By Jesper Lützen (auth.)
I first discovered the speculation of distributions from Professor Ebbe Thue Poulsen in an undergraduate path at Aarhus college. either his lectures and the textbook, Topological Vector areas, Distributions and Kernels by means of F. Treves, utilized in the path, opened my eyes to the wonder and summary simplicity of the idea. although my incomplete learn of many branches of classical research left me with the query: Why is the idea of distributions vital? In my endured reviews this question was once steadily responded, yet my becoming curiosity within the heritage of arithmetic triggered me to change my query to different questions equivalent to: For what objective, if any, was once the speculation of distributions initially created? Who invented distributions and while? I fast stumbled on solutions to the final questions: distributions have been invented via S. Sobolev and L. Schwartz round 1936 and 1950, respectively. realizing this solution, even if, simply created a brand new query: Did Sobolev and Schwartz build distributions from scratch or have been there previous tendencies and, if this is the case, what have been they? it really is this question, in regards to the pre heritage of the idea of distributions, which i try to resolution during this ebook. so much of my examine happened on the heritage of technology division of Aarhus college. I desire to thank this division for its monetary and highbrow aid. i'm particularly thankful to Lektors Kirsti Andersen from the background of technological know-how division and Lars Mejlbo from the math division, for his or her kindness, confident feedback, and encouragement.
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31. Bacher's ideas were continued by his fellow-countryman, G. C. Evans (born 1887), professor at Rice Institute. Evans worked a great deal in potential theory in which he was the first to use Lebesgue-Stieltjes integrals for studying potentials of arbitrary mass-or charge distributions in the plane and in space. In his first published account of his theories [1920] Evans gave the following sketch of the source of his ideas: These studies originated in 1907, when it first became apparent to me that the [potential] theory was unnecessarily complicated by the form of the Laplacian operator, but I did not work on the subject until 1913 when it occurred to me to use instead of the operator (36) the operator .
By a simple argument which used only Fubini's theorem and the identity (49) proved earlier he showed that: if in the definition of potential function of generalized derivatives the function is assumed to be continuous, as a point function, the specialized concept thus obtained is identical with the one formulated by Tonelli .... Thus it happens that the theorems proved by Tonelli in the last cited reference [Tonelli 1928/29J are in their essence special cases of those given earlier by the author.
As (2°) above. 28 With these new definitions at his disposal Tonelli could prove (A)11'Th~ s~rface described by (19) has finite Lebesgue area S,fis of bounded {l. VarIatIOn. (B) Ifone of the equivalent statements of A holds true, then H)1 + p2 + q2 Q exists and S ~ 11 )1 + p2 + q2. Q (C) If I in (19) has finite Lebesgue measure the following equivalence holds: area S = 11)1 + p2 + q2 ~I is absolutely continuous. Q In this way the question ofthe applicability of the formula (20) in the case of surfaces defined by (19) was completely settled.