The space of functions that asymptotically approach polynomials. (Q1432430)

A real-valued function \(x\) in \(C^ m=C^ m[t_ 0,\infty)\) is called asymptotically approaching an \(m\)-degree polynomial \(p_ x(t) = \sum_{k=0}^ ma_{k,x}t^ k\) if \(\lim_{t\to \infty}(x^{(k)}(t)-p_ x^{(k)}(t)) = 0\), \(k=0,1,\dots,m.\) The space of all these functions, denoted by \(PA^ m_ m\), is a Banach space with respect to the norm \(\| x\| _{PA^ m_ m} = \| x\| _{C^ m}\| + | a_ x| , \) where \(a_ x=(a_{0,x},\dots,a_{m,x})\). The author considers the related space \(PA^{m+1}_ m\) formed by all functions in \(PA^ m_ m\) with locally integrable generalized derivatives of order \(m+1\), and equips it with a complete metric derived from an almost norm. By an almost norm on a vector space \(X\) the author understands a positively definite sublinear functional \(\| \;\| \) such that \(\| -x\| = \| x\|,\) for all \(x\in X\), i.e. the absolute homogeneity is corrupted. The almost norm of a function \(x\) in \(PA^{m+1}_ m\) is defined by \[ \| x\| _{PA^{m+1}_ m} = \| x\| _{PA^{m}_ m} + \sum_{n=1}^\infty 2^{-n}\frac{\| x^{(m+1)}\| ^{(n)}_ 1}{1+ \| x^{(m+1)}\| _ 1^{(n)}}, \] where, for a locally summable function \(z\) on \([t_ 0,\infty)\) one puts \(\| z\| _ 1^{(n)} =\int_{t_ 0}^{t_ 0+n}| z(t)| \,dt.\) The author considers also \(L\)-polynomial asymptotics. The corresponding space \(LAP^{m+1}_ m\) is complete with respect to an almost norm defined in a similar way and continuously embedded in \(LAP^{m}_ m\). The study of these spaces is motivated by some applications to the solution of some systems of differential equations.