on the surface delta function. Fourier transform The delta function is a tempered distribution, and therefore it has a well-defined Fourier transform. Formally, one finds Properly speaking, the Fourier transform of a distribution is defined by imposing self-adjointness of the Fourier transform under the duality pairing of tempered distributions with Schwartz functions. Thus is defined as the unique tempered distribution satisfying for all Schwartz functions . And indeed it follows from this that As a result of this identity, the convolution of the delta function with any other tempered distribution S is simply S: That is to say that δ is an identity element for the convolution on tempered distributions, and in fact, the space of compactly supported distributions under convolution is an associative algebra with identity the delta function. This property is fundamental in signal processing, as convolution with a tempered distribution is a linear time-invariant system, and applying the linear time-invariant system measures its impulse response. The impulse response can be computed to any desired degree of accuracy by choosing a suitable approximation for δ, and once it is known, it characterizes the system completely. See . The inverse Fourier transform of the tempered distribution f(ξ) = 1 is the delta function. Formally, this is expressed and more rigorously, it follows since for all Schwartz functions f. In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on R. Formally, one has This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution is which again follows by imposing self-adjointness of the Fourier transform. By analytic continuation of the Fourier transform, the Laplace transform of the delta function is found to be Derivatives of the Dirac delta function The derivative of the Dirac delta distribution, denoted and also called the Dirac delta prime or Dirac delta derivative as described in Laplacian of the indicator, is defined on compactly supported smooth test functions by The first equality here is a kind of integration by parts, for if were a true function then The -th derivative of is defined similarly as the distribution given on test functions by In particular, is an infinitely differentiable distribution. The first derivative of the delta function is the distributional limit of the difference quotients: More properly, one has where is the translation operator, defined on functions by , and on a