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There are eight‐element orthogonal exponentials on the spatial Sierpinski gasket
Author(s) -
Wang Qi,
Li JianLin
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700471
Subject(s) - mathematics , exponential function , orthogonal basis , sierpinski triangle , combinatorics , conjecture , hilbert space , cardinality (data modeling) , element (criminal law) , discrete mathematics , pure mathematics , mathematical analysis , fractal , physics , quantum mechanics , computer science , law , political science , data mining
The self‐affine measure μ M , Dcorresponding to an expanding matrix M = d i a g [ p 1 , p 2 , p 3 ]and the digit set D = 0 , e 1 , e 2 , e 3in the space R 3 is supported on the spatial Sierpinski gasket, wheree 1 , e 2 , e 3are the standard basis of unit column vectors in R 3 andp 1 , p 2 , p 3 ∈ Z ∖ { 0 , ± 1 } . In the casep 1 ∈ 2 Z andp 2 , p 3 ∈ 2 Z + 1 , it is conjectured that the cardinality of orthogonal exponentials in the Hilbert spaceL 2 ( μ M , D )is at most “4”, where the number 4 is the best upper bound. That is, all the four‐element sets of orthogonal exponentials are maximal. This conjecture has been proved to be false by giving a class of the five‐element orthogonal exponentials inL 2 ( μ M , D ) . In the present paper, we construct a class of the eight‐element orthogonal exponentials in the corresponding Hilbert spaceL 2 ( μ M , D )to disprove the conjecture. We also illustrate that the constructed sets of orthogonal exponentials are maximal.