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- W2017650306 abstract "Making the assumption that the graph of a periodic function is given, the problem of the best way of determining the Fourier constants in the series equation which represents it is considered. The ordinary method of procedure is to neglect all the harmonics above a certain order and determine the coefficients of the harmonic terms by making the curve represented by this equation pass through a number of arbitrarily selected points on the given curve. This is the method used, for instance, by Runge and Grover. A serious defect in this solution is that the values found for the amplitudes of the harmonics, more especially for the higher harmonics, may be very different from their true Fourier values. The method gives no indication of the magnitude of these errors. Gauss pointed out many years ago that the solution of this limited problem could be written down at once mathematically, and that it was of importance in certain interpolation problems in astronomy. Another method has been suggested recently by Silvanus Thompson. He uses certain series formulae for finding the Fourier constants. The author suggests other series formulae of a similar kind. If the given curve be approximately sine shaped so that the amplitudes of the higher harmonics are small, this method is both simple and accurate. For distorted waves, however, the lack of accuracy is serious in practice. It has also the drawback that an error made in computing the value of one of the constants may introduce errors in the computed values of others. The author gives numerical examples to illustrate the accuracy attainable by the use of infinite series formulae. He concludes by pointing out that in the great majority of cases much the best method of procedure when determining the constants is to evaluate Fourier's integrals by the methods of mechanical quadrature given in books on the calculus of finite differences. In particular he has found that Weddle's rule is admirably adapted for the practical computation of the Fourier integrals. By means of this rule, a new and simple proof of which is given, each constant is determined separately to a high order of accuracy. Numerical examples are given to illustrate this. The series formulae used by Thompson can be usefully employed either for verifying the values found by mechanical quadrature or for indicating when the higher harmonics cannot be neglected." @default.
- W2017650306 created "2016-06-24" @default.
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- W2017650306 date "1914-12-01" @default.
- W2017650306 modified "2023-09-28" @default.
- W2017650306 title "Practical Harmonic Analysis" @default.
- W2017650306 doi "https://doi.org/10.1088/1478-7814/27/1/310" @default.
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