✦ LIBER ✦

On the nonomnipotence of regular summability methods

✍ Scribed by S Haber; O Shisha

Publisher: Elsevier Science
Year: 1978
Tongue: English
Weight: 74 KB
Volume: 28
Category: Article
ISSN: 0001-8708
DOI: 10.1016/0001-8708(78)90115-9

No coin nor oath required. For personal study only.

✦ Synopsis

Let T = (tm,J (m, n = I, 2 ,...; all t,,, , > 0) define a regular summability method. It is known [l] that there is a bounded divergent sequence whose T-transform is also divergent. Here we point out that one can say more: namely, that for some real, bounded, divergent sequence {a,}~=, , its T-transform diverges just a badly as itself. For every real sequence (a,}Ei , its T-transform {b,}~=, satisfies [2], so that, if {a,} is bounded, and, thus, the divergence of (blE} is not worse than that of (a,}. Our goal is a real, bounded, divergent sequence {a,> for which equality holds in (*). As such a sequence one can take the sequence, consisting of l's and -l's, defined in [l], as the argument given there does, in fact, establish the desired properties.

If the t,., are not assumed 20, but only real, matters are a bit worse. Inequality (*) is replaced by where

📜 SIMILAR VOLUMES

On Scales of Summability Methods

✍ Rüdiger Kiesel 📂 Article 📅 1995 🏛 John Wiley and Sons 🌐 English ⚖ 402 KB

In this paper we consider generalized Norlund methods (Nap), a > -1, power series methods (J,) and the iteration product of two such methods. A particular case is that of the Cesaro means (C,) with corresponding power series method (A), i.e., Abel's method. We obtain generalizations of inclusion, an

On the Strong Logarithmic Summabilities

On the Strong Logarithmic Summabilities of Fourier Series

✍ Włodzimierz Łenski 📂 Article 📅 1985 🏛 John Wiley and Sons 🌐 English ⚖ 270 KB

On H∞ -Summability and the Uniform Distr

On H∞ -Summability and the Uniform Distribution of Sequences

✍ Peter Schatte 📂 Article 📅 1983 🏛 John Wiley and Sons 🌐 English ⚖ 248 KB 👁 1 views

On the supersaturation of regular soluti

On the supersaturation of regular solutions

✍ Dr. habil. J. Bohm 📂 Article 📅 1982 🏛 John Wiley and Sons 🌐 English ⚖ 76 KB

On the Riemann Summability of Fourier In

On the Riemann Summability of Fourier Integrals and Real Hardy Spaces

✍ Ferenc Móricz 📂 Article 📅 2000 🏛 John Wiley and Sons 🌐 English ⚖ 213 KB 👁 2 views

We consider the Riemann means of single and multiple Fourier integrals of functions belonging to L 1 or the real Hardy spaces defined on IR n , where n ≥ 1 is an integer. We prove that the maximal Riemann operator is bounded both from H 1 (IR) into L 1 (IR) and from L 1 (IR) into weak -L 1 (IR). We

On the Regularity of Multivariate Hermit

On the Regularity of Multivariate Hermite Interpolation

✍ Hakop A. Hakopian 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 148 KB

In this paper a result due to Gevorgian, Sahakian, and the author concerning the regularity of bivariate Hermite interpolation is generalized in two directions: in the bivariate case and for arbitrary dimensions. Also a notion of independence (preregularity) of interpolation conditions is discussed