On Banach spaces of absolutely and strongly convergent Fourier series

Szalay, I.; Tanović-Miller, N.

doi:10.1007/bf01951398

Cited by 4 publications

(2 citation statements)

References 7 publications

(8 reference statements)

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“…[14] or [16] and the references cited there. These notions were applied to trigonometric and Fourier series in a series of recent papers, [16] through [18], which led to the study of the related spaces of functions, [14], [15], [19]: [a] S\ak\ = S\ab\ =s"=Sp = (Sp) for p>\.…”

Section: Examples and Applicationsmentioning

confidence: 99%

Absolute boundedness and absolute convergence in sequence spaces

Buntinas¹,

Tanović-Miller²

1991

Proc. Amer. Math. Soc.

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Abstract.Let ft* be the set of all sequences h = (hk)k*Ax of Os and Is. A sequence x in a topological sequence space E has the property of absolute boundedness \AB\ if ft* • x = {y\yk = hkxk , h € ft*} is a bounded subset of E . The subspace E,AB, of all sequences with absolute boundedness in E has a natural topology stronger than that induced by E. A sequence x has the property of absolute sectional convergence \AK\ if, under this stronger topology, the net {h • x} converges to x , where h ranges over all sequences in ft* with a finite number of Is ordered coordinatewise (h1 < h" iff V/c, hk < hk ). Absolute boundedness and absolute convergence are investigated. It is shown that, for an F.K-space E, we have E = E,AB, if and only if E = l°° • E, and every element of E has the property \AK\ if and only if E = c0 • E . Solid hulls and largest solid subspaces of sequence spaces are also considered. The results are applied to standard sequence spaces, convergence fields of matrix methods, classical Banach spaces of Fourier series and to more recently introduced spaces of absolutely and strongly convergent Fourier series.

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Section: Examples and Applicationsmentioning

confidence: 99%

Absolute boundedness and absolute convergence in sequence spaces

Buntinas¹,

Tanović-Miller²

1991

Proc. Amer. Math. Soc.

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show abstract

“…Extending Hardy-Littlewood's concept of strong (C, 1) summability to Cesàro methods (C, α) of order α ≥ 0, Hyslop [8] arrived at his notion of strong convergence. Subsequently, this was successfully applied to the study of trigonometric series in several papers written by N. Tanović-Miller and her co-workers [16,17,18,13,14]. Strong convergence of trigonometric series attracts attention because of its position between ordinary and absolute convergence [4,16,17].…”

Section: Introductionmentioning

confidence: 99%