Asymptotic Uncorrelation for Mexican Needlets

We recall Mexican needlets from [5]. We derive an estimate for certain types of Legendre series, which we apply to the statistical properties of Mexican needlets. More precisely, we shall show that, under isotropy and Gaussianity assumptions, the Mexican needlet coefficients of a random field on the sphere are asymptotically uncorrelated, as the frequency parameter goes to infinity. This property is important in the analysis of cosmic microwave background radiation.

[1]  Domenico Marinucci,et al.  On the dependence structure of wavelet coefficients for spherical random fields , 2008, 0805.4154.

[2]  Paolo Baldi,et al.  Spherical needlets for cosmic microwave background data analysis , 2008 .

[3]  P. Baldi,et al.  Asymptotics for spherical needlets , 2006, math/0606599.

[4]  Willi Freeden,et al.  Constructive Approximation on the Sphere: With Applications to Geomathematics , 1998 .

[5]  Pencho Petrushev,et al.  Localized Tight Frames on Spheres , 2006, SIAM J. Math. Anal..

[6]  Pencho Petrushev,et al.  Decomposition of Besov and Triebel–Lizorkin spaces on the sphere , 2006 .

[7]  Domenico Marinucci,et al.  The needlets bispectrum , 2008, 0802.4020.

[8]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[9]  P. Baldi,et al.  Subsampling needlet coefficients on the sphere , 2007, 0706.4169.

[10]  Fr'ed'eric Guilloux,et al.  Practical wavelet design on the sphere , 2007, 0706.2598.

[11]  Paolo Baldi,et al.  High frequency asymptotics for wavelet-based tests for Gaussianity and isotropy on the torus , 2006 .

[12]  P. Baldi,et al.  Spherical Needlets for CMB Data Analysis , 2007, 0707.0844.

[13]  Daryl Geller,et al.  Continuous wavelets on compact manifolds , 2008, 0811.4440.

[14]  Daryl Geller,et al.  Nearly tight frames and space-frequency analysis on compact manifolds , 2007, 0706.3642.

[15]  Norman Morrison,et al.  Introduction to Fourier Analysis , 1994, An Invitation to Modern Number Theory.