On the Convolution Powers of Complex Functions on $$\mathbb {Z}$$ Z

Randles, Evan; Saloff‐Coste, Laurent

doi:10.1007/s00041-015-9386-1

Cited by 19 publications

(80 citation statements)

References 21 publications

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“…For the general class of finitely supported complex valued functions on Z, [17] completely settles Questions (i) and (ii). For instance, consider the following theorem of [17].…”

Section: Introductionmentioning

confidence: 92%

“…In one dimension, Questions (i-iv) were recently addressed in the articles [4] and [17]. For the general class of finitely supported complex valued functions on Z, [17] completely settles Questions (i) and (ii). For instance, consider the following theorem of [17].…”

Section: Introductionmentioning

confidence: 97%

“…Theorem 1.1 (Theorem 1.1 of [17]). Let φ : Z → C have finite support consisting of more than one point.…”

Section: Introductionmentioning

confidence: 99%

“…In settling Question (ii), the article [17] gives an exhaustive account of local limit theorems in which the set of possible attractors includes the Airy function and the heat kernel evaluated at purely imaginary time. In addressing Question (iii), the article [4] contains a number of results concerning global space-time estimates for φ (n) for a finitely supported function φ -our results recapture (and extend in the case of Theorem 1.6) these results of [4].…”

Section: Introductionmentioning

confidence: 99%

“…To this end, we follow the article [4] and generalize the results therein. A complete theory for finitely supported functions on Z d , in which the results of [17] will fit, is not presently known. Not surprisingly, our results recapture the well-known results of random walk theory on Z d (see Subsection 7.6).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Convolution powers of complex functions on $\mathbb Z^d$

Randles¹,

Saloff‐Coste²

2017

Rev. Mat. Iberoam.

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The study of convolution powers of a finitely supported probability distribution φ on the d-dimensional square lattice is central to random walk theory. For instance, the nth convolution power φ (n) is the distribution of the nth step of the associated random walk and is described by the classical local limit theorem. Following previous work of P. Diaconis and the authors, we explore the more general setting in which φ takes on complex values. This problem, originally motivated by the problem of Erastus L. De Forest in data smoothing, has found applications to the theory of stability of numerical difference schemes in partial differential equations. For a complex valued function φ on Z d , we ask and address four basic and fundamental questions about the convolution powers φ (n) which concern sup-norm estimates, generalized local limit theorems, pointwise estimates, and stability. This work extends one-dimensional results of I. J. Schoenberg, T. N. E. Greville, P. Diaconis and the second author and, in the context of stability theory, results by V. Thomée and M. V. Fedoryuk.

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“…For the general class of finitely supported complex valued functions on Z, [17] completely settles Questions (i) and (ii). For instance, consider the following theorem of [17].…”

Section: Introductionmentioning

confidence: 92%