| 释义 |
Fourier SeriesFourier series are expansions of Periodic Functions  in terms of an infinite sum ofSines and Cosines
 | (1) |
 and  in the sum. The computation andstudy of Fourier series is known as Harmonic Analysis.
To compute a Fourier series, use the integral identities
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 | (3) |
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 is the Kronecker Delta. Now, expand your function as an infinite series of the form
where we have relabeled the  term for future convenience but set  and left  for  . Assumethe function is periodic in the interval  . Now use the orthogonality conditions to obtainandso
Plugging back into the original series then gives
for  , 2, 3, .... The series expansion converges to the function  (equal to the original function atpoints of continuity or to the average of the two limits at points of discontinuity)
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Near points of discontinuity, a ``ringing'' known as the Gibbs Phenomenon, illustrated above, occurs.For a function periodic on an interval  , use a change of variables to transform the interval of integrationto  . Let
Solving for  ,  . Plugging this in gives
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 | (18) |
 , then  is Odd. (This follows since  is Odd and an Even Function times an OddFunction is an Odd Function.) Therefore,  for all  . Similarly, if a function is Odd so that , then  is Odd. (This follows since  isEven and an Even Function times an Odd Function is an Odd Function.) Therefore,  for all.
Because the Sines and Cosines form a CompleteOrthogonal Basis, the Superposition Principle holds, and the Fourier series of alinear combination of two functions is the same as the linear combination of the corresponding two series. TheCoefficients for Fourier series expansions for a few common functions are given in Beyer (1987,pp. 411-412) and Byerly (1959, p. 51).
The notion of a Fourier series can also be extended to Complex Coefficients. Consider a real-valued function . Write
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For a function periodic in , these become
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 froma discrete variable to a continuous one as the length  .See also Dirichlet Fourier Series Conditions, Fourier Cosine Series, Fourier Sine Series, FourierTransform, Gibbs Phenomenon, Lebesgue Constants (Fourier Series), Legendre Series, Riesz-FischerTheorem
References
 Fourier TransformsArfken, G. ``Fourier Series.'' Ch. 14 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 760-793, 1985. Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987. Brown, J. W. and Churchill, R. V. Fourier Series and Boundary Value Problems, 5th ed. New York: McGraw-Hill, 1993. Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959. Carslaw, H. S. Introduction to the Theory of Fourier's Series and Integrals, 3rd ed., rev. and enl. New York: Dover, 1950. Davis, H. F. Fourier Series and Orthogonal Functions. New York: Dover, 1963. Dym, H. and McKean, H. P. Fourier Series and Integrals. New York: Academic Press, 1972. Folland, G. B. Fourier Analysis and Its Applications. Pacific Grove, CA: Brooks/Cole, 1992. Groemer, H. Geometric Applications of Fourier Series and Spherical Harmonics. New York: Cambridge University Press, 1996. Körner, T. W. Fourier Analysis. Cambridge, England: Cambridge University Press, 1988. Körner, T. W. Exercises for Fourier Analysis. New York: Cambridge University Press, 1993. Lighthill, M. J. Introduction to Fourier Analysis and Generalised Functions. Cambridge, England: Cambridge University Press, 1958. Morrison, N. Introduction to Fourier Analysis. New York: Wiley, 1994. Sansone, G. ``Expansions in Fourier Series.'' Ch. 2 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 39-168, 1991.
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