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Fourier级数和正交多项式（Fourier Series and Orthogonal Polynomials）（影印版）

样章
作者：
Dunham Jackson
定价：
135.00元
版面字数：
250.00千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2026-01-05
ISBN：
978-7-04-065927-6
物料号：
65927-00
出版时间：
2026-03-27
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
分析
暂无
  • 前辅文
  • I. FOURIER SERIES
    • 1. Definition of Fourier series
    • 2. Orthogonality of sines and cosines
    • 3. Determination of the coefficients
    • 4. Series of cosines and series of sines
    • 5. Examples
    • 6. Magnitude of coefficients under special hypotheses
    • 7. Riemann's theorem on limit of general coefficient
    • 8. Evaluation of a sum of cosines
    • 9. Integral formula for partial sum of Fourier series
    • 10. Convergence at a point of continuity
    • 11. Uniform convergence under special hypotheses
    • 12. Convergence at a point of discontinuity
    • 13. Sufficiency of conditions relating to a restricted neighborhood
    • 14. Weierstrass's theorem on trigonometric approximation
    • 15. Least-square property
    • 16. Parseval's theorem
    • 17. Summation of series
    • 18. Fejér's theorem for a continuous function
    • 19. Proof of Weierstrass's theorem by means of de la Vallée Poussin's integral
    • 20. The Lebesgue constants
    • 21. Proof of uniform convergence by the method of Lebesgue
  • II. LEGENDRE POLYNOMIALS
    • 1. Preliminary orientation
    • 2. Definition of the Legendre polynomials by means of the generating function
    • 3. Recurrence formula
    • 4. Differential equation and related formulas
    • 5. Orthogonality
    • 6. Normalizing factor
    • 7. Expansion of an arbitrary function in series
    • 8. Christoffel's identity
    • 9. Solution of the differential equation
    • 10. Rodrigues's formula
    • 11. Integral representation
    • 12. Bounds of Pn(x)
    • 13. Convergence at a point of continuity interior to the interval
    • 14. Convergence at a point of discontinuity interior to the interval
  • III. BESSEL FUNCTIONS
    • 1. Preliminary orientation
    • 2. Definition of J0(x)
    • 3. Orthogonality
    • 4. Integral representation of J0(x)
    • 5. Zeros of J0(x) and related functions
    • 6. Expansion of an arbitrary function in series
    • 7. Definition of Jn(x)
    • 8. Orthogonality: developments in series
    • 9. Integral representation of Jn(x)
    • 10. Recurrence formulas
    • 11. Zeros
    • 12. Asymptotic formula
    • 13. Orthogonal functions arising from linear boundary value problems
  • IV. BOUNDARY VALUE PROBLEMS
    • 1. Fourier series: Laplace's equation in an infinite strip
    • 2. Fourier series: Laplace's equation in a rectangle
    • 3. Fourier series: vibrating string
    • 4. Fourier series: damped vibrating string
    • 5. Polar coordinates in the plane
    • 6. Fourier series: Laplace's equation in a circle
    • 7. Transformation of Laplace's equation in three dimensions
    • 8. Legendre series: Laplace's equation in a sphere
    • 9. Bessel series: Laplace's equation in a cylinder
    • 10. Bessel series: circular drumhead
  • V. DOUBLE SERIES
    • 1. Boundary value problem in a cube
    • 2. General spherical harmonics
    • 3. Laplace series
    • 4. Harmonic polynomials
    • 5. Rotation of axes
    • 6. Integral representation for group of terms in the Laplace series
    • 7. Completeness of the Laplace series
    • 8. Boundary value problem in a cylinder
  • VI. THE PEARSON FREQUENCY FUNCTIONS
    • 1. The Pearson differential equation
    • 2. Quadratic denominator, real roots
    • 3. Quadratic denominator, complex roots
    • 4. Linear or constant denominator
    • 5. Finiteness of moments
  • VII. ORTHOGONAL POLYNOMIALS
    • 1. Weight function
    • 2. Schmidt's process
    • 3. Orthogonal polynomials corresponding to an arbitrary weight function
    • 4. Development of an arbitrary function in series
    • 5. Formula of recurrence
    • 6. Christoffel-Darboux identity
    • 7. Symmetry
    • 8. Zeros
    • 9. Least-square property
    • 10. Differential equation
  • VIII. JACOBI POLYNOMIALS
    • 1. Derivative definition
    • 2. Orthogonality
    • 3. Leading coefficients
    • 4. Normalizing factor
    • 5. Recurrence formula
    • 6. Differential equation
  • IX. HERMITE POLYNOMIALS
    • 1. Derivative definition
    • 2. Orthogonality and normalizing factor
    • 3. Hermite and Gram-Charlier series
    • 4. Recurrence formulas
    • 5. Generating function
    • 6. Wave equation of the linear oscillator
  • X. LAGUERRE POLYNOMIALS
    • 1. Derivative definition
    • 2. Orthogonality; normalizing factor
    • 3. Differential equation and recurrence formulas
    • 4. Generating function
    • 5. Wave equation of the hydrogen atom
  • XI. CONVERGENCE
    • 1. Scope of the discussion
    • 2. Magnitude of the coefficients
    • 3. Convergence
    • 4. Magnitude of the coefficients
    • 5. Convergence
    • 6. Special Jacobi polynomials
    • 7. Multiplication or division of the weight function by a polynomial
    • 8. Korous's theorem on bounds of orthonormal polynomials
  • EXERCISES
  • BIBLIOGRAPHY
  • INDEX

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