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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.


Approximants ℓj(x) employed to show bound (a) in lemma 6.1. For reference, we also show , ,  and .
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RSPA20140642F2: Approximants ℓj(x) employed to show bound (a) in lemma 6.1. For reference, we also show , , and .

Mentions:  Bound (a). Let{xj}j=03={0,16,13,12}and{yj}j=03={0,34,32,1}.For x∈[xj,xj+1), letℓj(x)=yj+1−yjxj+1−xj(x−xj)+yj for j=0,1andℓ2(x)=1(figure 2). Becausesin4/3,4/3−1⁡(y1)=(34) 2F 1(34,34;74;(34)4/3)<105100<110100<π24=π4/3,4/36and is an increasing function of t∈(0,πp,p/2), thensin4/3,4/3⁡(π4/3,4/3x1)>y1.


Basis properties of the p, q-sine functions.

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

Approximants ℓj(x) employed to show bound (a) in lemma 6.1. For reference, we also show , ,  and .
© Copyright Policy - open-access
Related In: Results  -  Collection

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

RSPA20140642F2: Approximants ℓj(x) employed to show bound (a) in lemma 6.1. For reference, we also show , , and .
Mentions:  Bound (a). Let{xj}j=03={0,16,13,12}and{yj}j=03={0,34,32,1}.For x∈[xj,xj+1), letℓj(x)=yj+1−yjxj+1−xj(x−xj)+yj for j=0,1andℓ2(x)=1(figure 2). Becausesin4/3,4/3−1⁡(y1)=(34) 2F 1(34,34;74;(34)4/3)<105100<110100<π24=π4/3,4/36and is an increasing function of t∈(0,πp,p/2), thensin4/3,4/3⁡(π4/3,4/3x1)>y1.

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.