On the Laplace Transform for Distributions

Price, Douglas B.

doi:10.1137/0506006

Cited by 10 publications

(8 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…holds for all b N, and is integrable iff there exists a positive integer K such that (5) I(, )l-<Kmaxsupl(t)l, tR, O<-]<=K, holds for all b @ (see [3]). Here denotes the ordinary ]th order derivative of .C learly, ', and we shall see (Prop.…”

mentioning

confidence: 97%

“…We shall make use of the well-known subspaces of tempered distributions '(R and 6'(), (on the two real lines R and ), as well as the not so well-known subspace of integrable distributions (R) [3]. We recall that a distribution f is tempered iff (if and only if) there exist positive integers M and N such that (4) …”

mentioning

confidence: 99%

“…Here, and throughout the paper, the readers are assumed to be familiar with elementary aspects of distributions and their Fourier transforms (such as given in [1], [9]). In 3, we consider some preliminary results which come principally from recent work in references [3] and [6]. This section is used mainly for motivation, but includes important definitions of rings of distributions and related functions.…”

mentioning

confidence: 99%

“…Preliminary results and definitions. We shall need several results from references [3] and [6]. 246 RAIMOND A. STRUBLE PROPOSITION 1.…”

mentioning

confidence: 99%

See 3 more Smart Citations

The Numerical-Valued Fourier Transform in the Two-Sided Operational Calculus

Struble¹

1977

SIAM J. Math. Anal.

View full text Add to dashboard Cite

The classical and distributional Fourier transform is extended to the ultimate setting in which it can be considered to be numerical-valued. It is extended as a ring isomorphism onto the ring of all measurable and finite almost everywhere functions under ordinary (pointwise) addition and multiplication of functions. The essential technique needed for this extension is the familiar algebraic procedure (first applied by Mikusifiski to the operational calculus in the late 40's) of imbedding a given ring in a larger ring of fractions, the denominators being nondivisors of zero. Where Mikusifiski's application resulted in a field of one-sided operators, the present application results in a ring of two-sided operators. Beyond this, only classical Fourier analysis is needed, though the extension of the latter to distributions is very useful in identifying many of the operators.A descriptive subtitle for this paper would be Basic definitions and theorems, with applications to be considered later. For reasons of motivation and practical emphasis, the operational calculus is developed in a slightly more restrictive setting where the Fourier transforms are continuous almost everywhere.

show abstract

mentioning

confidence: 97%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…Preliminary results and definitions. We shall need several results from references [3] and [6]. 246 RAIMOND A. STRUBLE PROPOSITION 1.…”

mentioning

confidence: 99%

See 2 more Smart Citations

The Numerical-Valued Fourier Transform in the Two-Sided Operational Calculus

Struble¹

1977

SIAM J. Math. Anal.

View full text Add to dashboard Cite

show abstract

“…Various generalizations of Stieltjes transform have been given by various authors viz., Widder [3], Pollard [4], Sumner [5], Mishra [6], Pathak([7]- [8]), Rao [9], Varma[lO], Arya([11]- [15]), Ghosh([16]- [18]), Boas & Widder[19], and Dube [20]. [27]. …”

Section: Introductionmentioning

confidence: 99%