| 11. | Trigonometric functions also prove to be useful in the study of general periodic functions.
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| 12. | Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry.
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| 13. | The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774.
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| 14. | This can also be achieved by requiring certain symmetries and that sine be a periodic function.
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| 15. | General mathematical techniques for analyzing non-periodic functions fall into the category of Fourier analysis.
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| 16. | :Briefly, Zygmund's proof starts by proving the statement for continuous periodic functions.
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| 17. | This means periodic functions such as the sine and cosine functions cannot exist in Hardy fields.
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| 18. | One common generalization of periodic functions is that of "'antiperiodic functions " '.
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| 19. | Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions.
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| 20. | Denoting the sine or cosine basis functions by, the expansion of the periodic function takes the form:
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