On Absolute Completeness of Systems of Exponentials on a Closed Interval

Krasichkov-Ternovskii, Igor Fedorovich

doi:10.1070/sm1988v059n02abeh003137

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…Therefore, using Remark 3 to the result of [11] cited above, we can claim that x expands into an absolutely convergent series in E m+1 , and all the more in E. Therefore, we obtain the following generalization of Theorem A by a method different from [2].…”

Section: Problemmentioning

confidence: 92%

“…According to Remark 2 to Theorem 2 of [11], an element x is approximated in E m+1 by finite linear combinations of the form x k ∈Xm c k…”

Section: Problemmentioning

confidence: 99%

“…We can simplify the statement of Theorem 3 if we pass to the realization of functionals via the transform (11). However, this procedure hits the problem of internal description of A c (K m , Λ) and the image F m = T (A c (K m , Λ)) of its dual space, a satisfactory solution to which for an arbitrarily located sequence Λ ⊂ C \ G is doubtful (see [16]).…”

Section: Application To Systems Of Partial Fractions In the Space Of mentioning

confidence: 99%

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Dual characterization of absolutely representing systems in inductive limits of Banach spaces

Sherstyukov

2010

Sib Math J

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Section: Problemmentioning

confidence: 92%

“…According to Remark 2 to Theorem 2 of [11], an element x is approximated in E m+1 by finite linear combinations of the form x k ∈Xm c k…”

Section: Problemmentioning

confidence: 99%

Section: Application To Systems Of Partial Fractions In the Space Of mentioning

confidence: 99%

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Dual characterization of absolutely representing systems in inductive limits of Banach spaces

Sherstyukov

2010

Sib Math J

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A Uniqueness Theorem for Subharmonic Functions of Finite Order

Хабибуллин¹

1992

Math. USSR Sb.

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Absolute completeness of systems of exponentials on convex compact sets

Krasichkov-Ternovskii¹,

Shilova²

2005

Sb. Math.

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We determined the relationship between intra-fractional breathing motion and safety margins, using daily real-time tumour tracking data of 40 patients (43 tumour locations), treated with radiosurgery at Hokkaido University. We limited our study to the dose-blurring effect of intra-fractional breathing motion, and did not consider differences in positioning accuracy or systematic errors. The additional shift in the prescribed isodose level (e.g. 95%) was determined by convolving a one-dimensional dose profile, having a dose gradient representing an 8 MV beam through either lung or water, with the probability density function (PDF) of breathing. This additional shift is a measure for the additional margin that should be applied in order to maintain the same probability of tumour control as without intra-fractional breathing. We show that the required safety margin is a nonlinear function of the peak-to-peak breathing motion. Only a small reduction in the shift of isodose curves was observed for breathing motion up to 10 mm. For larger motion, 20 or 30 mm, control of patient breathing during irradiation, using either gating or breath hold, can allow a substantial reduction in safety margins of about 7 or 12 mm depending on the dose gradient prior to blurring. Clinically relevant random setup uncertainties, which also have a blurring effect on the dose distribution, have only a small effect on the margin needed for intra-fractional breathing motion. Because of the one-dimensional nature of our analysis, the resulting margins are mainly applicable in the superior-inferior direction. Most measured breathing PDFs were not consistent with the PDF of a simple parametric curve such as cos 4 , either because of irregular breathing or base-line shifts. Instead, our analysis shows that breathing motion can be modelled as Gaussian with a standard deviation of about 0.4 times the peak-to-peak breathing motion.

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On Absolute Completeness of Systems of Exponentials on a Closed Interval

Cited by 3 publications

References 15 publications

Dual characterization of absolutely representing systems in inductive limits of Banach spaces

Dual characterization of absolutely representing systems in inductive limits of Banach spaces

A Uniqueness Theorem for Subharmonic Functions of Finite Order

Absolute completeness of systems of exponentials on convex compact sets

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