Interpolation Problem in Paley-Wiener Weighted Spaces

Yu. Lyubarskii and K. Seip K. (Revista Matematica Iberoamericana, 1997 (13), № 2) found the criterion for the existence of a uniqe solution of the interpolation problem f(λk) = bk in the Paley-Wiener spaces in terms of Makenhoupt’s (Ap) condition of entire functions of the exponential type at most π, where p is a real number more than 1, whose restriction to the real line coincides with the class of functions which module degree number p is integrable on R whith p-norm. These results make it possible to obtain the criterion of the unconditional basisness of the system of exponentials in the space of functions which module degree number p is an integrable on (–π; π). At the same time the sequence (λk) with a unique limit point at the infinity, for which mentioned interpolation problem has a uniqe solution is called a complete interpolating sequence in the Paley-Wiener spaces. For p = 2 those descriptions coincides with those given by Pavlov (1979), Nikolsky (1980), and Minkin (1982). We generalize these results to the weighted spaces of entire functions of the exponential type at most σ with the p-norm (there is a power function with exponent ω as a weight), where σ is a nonnegative real number, ω – real number more then –1. That is, we find conditions for the completeness of the interpolating sequence (λk) in the Paley-Wiener weighted space. Different forms of these conditions are considered. Among them there are Mackenhoupt’s conditions (continuous and discrete (Ap) conditions). There is proved that if (λk) is a complete interpolation sequence in the Paley-Wiener weighted space, then it is a relatively dense set in the space C. Also there constructed an example of a complete interpolating sequence for σ = π