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On Representation of Functions in 𝑳𝟐 𝟎, 𝟏 By Using Affine System of Walsh ?Paley System Type

Khalid Hadi Hameed AL-Jourany

In this paper , we introduced a notion of affine system of Walsh-Paley system type .We considerbiorthogonal series of the form :𝒇 = (𝒇, Ѱ𝒏 ∞𝒏 =𝟎 )𝝋𝒏,where , 𝒇 belongs to space 𝐿2 0,1 = 𝐻 , (𝐻 is a Hilbert space) , which we are represented by :𝑯 = 𝑬 ⊕ 𝑾𝟎𝑯 ⊕ 𝑾𝟏𝑯,where , 𝑬 = 𝒔𝒑𝒂𝒏 (𝒘) , (𝒘is Walsh- Paley function ) , and 𝑾𝟎 , 𝑾𝟏are two operators which are defined . Some properties are given with proves for this representation , as well as we gave a general forms for this space with its prove by using induction rule . Also 𝝋𝒏 n≥0 is the affine system of Walsh –Paley system type of a function 𝝋. We introduced this function by using Walsh-Paley function . Also ,(𝒇, Ѱ𝒏)are the Fourier coefficients of a function 𝒇 ∈ 𝐿2 0,1 in the Walsh -Paley system . We are proving that these coefficients formsbiorthogonal conjugate to the system 𝝋𝒏 n≥0 of a function 𝝋 . Finally , we showed that , the affine system of Walsh –Paley system type of a function 𝝋 is Bessel system.