About: Convolution quotient

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In mathematics, a convolution quotient is to the operation of convolution as a quotient of integers is to multiplication. Convolution quotients were introduced by Mikusiński, and their theory is sometimes called Mikusiński's operational calculus. The kind of convolution with which this theory is concerned is defined by Convolution quotients are used in an approach to making Dirac's delta function and other generalized functions logically rigorous.

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  • In mathematics, a convolution quotient is to the operation of convolution as a quotient of integers is to multiplication. Convolution quotients were introduced by Mikusiński, and their theory is sometimes called Mikusiński's operational calculus. The kind of convolution with which this theory is concerned is defined by It follows from the Titchmarsh convolution theorem that if the convolution of two functions that are continuous on is equal to 0 everywhere on that interval, then at least one of is 0 everywhere on that interval. A consequence is that if are continuous on then only if This fact makes it possible to define convolution quotients by saying that for two functions ƒ, g, the pair (ƒ, g) has the same convolution quotient as the pair (h * ƒ,h * g). Convolution quotients are used in an approach to making Dirac's delta function and other generalized functions logically rigorous. (en)
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  • In mathematics, a convolution quotient is to the operation of convolution as a quotient of integers is to multiplication. Convolution quotients were introduced by Mikusiński, and their theory is sometimes called Mikusiński's operational calculus. The kind of convolution with which this theory is concerned is defined by Convolution quotients are used in an approach to making Dirac's delta function and other generalized functions logically rigorous. (en)
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  • Convolution quotient (en)
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