The Newton polynomials are defined as the coefficients in the expansion \[ (1-w)^ z=\sum^{\infty}_{n=0}N_ n(z)w^ n \] where \(| w| <1\) and z is arbitrary. Let \(w_ h(t)=e^{-t}t^{h-1}\) on (0,\(\infty)\), h a positive real number and \(L^ 2(w_ h)\) the weighted Lebesgue space. The weighted Mellin transform is a mapping from \(L^ 2(w_ h)\) onto a linear space \({\mathcal N}_ h\) of holomorphic functions on the half-plane Re z\(>-h/2\). Under some inner product using the weighted Mellin transform, \({\mathcal N}_ h\) is a Hilbert space which contains the Newton polynomials as a complete orthogonal set. \({\mathcal N}_ h\) is called a Newton space. The inversion of the weighted Mellin transform yields a kind of Plancherel theory for the Newton spaces. In this paper the Plancherel theorem is proved.
Summary: The purpose of this paper is to provide a selfcontained introduction to Newton spaces and explore the associated function theory. It is shown that Newton spaces have certain properties reminiscent of Hardy classes and Fourier analysis. However, the function theory is different from that of Hardy classes and more closely related to the calculus of finite differences. The calculation of shift operators is equivalent to the problem of giving an analytical solution of the functional equation \(F(z+1)-F(z)=G(z)\). Connections are also shown with integer valued functions and singularities of analytic functions.