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The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called a ramp waveform. The convention is that a sawtooth wave ramps upward and then sharply drops.
The Fourier series expansion of the sawtooth function (below) looks more complicated than the simple formula () =, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the heat equation .
The first polynomial is a sawtooth function. Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions, and P 0 (x) is not even a function, being the derivative of a sawtooth and so a Dirac comb. The following properties are of interest, valid for all :
The six arrows represent the first six terms of the Fourier series of a square wave. The two circles at the bottom represent the exact square wave (blue) and its Fourier-series approximation (purple). (Odd) harmonics of a 1000 Hz square wave Graph showing the first 3 terms of the Fourier series of a square wave
Inspired by correspondence in Nature between Michelson and A. E. H. Love about the convergence of the Fourier series of the square wave function, J. Willard Gibbs published a note in 1898 pointing out the important distinction between the limit of the graphs of the partial sums of the Fourier series of a sawtooth wave and the graph of the limit ...
These are called Fourier series coefficients. The term Fourier series actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients. When the non-zero portion of the input function has finite duration, the Fourier transform is continuous and finite-valued.
The Fourier transform of a periodic function cannot be defined using the integral formula directly. In order for integral in Eq.1 to be defined the function must be absolutely integrable. Instead it is common to use Fourier series. It is possible to extend the definition to include periodic functions by viewing them as tempered distributions.
The basis functions are sine waves with wavelengths λ / n (for integer n) shorter than the wavelength λ of the sawtooth itself (except for n = 1, the fundamental wave). A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function f.