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Walsh function
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The system of Walsh functions (or, simply, Walsh system) may be viewed as a discrete, digital counterpart of continuous, analog system of trigonometric functions on the unit interval. Unlike trigonometric functions, Walsh functions are only piecewise-continuous, and, in fact, are piecewise constant. The functions take the values -1 and +1 only, on sub-intervals defined by dyadic fractions.
Both systems form a complete, orthonormal set of functions, an orthonormal basis in Hilbert space L2[0,1] of the square-integrable functions on the unit interval. Both are systems of bounded functions, unlike, say, Haar system or Franklin system.
Both trigonometric and Walsh systems admit natural extension by periodicity from the unit interval to the real line R. Furthermore, both Fourier analysis on the unit interval (Fourier series) and on the real line (Fourier transform) have thei
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