Abstract
A common requirement in various applied mathematical problems is to express a given function as a linear combination of other known functions. The reasons for this vary from analytical necessity to computational convenience.
The original version of this chapter was revised. An erratum to this chapter can be found at DOI http://dx.doi.org/10.1007/978-3-319-43370-7_10
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-43370-7_10
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- 1.
We adopt the notation i 2 = −1 here.
- 2.
There is a subtle point (which we largely bypass) involved in this example. Strictly speaking, the indicated integrals should be understood as Lebesgue type integrals rather than conventional Riemann integrals. Further, the vectors should be regarded as equivalence classes of functions whose difference on sets of points is negligible as far as integration is concerned.
- 3.
Technically, g coincides in the L 2[0, 2π] sense with a 2π-periodic absolutely continuous function with square-integrable derivative. The notion of “absolutely continuous function” is required for the validity of the “fundamental theorem of calculus” in a Lebesgue integral context.
- 4.
since e in ζ − 1 vanishes when n = 0.
- 5.
The sum is “half the terms” of a 2n length Riemann sum for the integral.
- 6.
The technical statement of the required condition is given above for the case of 2π periodic functions.
- 7.
A more careful discussion of this result is in Section 2.3 above.
References
I.N. Gelfand, C.B. Shilov, Generalized Functions, vol. 2 (Academic, New York, 1968)
J.E. Marsden, Elementary Classical Analysis (W.H. Freeman and Company, San Francisco, 1974)
W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966)
Further Reading
H. Anton, C. Rorres, Applications of Linear Algebra (Wiley, New York, 1979)
C.M. Close, The Analysis of Linear Circuits (Harcourt, Brace and World, New York, 1966)
P.M. DeRusso, R.J. Roy, C.M. Close, State Variables for Engineers (Wiley, New York, 1965)
K. Hoffman, R. Kunze Linear Algebra (Prentice-Hall, Englewood Cliffs, 1960)
A. Kolmogorov, S. Fomin, Metric and Normed Spaces (Graylock Press, Rochester, 1957)
Technical Staff, CMOS Databook (National Semiconductor Corporation, Santa Clara, 1977)
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Davis, J.H. (2016). Fourier Series. In: Methods of Applied Mathematics with a Software Overview. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43370-7_2
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