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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 L 2 0,1 = H , (H is a Hilbert space) , which we are represented by :H = E⊕W0H⊕W1H,where , E = span (w) , (wis WalshPaley 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 ∈ L 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.

international journal of innovative research in science engineering and technology/international journal of innovative research in science, engineering and technology

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