Abstract
We describe the computations of some intrinsic constants associated to an n-dimensional normed space \({\mathcal{V}}\), namely the N-th “allometry” constants
These are related to Banach–Mazur distances and to several types of projection constants. We also present the results of our computations for some low-dimensional spaces such as sequence spaces, polynomial spaces, and polygonal spaces. An eye is kept on the optimal operators T and T′, or equivalently, in the case N = n, on the best conditioned bases. In particular, we uncover that the best conditioned bases of quadratic polynomials are not symmetric, and that the Lagrange bases at equidistant nodes are best conditioned in the spaces of trigonometric polynomials of degree at most one and two.
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References
Asplund E.: Comparison between plane symmetric convex bodies and parallelograms. Math. Scand. 8, 171–180 (1960)
Chalmers, B.L.: n-dimensional spaces with maximal projection constant. In: 12th Int. Conf. on Approximation Theory (2007)
Chalmers B.L., Metcalf F.T.: Minimal generalized interpolation projections. JAT 20, 302–313 (1977)
Cheney E.W., Price K.H.: Minimal projections. In: Talbot, A. (eds) Approximation Theory., pp. 261–289. Academic Press, London (1970)
Cheney E.W., Hobby C.R., Morris P.D., Schurer F., Wulbert D.E.: On the minimal property of the Fourier projection. Trans. Am. Math. Soc. 143, 249–258 (1969)
de Boor C., Pinkus A.: Proof of the conjectures of Bernstein and Erdős concerning the optimal nodes for polynomial interpolation. JAT 24, 289–303 (1978)
Foucart S.: On the best conditioned bases of quadratic polynomials. JAT 130, 46–56 (2004)
Foucart, S.: Some comments on the comparison between condition numbers and projection constants. In: Neamtu, M., Schumaker, L.L. (eds.) Approximation Theory XII: San Antonio 2007, pp. 143–156. Nashboro Press (2008)
König H., Tomczak-Jaegermann N.: Norms of minimal projections. J. Funct. Anal. 119, 253–280 (1994)
Konheim A.G., Rivlin T.J.: Extreme points of the unit ball in a space of real polynomials. Am. Math. Mon. 73, 505–507 (1966)
Morris P.D., Price K.H., Cheney E.W.: On an approximation operator of de la Vallée Poussin. JAT 13, 375–391 (1975)
Morris P.D., Cheney E.W.: On the existence and characterization of minimal projections. J. Reine Angew. Math. 270, 61–76 (1974)
Pan K.C., Shekhtman B.: On minimal interpolating projections and trace duality. JAT 65, 216–230 (1991)
Rivlin T.J.: Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory. Wiley, New York (1990)
Szarek S.J.: Spaces with large distance to \({l\sp n\sb \infty}\) and random matrices. Am. J. Math. 112, 899–942 (1990)
Vale, R., Waldron, S.: The vertices of the platonic solids are tight frames. In: Advances in Constructive Approximation: Vanderbilt 2003, pp. 495–498. Nashboro Press, Brentwood (2004)
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Foucart, S. Allometry constants of finite-dimensional spaces: theory and computations. Numer. Math. 112, 535–564 (2009). https://doi.org/10.1007/s00211-009-0225-7
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DOI: https://doi.org/10.1007/s00211-009-0225-7