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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.


Region of the (p,q)-plane where theorem 5.2(c) applies. Even when we know A is invertible in this region as a consequence of theorem 5.2(a), the upper bound on the Riesz constant provided by (4.2) improves upon that provided by (1.2) (case r=2). In this graph, 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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RSPA20140642F5: Region of the (p,q)-plane where theorem 5.2(c) applies. Even when we know A is invertible in this region as a consequence of theorem 5.2(a), the upper bound on the Riesz constant provided by (4.2) improves upon that provided by (1.2) (case r=2). In this graph, p corresponds to the horizontal axis and q to the vertical axis and the dotted line shows p=q. (Online version in colour.)

Mentions: We recover [3], corollary 4.3 from the part (a) of this theorem by observing that for all p,q>1,a1≥22∫01/22xsin⁡(πx) dx=42π2.In fact, for (p,q)∈(1,2)2, the better estimatea1≥22∫01sin2⁡(πx) dx=22,ensures invertibility of A for all r>1 whenever5.8πp,q<2π2π2−8.See figures 4 and 5.


Basis properties of the p, q-sine functions.

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

Region of the (p,q)-plane where theorem 5.2(c) applies. Even when we know A is invertible in this region as a consequence of theorem 5.2(a), the upper bound on the Riesz constant provided by (4.2) improves upon that provided by (1.2) (case r=2). In this graph, 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

RSPA20140642F5: Region of the (p,q)-plane where theorem 5.2(c) applies. Even when we know A is invertible in this region as a consequence of theorem 5.2(a), the upper bound on the Riesz constant provided by (4.2) improves upon that provided by (1.2) (case r=2). In this graph, p corresponds to the horizontal axis and q to the vertical axis and the dotted line shows p=q. (Online version in colour.)
Mentions: We recover [3], corollary 4.3 from the part (a) of this theorem by observing that for all p,q>1,a1≥22∫01/22xsin⁡(πx) dx=42π2.In fact, for (p,q)∈(1,2)2, the better estimatea1≥22∫01sin2⁡(πx) dx=22,ensures invertibility of A for all r>1 whenever5.8πp,q<2π2π2−8.See figures 4 and 5.

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.