Spectrum of Selfaffine Fractal Measures

Abstract

A Borel probability measure and#956;M,D is said to be a spectral measure if it has a countable newlineexponential orthonormal basis {e2and#960;iand#10216;and#955;,xand#10217;}and#955;and#8712;and#923; for the Hilbert space L2(and#956;M,D). The set and#923; newlineis called a spectrum for and#956;M,D. This thesis is focused on the spectrality of a few self-affine newlinemeasures.Let and#8467;1, and#8467;2 and#8712; Z\{0}, M = newlineand#63726; newlineand#63728;3and#8467;1 0 newline0 3and#8467;2 newlineand#63737; newlineand#63739;, and D= newlineand#63729;and#63730; newlineand#63731; newlineand#63723; newlineand#63725;0 newline0 newlineand#63734; newlineand#63736;, newlineand#63723; newlineand#63725;1 newline0 newlineand#63734; newlineand#63736;, newlineand#63723; newlineand#63725;0 newline1 newlineand#63734; newlineand#63736; newlineand#63740;and#63741; newlineand#63742;. The spectra newlinefor the planar Sierpinski measures and#956;M,D are derived in this thesis. The spectra for and#956;M,D newlinewhen M = newlineand#63726; newlineand#63728; 0 3and#8467; newline3and#8467; 0 newlineand#63737; newlineand#63739;, and#8467; and#8712; Z\ {0} are also obtained.The spectra of and#956;M,D for a given M newlineand generalised digit set D are also determined.The method used in the planar Sierpinski newlinemeasure is extended to spatial measures. The spectra of and#956;M,D on spatial Sierpinski newlinegasket are obtained when M is a 3×3 diagonal matrix with elements and#8467;i and#8712; 2Z\ {0}, and newlineD= {0,e1,e2,e3},where ei s are the standard basis in R3. Further,the spectra ofand#956;M,D newlinefor off-diagonal 3×3 matrices are also secured.Unlikethe one dimension,the planar Sierpinski measure is the most studied in higher newlinedimensions.Apart from that, in R2, there are a few studies of spectral measure that are newlinerelated to the four-element digit set D = newlineand#63729;and#63730; newlineand#63731; newlineand#63723; newlineand#63725;0 newline0 newlineand#63734; newlineand#63736;, newlineand#63723; newlineand#63725;1 newline0 newlineand#63734; newlineand#63736;, newlineand#63723; newlineand#63725;0 newline1 newlineand#63734; newlineand#63736;, newlineand#63723; newlineand#63725;and#8722;1 newlineand#8722;1 newlineand#63734; newlineand#63736; newlineand#63740;and#63741; newlineand#63742;.In this thesis,the existence of a spectral set and#923; for the measure and#956;M,D with the four element digit set D and an expansive matrix M and#8712; M2(Z) with all even entries is established. Additionally,the spectra of L2(and#956;M,D) for some particular M and D are obtained.The spectrum is invariant under translation. The spectral measure can have uncountable spectra containing zero.It is crucial to characterise each spectrum or maximal family of the orthogonal exponentials of L2(and#956;M,D) as it aids in the understanding of convergent problems,function approximation,and structural properties of the space.Let and#956;M,D be the measure corresponding to an expansive matrix M = newlineand#63726; newlineand#63728;2and#8467;1 0 newline0 2and#8467;2 newlineand#63737; newlineand#63739; with newline|and#8467;i| and#8805; 2, and#8467;1and#8467;2 gt 0 and the four element digit set D. A complete characterisation of all newlinemaximal orthogonal exponential sets of and#956;M,D demonstrated.