Fourier Series

Description

Richard A. Silverman's series of translations of outstanding Russian textbooks and monographs is well-known to people in the fields of mathematics, physics, and engineering. The present book is another excellent text from this series, a valuable addition to the English-language literature on Fourier series. This edition is organized into nine well-defined chapters: Trigonometric Fourier Series, Orthogonal Systems, Convergence of Trigonometric Fourier Series, Trigonometric Series with Decreasing Coefficients, Operations on Fourier Series, Summation of Trigonometric Fourier Series, Double Fourier Series and the Fourier Integral, Bessel Functions and Fourier-Bessel Series, and the Eigenfunction Method and its Applications to Mathematical Physics. Every chapter moves clearly from topic to topic and theorem to theorem, with many theorem proofs given. A total of 107 problems will be found at the ends of the chapters, including many specially added to this English-language edition, and answers are given at the end of the text.Richard Silverman's excellent translation makes this book readily accessible to mathematicians and math students, as well as workers and students in the fields of physics and engineering. He has also added a bibliography, containing suggestions for collateral and supplementary reading. 1962 edition. CONTENTS 1 TRIGONOMETRIC FOURIER SERIES Page 1. 1: Periodic Functions, 1. 2: Harmonies, 3. 3: Trigonometric Polynomials and Series, 6. 4: A More Precise Terminology. Integrability. Series of Functions, 8. 5: The Basic Trigo- nometric System. The Orthogonality of Sines and Cosines, 10. 6: Fourier Series for Functions of Period 21:, 12. 7: Fourier Series for Functions Defined on an Interval of Length 21:, 15. 8: Right-hand and Left-hand Limits. Jump Discontinuities, 17. 9: Smooth and Piecewise Smooth Functions, 18. 10: A Criterion for the Convergence of Fourier Series, 19. 11: Even and Odd Functions, 21. 12: Cosine and Sine Series, 22. 13: Examples of Expansions in Fourier Series, 24. 14: The Complex Form of a Fourier Series, 32. 15: Functions of Period 21, 35. Problems, 38. 2 ORTHOGONAL SYSTEMS Page 41. 1: Definitions, 41. 2: Fourier Series with Respect to an Orthogonal System, 42. 3: Some Simple Orthogonal Systems, 44. 4: Square Inte- grable Functions. The Schwarz Inequality, 50. 5: The Mean Square Error and its Minimum, 51. 6: Bessel’s Inequality, 53. 7: Complete Systems. Convergence in the Mean, 54. 8: Important Properties of Complete Systems, 57. 9: A Criterion for the Completeness of a System, 58. ”'10: The Vector Analogy, 60. Problems, 63. 3 CONVERGENCE OF TRIGONOMETRIC FOURIER SERIES Page 66. l : A Consequence of Bessel’s Inequality, 66. 2: The Limit as n—->oo of the Trigonometric Integrals L”f(x) cos nx dx and 1370:) sin nx dx, 67. 3: Formula for the Sum of Cosines. Auxiliary Integrals, 71. 4: The Integral Formula for the Partial Sum of a Fourier Series, 72. 5: Right-Hand and Left-Hand Derivatives, 73. 6: A Sufficient Condition for Convergence of a Fourier Series at a Con- tinuity Point, 75. 7: A Sufficient Condition for Convergence of a Fourier Series at a Point of Discontinuity, 77. 8: Generalization of the Sufficient Conditions Proved in Secs. 6 and 7, 78. 9: Convergence of the Fourier Series of a Piecewise Smooth Function (Continuous or Discontinuous), 79. 10: Absolute and Uniform Convergence of the Fourier Series of a Continuous, Piecewise Smooth Function of Period 21:, 80. 11: Uniform Convergence of the Fourier Series of a Continuous Function of Period 21: with an Absolutely Integrable Derivative, 82. 12: Generalization of the Results of Sec. 11, 85. 13: The Localization Principle, 90. 14: Examples of Fourier Series Expansions of Unbounded Func- tions, 91. 15: A Remark Concerning Functions of Period 21, 94. Problems, 94. 4 TRIGONOMETRIC SERIES WITH DECREASING COEFFICIENTS Page 97. l: Abel’s Lemma, 97. 2: Formula for the Sum of Sines. Auxiliary Inequalities, 98. 3: Convergence of Trigonometric Series with Monotonically Decreasing Coefficients, 100. *4: Some Consequences of the Theorems of Sec. 3, 103. 5: Applications of Functions of a Complex Variable to the Evaluation of Certain Trigono- metric Series, 105. 6: A Stronger Form of the Results of Sec. 5, 108. Problems, 112. 5 OPERATIONS ON FOURIER SERIES Page 115. 1: Approximation of Functions by Trigonometric Polynomials, 115. 2: Completeness of the Trigonometric System, 117. 3: Parseval’s Theorem. The Most Important Consequences of the Completeness of the Trigonometric System, 119. *4: Approximation of Functions by Polynomials, 120. 5: Addition and Subtraction of Fourier Series, Multiplication of a Fourier Series by a Number, 122. ”'6: Products of Fourier Series, 123. 7: Integration of Fourier Series, 125. 