bed-of-nails function | comb function | replicating symbol | sampling function

The shah function is defined by ш(x) | congruent | sum_(n = - ∞)^∞ δ(x - n) | congruent | sum_(n = - ∞)^∞ δ(x + n), where δ(x) is the delta function, so ш(x) = 0 for x not element Z (i.e., x is not an integer). The shah function is also called the sampling symbol or replicating symbol, and is implemented in the Wolfram Language as DiracComb[x]. It obeys the identities ш(a x) | = | 1/( left bracketing bar a right bracketing bar ) sum_(n = - ∞)^∞ δ(x - n/a) ш(-x) | = | ш(x) ш(x + n) | = | ш(x) ш(x - 1/2) | = | ш(x + 1/2).

convolution | delta function | impulse pair | sinc function