ABSTRACT
This paper gives results on weak and strong convergence in https://www.w3.org/1998/Math/MathML"> L 2 ( R s ) r https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419839/4baaeab4-d6e3-45ed-9dec-1deb90c341dd/content/inline-math89.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 2(IR s ) r of the cascade sequence (φ k,n ) generated by the nonstationary matrix cascade algorithm https://www.w3.org/1998/Math/MathML"> ϕ k , n = | M | Σ j h k + 1 ( j ) ϕ k + 1 , n - 1 ( M ⋅ - j ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419839/4baaeab4-d6e3-45ed-9dec-1deb90c341dd/content/inline-math90.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where for each k = 1,2,..., hk is a finite sequence of r x r matrices and M is an integer dilation matrix. The limit as n → ∞ of the cascade sequence is an r-vector of functions (φ k ) that is a solution of the nonstationary matrix refinement equations https://www.w3.org/1998/Math/MathML"> ϕ k = | M | Σ j h k + 1 ( j ) ϕ k + 1 ( M ⋅ - j ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419839/4baaeab4-d6e3-45ed-9dec-1deb90c341dd/content/inline-math91.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . We give a characterization of weak convergence of (φ k,n ) as n → ∞ under a weak assumption. Further assumption that there is a stationary refinement equation at ∞ with matrix filter h satisfying https://www.w3.org/1998/Math/MathML"> Σ k | h k ( j ) - h ( j ) | - < ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419839/4baaeab4-d6e3-45ed-9dec-1deb90c341dd/content/inline-math92.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for all j, gives convergence of the nonstationary cascade sequence (φ k,n ) as both k and n tend to infinity. The convergence is completely determined by the spectral properties of the transition operator associated with h.
