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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.


(a–e) Different relations and boundaries between the regions of the (p,q)-plane where theorem 5.2(a) and (b) as well as proposition 7.1 (with different values of k) apply. In all graphs, p corresponds to the horizontal axis and q to the vertical axis and the dotted line shows p=q. (Online version in colour.)
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RSPA20140642F4: (a–e) Different relations and boundaries between the regions of the (p,q)-plane where theorem 5.2(a) and (b) as well as proposition 7.1 (with different values of k) apply. In all graphs, p corresponds to the horizontal axis and q to the vertical axis and the dotted line shows p=q. (Online version in colour.)

Mentions: Because a1>0, (4.1) supersedes (1.2), only when the pair (p,q) is such that a9>0. From this corollary, we see below that the change of coordinates is invertible in a neighbourhood of the threshold set by the condition (1.3). See proposition 7.1 and figures 3 and 4].


Basis properties of the p, q-sine functions.

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

(a–e) Different relations and boundaries between the regions of the (p,q)-plane where theorem 5.2(a) and (b) as well as proposition 7.1 (with different values of k) apply. In all graphs, p corresponds to the horizontal axis and q to the vertical axis and the dotted line shows p=q. (Online version in colour.)
© Copyright Policy - open-access
Related In: Results  -  Collection

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

RSPA20140642F4: (a–e) Different relations and boundaries between the regions of the (p,q)-plane where theorem 5.2(a) and (b) as well as proposition 7.1 (with different values of k) apply. In all graphs, p corresponds to the horizontal axis and q to the vertical axis and the dotted line shows p=q. (Online version in colour.)
Mentions: Because a1>0, (4.1) supersedes (1.2), only when the pair (p,q) is such that a9>0. From this corollary, we see below that the change of coordinates is invertible in a neighbourhood of the threshold set by the condition (1.3). See proposition 7.1 and figures 3 and 4].

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.