Classes of functions defined on the real line and their approximation by entire functions. I (Q2640079)

The author introduces and investigates the class of functions \(\hat L^{\psi}_{\beta}{\mathfrak N}\) defined by \(\int^{\infty}_{- \infty}h(x+t)\Psi_{\beta}(t)dt,\) where \(\Psi_{\beta}(t)\) is the Fourier transform of \(\psi\) and \(h\in {\mathfrak N}\). If h is \(2\pi\)- periodic, this class reduces to one which was intensively studied by the author. The central problem is the approximation of the functions in this class by some operators, which in periodical case reduces to partial sums of the Fourier series and in the general case to some entire functions of exponential type.