NDLI: Tight Gaussian 4-Designs

Content Provider	Springer Nature Link
Author	Bannai, Etsuko Bannai, Eiichi
Copyright Year	2005
Abstract	A Gaussian t-design is defined as a finite set X in the Euclidean space ℝ$^{n}$ satisfying the condition: $$\frac{1}{V({\mathbb R}^n)}\int_{{\mathbb R}^n} f(x)e^{-\alpha^2\|\|x\|\|^2}dx=\sum_{u\in X}\omega(u)f(u)$$ for any polynomial f(x) in n variables of degree at most t, here α is a constant real number and ω is a positive weight function on X. It is easy to see that if X is a Gaussian 2e-design in ℝ$^{n}$, then $$\|X\|\geq {n+e\choose e}$$ . We call X a tight Gaussian 2e-design in ℝ$^{n}$ if $$\|X\|={n+e\choose e}$$ holds. In this paper we study tight Gaussian 2e-designs in ℝ$^{n}$. In particular, we classify tight Gaussian 4-designs in ℝ$^{n}$ with constant weight $$\omega=\frac{1}{\|X\|}$$ or with weight $$\omega(u)=\frac{e^{-\alpha^2\|\|u\|\|^2}} {\sum_{x\in X}e^{-\alpha^2\|\|x\|\|^2}}$$ . Moreover we classify tight Gaussian 4-designs in ℝ$^{n}$ on 2 concentric spheres (with arbitrary weight functions).
Ending Page	63
Page Count	25
Starting Page	39
File Format	PDF
ISSN	09259899
e-ISSN	15729192
Journal	Journal of Algebraic Combinatorics
Issue Number	1
Volume Number	22
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2005-01-01
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Euclidean design Convex and Discrete Geometry Computer Science 2-distance set quadrature formula addition formula Group Theory and Generalizations Order, Lattices, Ordered Algebraic Structures Gaussian design Combinatorics tight design spherical design
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Algebra and Number Theory

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