Developments of the theory of generalized functions or distributions – A vision of Paul Dirac
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Lokenath Debnath
Abstract
This paper deals with six major developments of the theory of generalized functions or distributions as a whole new branch of modern analysis. These include (i) The Dirac delta function as the limit of sequences of ordinary functions, (ii) Schwartz´s new theory of distributions based on test functions, (iii) Temple–Lighthill´s simpler analytical approach to generalized functions based on good functions, (iv) Mikusinski´s algebraic approach to generalized functions, (v) Cauchy´s representation of distributions by analytic functions, and (vi) Sato´s less abstract and computationally more effective approach to generalized functions (or hyperfunctions) based on complex variables. Included are basic features, properties and examples of generalized functions including the Dirac delta function and the Heaviside function. Some additional properties of convolution are discussed in the end.
© by Oldenbourg Wissenschaftsverlag, München, Germany
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Articles in the same Issue
- Uniform spacing of zeros of orthogonal polynomials for locally doubling measures
- A uniqueness polynomial for equi-polar meromorphic functions
- The spectrum of the Hausdorff operator on a subspace of L2(ℝ)
- An application of q-mathematics on absolute summability of orthogonal series
- On proofs for monotonicity of a function involving the psi and exponential functions
- On the (M,λn) method of summability
- Developments of the theory of generalized functions or distributions – A vision of Paul Dirac
