Power boundedness in Fourier and Fourier–Stieltjes algebras and other commutative Banach algebras

Kaniuth, Eberhard; Lau, Anthony To-Ming; Ülger, A.

doi:10.1016/j.jfa.2010.11.012

Cited by 7 publications

(5 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In view of (13) and the preceding discussion, one expects an estimate of the form (14) to hold for φ (n) , although, we note that no such estimate can be established on these grounds (this is due to the error term in (13)). This however motivates the correct form and we are able to establish the following result which captures, as a special case, the situation described above in which Ω(φ) = {ξ 0 }.…”

Section: Introductionmentioning

confidence: 74%

“…For an approximate difference scheme to an initial value problem, the property (1) is necessary and sufficient for convergence to a classical solution; this is the so-called Lax equivalence theorem [18] (see Section 6). Property (1) is also called power boundedness and can be seen in the context of Banach algebras where φ is an element of the Banach algebra ( 1 (Z d ), • 1 ) equipped with the convolution product [14,21].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Randles¹,

Saloff‐Coste²

2017

Rev. Mat. Iberoam.

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 99%

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

Randles¹,

Saloff‐Coste²

2017

Rev. Mat. Iberoam.

View full text Add to dashboard Cite

show abstract

“…In particular, it is shown that if u is a power bounded element of B(G), then the set of all x in G such that u(x) = 1 is in the closed coset ring of G, which was completely characterized by Forrest [9] (and previously by Gilbert [13] and Schreiber [40] in the abelian case). Further related investigations are carried out in the articles [19] and [20], continuing Tony's collaboration with E. Kaniuth and A.Ülger.…”

Section: Multipliers Of Commutative Banach Algebras With Applications Tomentioning

confidence: 99%

The mathematical work of Anthony To-Ming Lau

Moslehian¹,

Neufang²

2016

Ann. Funct. Anal.

View full text Add to dashboard Cite

show abstract

“…In the first case, by [24, Proposition 1.1], γ extends to some continuous positive definite function σ on G, and so λσ is the required extension of u. Alternatively, instead of using [24], we could proceed as follows. Since |u(x)| = 1 for all x ∈ G 0 , by [21,Proposition 4.5] either u is constant or w * -lim n→∞ u n = 0. Therefore, to establish the theorem, we have to show that if w * -lim n→∞ u n = 0, then there exists w ∈ B(G) such that w| G 0 = u and w * -lim n→∞ w n = 0.…”

Section: This Formula In Turn Impliesmentioning

confidence: 99%

“…for all n. It therefore suffices to verify that v is also power bounded. This can be done by using precisely the same arguments as in the proof of [21,Theorem 3.4].…”

mentioning

confidence: 99%

Power boundedness in Banach algebras associated with locally compact groups

2014

View full text Add to dashboard Cite

Let G be a locally compact group and B(G) the Fourier-Stieltjes algebra of G. Pursuing our investigations of power bounded elements in B(G), we study the extension property for power bounded elements and discuss the structure of closed sets in the coset ring of G which appear as 1-sets of power bounded elements. We also show that L 1 -algebras of noncompact motion groups and of noncompact IN-groups with polynomial growth do not share the so-called power boundedness property. Finally, we give a characterization of power bounded elements in the reduced Fourier-Stieltjes algebra of a locally compact group containing an open subgroup which is amenable as a discrete group.Introduction. An element a of a Banach algebra A is said to be power bounded if sup n∈N a n < ∞. Power bounded elements in Banach algebras, especially power bounded operators on Banach spaces, have been studied by several authors, with emphasis on the impact on spectra [1], [23], [30], [31]. Power boundedness of measures on the real line has first been dealt with in [4] (see also [2] and [5] for related problems). For general locally compact abelian groups G, the most comprehensive work on power boundedness in the measure algebra M (G) and the L 1 -algebra L 1 (G) is due to Schreiber [34]. Actually, [34] has substantially influenced and inspired our previous investigations [20]-[22] on power boundedness in Fourier and Fourier-Stieltjes algebras of locally compact groups, and also the present study. Naturally, for nonabelian groups the proofs turn out to be much more involved.In [20, Theorem 4.1] we have shown that if G is an arbitrary locally compact group and u is any power bounded element of B(G), then the sets

show abstract

Power boundedness in Fourier and Fourier–Stieltjes algebras and other commutative Banach algebras

Cited by 7 publications

References 23 publications

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

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

The mathematical work of Anthony To-Ming Lau

Power boundedness in Banach algebras associated with locally compact groups

Contact Info

Product

Resources

About