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University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > $t$-design curves and mobile sampling on the sphere

## $t$-design curves and mobile sampling on the sphereAdd to your list(s) Download to your calendar using vCal - Karlheinz Groechenig (University of Vienna)
- Thursday 18 July 2024, 09:30-10:10
- Seminar Room 1, Newton Institute.
If you have a question about this talk, please contact nobody. DREW01 - Multivariate approximation, discretization, and sampling recovery A spherical $t$-design curve is a curve on the $d$-dimensional sphere such that the corresponding line integral integrates polynomials of degree $t$ exactly. Spherical $t$-design curves can be used for mobile sampling and reconstruction of functions on the sphere. (i) We derive lower asymptotic bounds for the length of $t$-design curves. (ii) For the unit sphere in $\mathbb{R}^3$ and small degrees, we present examples of $t$-design curves with small $t$. (iii) We prove the existence of asymptotically optimal $t$-design curves in the Euclidean $2$-sphere. This construction isbased on and uses the existence of $t$-design points verified by Bondarenko, Radchenko, and Viazovska (2013). For higher-dimensionalspheres we inductively prove the existence of $t$-design curves. More generally, one can study the concept of t-design curves on a compact Riemannian manifold. This means that the line integral along a $t$-design curve integrates “polynomials” of degree $t$ exactly. For the $d$-dimensional tori, we construct $t$-design curves with asymptotically optimal length. This is joint work with Martin Ehler and Clemens Karner, Univ. Vienna. This talk is part of the Isaac Newton Institute Seminar Series series. ## This talk is included in these lists:- All CMS events
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