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finite Fourier series

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1: 27.10 Periodic Number-Theoretic Functions
Every function periodic (mod k ) can be expressed as a finite Fourier series of the form … is a periodic function of n ( mod k ) and has the finite Fourier-series expansion …
2: 29.20 Methods of Computation
The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …
3: 1.8 Fourier Series
§1.8 Fourier Series
Uniqueness of Fourier Series
§1.8(ii) Convergence
4: Bibliography T
  • A. Takemura (1984) Zonal Polynomials. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA.
  • I. C. Tang (1969) Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.
  • A. Terras (1999) Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts, Vol. 43, Cambridge University Press, Cambridge.
  • E. C. Titchmarsh (1986a) Introduction to the Theory of Fourier Integrals. Third edition, Chelsea Publishing Co., New York.
  • G. P. Tolstov (1962) Fourier Series. Prentice-Hall Inc., Englewood Cliffs, N.J..
  • 5: 20.14 Methods of Computation
    The Fourier series of §20.2(i) usually converge rapidly because of the factors q ( n + 1 2 ) 2 or q n 2 , and provide a convenient way of calculating values of θ j ( z | τ ) . … Hence the first term of the series (20.2.3) for θ 3 ( z τ | τ ) suffices for most purposes. In theory, starting from any value of τ , a finite number of applications of the transformations τ τ + 1 and τ - 1 / τ will result in a value of τ with τ 3 / 2 ; see §23.18. …
    6: 18.17 Integrals
    §18.17(v) Fourier Transforms
    Jacobi
    Ultraspherical
    Legendre
    Hermite
    7: 1.17 Integral and Series Representations of the Dirac Delta
    §1.17 Integral and Series Representations of the Dirac Delta
    §1.17(ii) Integral Representations
    Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)): …
    §1.17(iii) Series Representations
    Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): …
    8: Bibliography H
  • G. H. Hardy (1949) Divergent Series. Clarendon Press, Oxford.
  • M. Heil (1995) Numerical Tools for the Study of Finite Gap Solutions of Integrable Systems. Ph.D. Thesis, Technischen Universität Berlin.
  • P. Henrici (1986) Applied and Computational Complex Analysis. Vol. 3: Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons Inc.], New York.
  • E. Hille (1929) Note on some hypergeometric series of higher order. J. London Math. Soc. 4, pp. 50–54.
  • C. J. Howls (1992) Hyperasymptotics for integrals with finite endpoints. Proc. Roy. Soc. London Ser. A 439, pp. 373–396.
  • 9: 2.3 Integrals of a Real Variable
    For the Fourier integral …assume a and b are finite, and q ( t ) is infinitely differentiable on [ a , b ] . … Since q ( t ) need not be continuous (as long as the integral converges), the case of a finite integration range is included. … Then … If p ( b ) is finite, then both endpoints contribute: …
    10: Bibliography C
  • S. M. Candel (1981) An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.
  • H. S. Carslaw (1930) Introduction to the Theory of Fourier’s Series and Integrals. 3rd edition, Macmillan, London.
  • I. Cherednik (1995) Macdonald’s evaluation conjectures and difference Fourier transform. Invent. Math. 122 (1), pp. 119–145.
  • W. W. Clendenin (1966) A method for numerical calculation of Fourier integrals. Numer. Math. 8 (5), pp. 422–436.
  • J. W. Cooley and J. W. Tukey (1965) An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19 (90), pp. 297–301.