ABSTRACT
We prove that if https://www.w3.org/1998/Math/MathML"> { ϕ ^ j # ( 2 π n ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419839/4baaeab4-d6e3-45ed-9dec-1deb90c341dd/content/inline-math541.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> { ψ ^ j # ( 2 π n ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419839/4baaeab4-d6e3-45ed-9dec-1deb90c341dd/content/inline-math542.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> are in l 2(Z), then https://www.w3.org/1998/Math/MathML"> { ϕ j ( x - l 2 j ) , ψ j ( x − l 2 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-math543.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (l = 0,...,2 j ,- 1) is biorthogonal if and only if for each l=0, • •.,2j - 1, there is a positive constant C such that https://www.w3.org/1998/Math/MathML"> Σ α ∈ Z | ϕ ^ j # ( 2 π ( l + 2 j α ) ) ψ ^ ¯ j # ( 2 π ( l + 2 j α ) ) | ≥ C , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419839/4baaeab4-d6e3-45ed-9dec-1deb90c341dd/content/inline-math544.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where https://www.w3.org/1998/Math/MathML"> ϕ j = 2 − j 2 ϕ 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-math545.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> ψ j = 2 − j 2 ψ 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-math546.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
