# finite Fourier series

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## 10 matching pages

##### 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 $n\pmod{k}$ 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. …
##### 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 $q^{(n+\frac{1}{2})^{2}}$ or $q^{n^{2}}$, and provide a convenient way of calculating values of $\theta_{j}\left(z\middle|\tau\right)$. … Hence the first term of the series (20.2.3) for $\theta_{3}\left(z\tau^{\prime}\middle|\tau^{\prime}\right)$ suffices for most purposes. In theory, starting from any value of $\tau$, a finite number of applications of the transformations $\tau\to\tau+1$ and $\tau\to-1/\tau$ will result in a value of $\tau$ with $\Im\tau\geq\sqrt{3}/2$; see §23.18. …
##### 7: 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.