilas

Title page......Page 1
Preface......Page 3
1. Definition of Fourier series......Page 12
2. Orthogonality of sines and cosines......Page 13
3. Determination of the coefficients......Page 14
4. Series of cosines and series of sines......Page 17
5. Examples......Page 19
6. Magnitude of coefficients under special hypotheses......Page 22
7. Riemann's theorem on limit of general coefficient......Page 25
9. Integral formula for partial sum of Fourier series......Page 28
10. Convergence at a point of continuity......Page 29
11. Uniform convergence under special hypotheses......Page 32
12. Convergence at a point of discontinuity......Page 33
13. Sufficiency of conditions relating to a restricted neighborhood......Page 35
14. Weierstrass's theorem on trigonometric approximation......Page 36
15. Least-square property......Page 38
16. Parseval's theorem......Page 40
17. Summation of series......Page 42
18. Fejér's theorem for a continuous function......Page 43
19. Proof of Weierstrass's theorem by means of de la Vallée Poussin's integral......Page 46
20. The Lebesgue constants......Page 51
21. Proof of uniform convergence by the method of Lebesgue......Page 53
2. Definition of the Legendre polynomials by means of the generating function......Page 56
3. Recurrence formula......Page 57
4. Differential equation and related formulas......Page 59
5. Orthogonality......Page 61
6. Normalizing factor......Page 62
7. Expansion of an arbitrary function in series......Page 64
8. Christoffel's identity......Page 65
9. Solution of the differential equation......Page 66
10. Rodrigues's formula......Page 68
11. Integral representation......Page 69
12. Bounds of P_n(X)......Page 72
13. Convergence at a point of continuity interior to the interval......Page 74
14. Convergence at a point of discontinuity interior to the interval......Page 76
2. Definition of J₀(x)......Page 80
3. Orthogonality......Page 82
4. Integral representation of Jo(x)......Page 85
5. Zeros of J₀(x) and related functions......Page 87
6. Expansion of an arbitrary function in series......Page 90
7. Definition of J_n(x)......Page 91
8. Orthogonality: developments in series......Page 93
9. Integral representation of J_n(x)......Page 95
10. Recurrence formulas......Page 96
12. Asymptotic formula......Page 98
13. Orthogonal functions arising from linear boundary value problems......Page 99
1. Fourier series: Laplace's equation in an infinite strip......Page 102
2. Fourier series: Laplace's equation in a rectangle......Page 106
3. Fourier series: vibrating string......Page 107
4. Fourier series: damped vibrating string......Page 111
5. Polar coordinates in the plane......Page 112
6. Fourier series: Laplace's equation in a circle; Poisson's integral......Page 114
7. Transformation of Laplace's equation in three dimensions......Page 116
8. Legendre series: Laplace's equation in a sphere......Page 118
9. Bessel series: Laplace's equation in a cylinder......Page 120
10. Bessel series: circular drumhead......Page 123
1. Boundary value problem in a cube; double Fourier series......Page 126
2. General spherical harmonies......Page 129
3. Laplace series......Page 132
4. Harmonic polynomials......Page 137
5. Rotation of axes......Page 140
6. Integral representation for group of terms in the Laplace series......Page 143
7. Completeness of the Laplace series......Page 148
8. Boundary value problem in a cylinder; series involving Bessel functions of positive order......Page 149
2. Quadratic denominator, real roots......Page 153
3. Quadratic denominator, complex roots......Page 156
4. Linear or constant denominator......Page 157
5. Finiteness of moments......Page 158
1. Weight function......Page 160
2. Schmidt's process......Page 162
3. Orthogonal polynomials corresponding to an arbitrary weight function......Page 164
4. Development of an arbitrary function in series......Page 166
5. Formula of recurrence......Page 167
6. Christoffel-Darboux identity......Page 168
7. Symmetry......Page 169
8. Zeros......Page 170
9. Least-square property......Page 171
10. Differential equation......Page 172
1. Derivative definition......Page 177
2. Orthogonality......Page 178
3. Leading coefficients......Page 180
4. Normalizing faetor; series of Jacobi polynomials......Page 182
5. Recurrence formula......Page 183
6. Differential equation......Page 184
1. Derivative definition......Page 187
2. Orthogonality and normalizing factor......Page 188
3. Hermite and Gram-Charlier series......Page 189
4. Recurrence formulas; differential equation......Page 190
6. Wave equation of the linear oscillator......Page 192
2. Orthogonality; normalizing factor; Laguerre series......Page 195
3. Differential equation and recurrenee formulas......Page 197
4. Generating function......Page 198
5. Wave equation of the hydrogen atom......Page 199
1. Scope of the discussion......Page 202
2. Magnitude of the coefficients; first hypothesis......Page 203
3. Convergence; first hypothesis......Page 205
4. Magnitude of the coefficients; second hypothesis......Page 208
5. Convergence; second hypothesis......Page 210
6. Special Jacobi polynomials......Page 211
7. Multiplication or division of the weight function by a polynomial......Page 212
8. Korous's theorem on bounds of orthonormal polynomials......Page 216
EXERCISES......Page 220
BIBLIOGRAPHY......Page 240
INDEX......Page 242