On Spectra and Spectral Eigenmatrix Problems of the Planar Sierpinski Measures

作者：An, Li-Xiang^*; Dong, Xin-Han; He, Xing-Gang

来源：Indiana University Mathematics Journal, 2022, 71(2): 913-952.

• Fourier orthogonal family
• Sierpinski measures
• spectral measures
• spectral eigenmatrix problems
• spectral structures

A Sierpinski-type measure mu is a self-similar measure which satisfies that @@@ mu(E) = 1/3 mu(3qE) + 1/3 mu(3qE - e(1)) + 1/3 mu(3qE - e(2)), @@@ for any Borel set E subset of R-2, where q >= 1 is an integer, and {e(1), e(2)} is a basis of R-2. Another form of expression for Sierpinski-type measures mu is the infinite convolution @@@ mu = delta((3q)-1 {0,e1,e2}) * delta((3q)-1{0,e1,e2}) * delta((3q)-1 {0,e1,e2}) * ... , @@@ where delta(E) = (1/#E) Sigma(e is an element of E) delta(e) is the linear combination of standard Dirac measures with equality weight and the convergence is in a weak sense. Let @@@ E-Lambda = {e(-2 pi i <lambda,x >) : lambda is an element of Lambda} @@@ be a family of exponentials in R-2 for any set Lambda subset of R-2. This paper gives a characterization for E-Lambda to be a maximal orthogonal family in L-2(mu). As its applications, some sufficient conditions are obtained for a maximal orthogonal family E-Lambda to be or not to be an orthogonal basis of L-2(mu). Moreover, several simple criteria are given for a matrix R is an element of M-2(R) so that there exists an orthogonal basis E Lambda satisfying that the family E-R Lambda is also an orthogonal basis of L-2(mu).