I will talk about J.P. May's axioms of a triangulated category, present some properties and some basic examples (K(mathcal{A}) and D(mathcal{A})). This will be a start to the generalized stable homotopy theory in the sense of HPS (Hovey, Palmieri and Strickland).

A reproducing kernel Hilbert space $H$ of functions on $\mathbb R$ which has a total orthogonal set of point evaluation vectors $(\delta_{x_{n}})$, $n\in Z$, is said to have the sampling property, since any $\phi\in H$ can be perfectly reconstructed from its `samples' or values taken on the set of points $(x_{n})$. The classic example of such a space is the Paley-Wiener space of $\Omega$-bandlimited functions. Such spaces are used extensively in applications including signal processing. In this talk we will apply the theory of self-adjoint extensions of symmetric operators to the study of such spaces. In particular, a sufficient operator-theoretic condition for a subspace of $L^{2}$ of the real line to be a reproducing kernel Hilbert space with the sampling property will be presented. Potential consequences for signal processing will be discussed.