8: Differentiation of Fourier Series. The Case of a Con- tinuous Function of Period 21:, 129. I"9: Difi‘erentiation of Fourier Series. The Case of a Function Defined on the Interval [— 1r, 1:], 132. ‘10: Difi‘erentiation of Fourier Series. The Case of a Function Defined on the Interval [0, 1:], 137. 11: Improving the Convergence of Fourier Series, 144. 12: A List of Trigonometric Expansions, 147. 13: Approximate Calculation of FOurier Coefiicients, 150. Problem, 152. 6 SUMMATION OF TRIGONOMETRIC FOURIER SERIES Page 155. 1: Statement of the Problem, 155. 2: The Method of Arithmetic Means, 156. 3: The Integral commrs ix Formula for the Arithmetic Mean of the Partial Sums of a Fourier Series, 157. 4: Summation of Fourier Series by the Method of Arithmetic Means, 158. 5: Abel’s Method of Summation, 162. 6: Poisson’s Kernel, 163. 7: Application of Abel’s Method to the Summation of Fourier Series, 164. Problems, 170. 7 DOUBLE FOURIER SERIES. THE FOURIER INTEGRAL Page 173. 1: Orthogonal Systems in Two Variables, 173. 2: The Basic Trigonometric System in Two Variables. Double Trigonometric Fourier Series, 175. 3: The Integral Formula for the Partial Sums of a Double Trigonometric Fourier Series. A Convergence Criterion, 178. 4: Double Fourier Series for a Function with Different Periods in x and y, 180. 5: The Fourier Integral as a Limiting Case of the Fourier Series, 180. 6: Improper Integrals Depending on a Parameter, 182. 7: Two Lemmas, 185. 8: Proof of the Fourier Integral Theorem, 188. 9: Different Forms of the Fourier Integral Theorem, 189. *10: The Fourier Transform, 190. ’11: The Spectral Function, 193. Problems, 195. 8 BESSEL FUNCTIONS AND FOURIER-BESSEL SERIES Page 197. l: Bessel’s Equation, 197. 2: Bessel Functions of The First Kind of Nonnegative Order, 198. 3: The Gamma Function, 20]. 4: Bessel Functions of the First Kind of Negative Order, 202. 5: The General Solution of Bessel’s Equation, 203. 6: Bessel Functions of the Second Kind, 204. 7: Relations between Bessel Functions of Different Orders, 205. 8: Bessel Functions of the First Kind of Half-Integral Order, 207. 9: Asymptotic Formulas for the Bessel Functions 208. 10: Zeros of the Bessel Functions and Related Func- tions, 213. 11: Parametric Form of Bessel’s Equation, 215. 12: Orthogonality of the Functions Jpotx), 216. 13: Evalua- tion of the Integral £x1p2(Xx)dx, 218. I'14: Bounds for the Integral L’ xJPZQx) dx, 219. 15: Definition of Fourier-Bessel Series, 220. 16: Criteria for the Convergence of Fourier- Bessel Series, 221. ‘17: Bessel’s Inequality and its Conse- quences, 223. ‘18: The Order of Magnitude of the Coeffi- cients which Guarantees Uniform Convergence of a Fourier- Bessel Series, 225. *19: The Order of Magnitude of the Fourier-Bessel Coefficients of a Twice Differentiable Function, 228. ‘20: The Order of Magnitude of the Fourier-Bessel Coefficients of a Function Which ix com Times, 231. ‘21: Term by Term Difi’erentiation of Fourier- Bessel Series, 234. 22: Fourier-Bessel Series of the Second Type, 237. I"23: Extension of the Results of Secs. 17—21 to Fourier-Bessel Series of the Second Type, 239. 24: Fourier- Bessel Expansions of Functions Defined on the Interval [0, I], 241. Problems, 243. 9 THE EIGENFUNCTION METHOD AND ITS APPLICA- TIONS T0 MATHEMATICAL PHYSICS Page 245. Part I: THEORY. 1: The Gist of the Method, 245. 2: The Usual Statement of the Boundary Value Problem, 250. 3: The Existence of Eigenvalues, 250. 4: Eigenfunctions and Their Orthogonality, 251. 5: Sign of the Eigenvalues, 254. 6: Fourier Series with Respect to the Eigenfunctions, 255. 7: Does the Eigenfunction Method Always Lead to a Solution of the Problem ?, 258. 8: The Generalized Solution, 261. 9: The Inhomogeneous Problem, 264. 10: Supplementary Remarks, 266. Part II: APPLICATIONS, 268. 11: Equation of a Vibrating String, 268. 12: Free Vibrations of a String, 269. 13: Forced Vibrations of a String, 273. 14: Equation of the Longitudinal Vibrations of a Rod, 275. 15: Free Vibrations of a Rod, 277. 16: Forced Vibrations of a Rod, 280. 17: Vibrations of a Rectangular Membrane, 282. 18: Radial Vibrations of a Circular Membrane, 288. 19: Vibrations of a Circular Membrane (General Case), 291. 20: Equation of Heat Flow in a Rod, 296. 21: Heat Flow in a Rod with Ends Held at Zero Temperature, 297. 22: Heat Flow in a Rod with Ends Held at Constant Temperature, 299. 23: Heat Flow in a Rod Whose Ends are at Specified Variable Temperatures, 301. 24. Heat Flow in a Rod Whose Ends Exchange Heat Freely with the Surrounding Medium, 301. 25: Heat Flow in an Infinite Rod, 306. 26: Heat Flow in a Circular Cylinder Whose Surface is Insulated, 310. 27: Heat Flow in a Circular Cylinder Whose Surface Exchanges Heat with the Surrounding Medium, 312. 28: Steady-State Heat Flow in a Circular Cylinder, 313. Problems, 316. ANSWERS T0 PROBLEMS Page 319. BIBLIOGRAPHY Page 331. INDEX Page 333.