The maximal cardinality of μM,D-orthogonal exponentials on the spatial Sierpinski gasket

• Published in: Monatshefte für Mathematik 2020, Vol.191 (1), p.203-224
• Main Authors: Wang, Qi, Li, Jian-Lin
• Format: Article
• Language: English
• Subjects: Exponential functions; Hilbert space; Mathematics; Mathematics and Statistics; Upper bounds
• ISSN: 0026-9255; 1436-5081
• DOI: 10.1007/s00605-019-01348-9

The self-affine measure μ M , D corresponding to an expanding integer matrix M = d i a g [ p 1 , p 2 , p 3 ] and the digit set D = 0 , e 1 , e 2 , e 3 in the space R 3 is supported on the spatial Sierpinski gasket, where e 1 , e 2 , e 3 are the standard basis of unit column vectors in R 3 and p 1 , p 2 , p 3 ∈ Z \ { 0 , ± 1 } . In the case p 1 ∈ 2 Z and p 2 , p 3 ∈ 2 Z + 1 , it is conjectured that all the four-element orthogonal exponentials in the Hilbert space L 2 ( μ M , D ) are maximal in the class of exponential functions. This conjecture has been proved to be false by giving a class of the five-element (and later, the eight-element) orthogonal exponentials in L 2 ( μ M , D ) . In the present paper, we completely determine the maximal cardinality of μ M , D -orthogonal exponentials on the spatial Sierpinski gasket. The main result shows that (i) if p 3 ≠ ± p 2 , then for any l ∈ N , there exist ( 2 l + 6 ) -element orthogonal exponentials in the Hilbert space L 2 ( μ M , D ) , which is also maximal in the class of exponential functions; (ii) if p 3 = - p 2 , then there exist at most eight mutually orthogonal exponential functions in L 2 ( μ M , D ) , where the number eight is the best upper bound.