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Basis properties of the p, q-sine functions.

Boulton L, Lord GJ - Proc. Math. Phys. Eng. Sci. (2015)

Bottom Line: Our findings rely on a Beurling decomposition of the corresponding change of coordinates in terms of shift operators of infinite multiplicity.We also determine refined bounds on the Riesz constant associated with this family.These results seal mathematical gaps in the existing literature on the subject.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics and Maxwell Institute for Mathematical Sciences , Heriot-Watt University , Edinburgh EH14 4AS, UK.

ABSTRACT

We improve the currently known thresholds for basisness of the family of periodically dilated p,q-sine functions. Our findings rely on a Beurling decomposition of the corresponding change of coordinates in terms of shift operators of infinite multiplicity. We also determine refined bounds on the Riesz constant associated with this family. These results seal mathematical gaps in the existing literature on the subject.

No MeSH data available.


Relation between the various statements of this paper with those of references [4,5], for the case p=q. The positions of p1,  and the value of ε are set only for illustration purposes, as we are certain only that . (Online version in colour.)
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RSPA20140642F3: Relation between the various statements of this paper with those of references [4,5], for the case p=q. The positions of p1, and the value of ε are set only for illustration purposes, as we are certain only that . (Online version in colour.)

Mentions: Section 6 is concerned with particular details of the case of equal indices p=q, and it involves results on both the general case r>1 and the specific case r=2. Rather curiously, we have found another gap which renders incomplete the proof of invertibility of A for p1<p<2 originally published in [5]. See remark 6.3. Moreover, the application of Bushell & Edmunds [4], theorem 4.5 only gets to a basisness threshold of , where is defined by the identity1.6πp~1,p~1=2π2π2−8.See also [2], remark 2.1. In theorem 6.5, we show that is indeed a Schauder basis of Lr for where , see [14], problem 1. As , basisness is now guaranteed for all p=q>p3 (figure 3).


Basis properties of the p, q-sine functions.

Boulton L, Lord GJ - Proc. Math. Phys. Eng. Sci. (2015)

Relation between the various statements of this paper with those of references [4,5], for the case p=q. The positions of p1,  and the value of ε are set only for illustration purposes, as we are certain only that . (Online version in colour.)
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4309129&req=5

RSPA20140642F3: Relation between the various statements of this paper with those of references [4,5], for the case p=q. The positions of p1, and the value of ε are set only for illustration purposes, as we are certain only that . (Online version in colour.)
Mentions: Section 6 is concerned with particular details of the case of equal indices p=q, and it involves results on both the general case r>1 and the specific case r=2. Rather curiously, we have found another gap which renders incomplete the proof of invertibility of A for p1<p<2 originally published in [5]. See remark 6.3. Moreover, the application of Bushell & Edmunds [4], theorem 4.5 only gets to a basisness threshold of , where is defined by the identity1.6πp~1,p~1=2π2π2−8.See also [2], remark 2.1. In theorem 6.5, we show that is indeed a Schauder basis of Lr for where , see [14], problem 1. As , basisness is now guaranteed for all p=q>p3 (figure 3).

Bottom Line: Our findings rely on a Beurling decomposition of the corresponding change of coordinates in terms of shift operators of infinite multiplicity.We also determine refined bounds on the Riesz constant associated with this family.These results seal mathematical gaps in the existing literature on the subject.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics and Maxwell Institute for Mathematical Sciences , Heriot-Watt University , Edinburgh EH14 4AS, UK.

ABSTRACT

We improve the currently known thresholds for basisness of the family of periodically dilated p,q-sine functions. Our findings rely on a Beurling decomposition of the corresponding change of coordinates in terms of shift operators of infinite multiplicity. We also determine refined bounds on the Riesz constant associated with this family. These results seal mathematical gaps in the existing literature on the subject.

No MeSH data available.