Integral representations and transforms of $N$-functions. I

The author has proposed a new approach to extrapolation of operators from the scale of Lebesgue spaces to the Orlicz spaces beyond this scale. In this article comprising two parts we develop some mathematical method that enables us to prove extrapolation theorems for arbitrary behavior of an operator in the Lebesgue scale (i.e., in the case when the norm of the operator is an arbitrary function of $p$) and also in the case when the basic scale is an interval of the Lebesgue scale with exponents separated from 1 or $+\infty$. In this event, we face ill-posed problems of inversion of the classical Mellin and Laplace type integral transforms over nonanalytic functions in terms of their asymptotic behavior on the real axis and also the question about the properties of convolution type integral transforms on classes of $N$-functions. In the first part of the article we study integral representations for $N$-functions by expansions in power functions with a positive weight and the behavior of convolution type integral transforms on classes of $N$-functions.