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