Unboundedness of the shift operator with respect to the Franklin system in the space L1

Gevorkyan, G. G.

doi:10.1007/bf01158404

Cited by 8 publications

(7 citation statements)

References 4 publications

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“…O<s<ll s --r (10) Item 1) of the lemma follows immediately from (10). Further, let t ~ [a -6, b + e I .…”

Section: ) Isn(f T)tmentioning

confidence: 84%

“…Let the series (6) satisfy condition (10). Then for any bounded sequence {e.}~=0 the sedes (7)/s a Fourier-IZ~ankl;n series in the sense of A-integration.…”

Section: ~)O ~- Laf(t)l Dt < Cle2r/22k#(j~ R)) (:()mentioning

confidence: 97%

“…]=k Obviously (see (19), (24), (25), (12), (10) Using Ciesielski's theorem [9,14], according to which a majorant of the partial sums of a Fourier-Franklin series is an operator of the type (2.2), and taking into account relations (14), (18) Taking into account the fact that 2i/2#(Bi) ---* 0 as i --* +oo (also see (10), (12), (13) (6) is the Fourler-Fr-nklin series of a Lebesgue integrable function, but rather the fact that the series (6) satisfies condition (10). Therefore, the following theorem holds.…”

Section: ~)O ~- Laf(t)l Dt < Cle2r/22k#(j~ R)) (:()mentioning

confidence: 99%

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Majorants and uniqueness of series in the Franklin system

Gevorkyan¹

1996

Math Notes

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ABSTRACT. It is proved that if a series in the Franklin system converges almost everywhere to a function f (t) and the distribution function of the majorant of partial sums satisfies the conditionas A -* co, then this series is a Fourier series for Lebesgue integrable functions f(t). In the general case the coefficients of the series are reconstructed by means of an A-integral. (1) A--*-i-ooThe following question arises in connection with Theorem I. Let the Frank!in series aJ.Ct)satisfy condition (1) and converge almost everywhere (a.e.) to some function f(t). Can the coefBcients of the series be expressed in terms of the function f(t)? This problem is treated in w of this article. It can be shown that if the sequence {At}k~=1 that provides the lower bound in condition (1) is known, then the coefficients an can be expressed in terms of f(t). Namely, the following theorem holds.

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“…O<s<ll s --r (10) Item 1) of the lemma follows immediately from (10). Further, let t ~ [a -6, b + e I .…”

Section: ) Isn(f T)tmentioning

confidence: 84%

“…Let the series (6) satisfy condition (10). Then for any bounded sequence {e.}~=0 the sedes (7)/s a Fourier-IZ~ankl;n series in the sense of A-integration.…”

Section: ~)O ~- Laf(t)l Dt < Cle2r/22k#(j~ R)) (:()mentioning

confidence: 97%

Section: ~)O ~- Laf(t)l Dt < Cle2r/22k#(j~ R)) (:()mentioning

confidence: 99%

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Majorants and uniqueness of series in the Franklin system

Gevorkyan¹

1996

Math Notes

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“…(8) Note that if I = N then P n = 0 for n 0 but the equality (8) remains still valid for all k, n ∈ Z. Therefore, applying (8) and…”

Section: Time Operators On Banach Spacesmentioning

confidence: 93%

Evolution semigroups and time operators on Banach spaces

Suchanecki

Gómez-Cubillo

2010

Journal of Mathematical Analysis and Applications

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We present new general methods to obtain shift representation of evolution semigroups defined on Banach spaces. We introduce the notion of time operator associated with a generalized shift on a Banach space and give some conditions under which time operators can be defined on an arbitrary Banach space. We also tackle the problem of scaling of time operators and obtain a general result about the existence of time operators on Banach spaces satisfying some geometric conditions. The last part of the paper contains some examples of explicit constructions of time operators on function spaces.

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“…The boundedness of the shift operator for the Franklin system in L p (I) (1 < p < ∞) arises from [12], while the unboundedness in L 1 (I) is proved in [18].…”

Section: Some Classical Examplesmentioning

confidence: 99%