Compressed sensing has been proposed as a computationally efficient method to estimate the finite-dimensional signals. The idea is to develop an undersampling operator that can sample the large but finite-dimensional sparse signals with a rate much below the required Nyquist rate. In other words, considering the sparsity level of the signal, the compressed sensing samples the signal with a rate proportional to the amount of information hidden in the signal. In this dissertation, first, we employ compressed sensing for physical layer signal processing of directional millimeter-wave communication. Second, we go through the theoretical aspect of compressed sensing by running a comprehensive theoretical analysis of compressed sensing to address two main unsolved problems, (1) continuous-extension compressed sensing in locally convex space and (2) computing the optimum subspace and its dimension using the idea of equivalent topologies using Köthe sequence. In the first part of this thesis, we employ compressed sensing to address various problems in directional millimeter-wave communication. In particular, we are focusing on stochastic characteristics of the underlying channel to characterize, detect, estimate, and track angular parameters of doubly directional millimeter-wave communication. For this purpose, we employ compressed sensing in combination with other stochastic methods such as Correlation Matrix Distance (CMD), spectral overlap, autoregressive process, and Fuzzy entropy to (1) study the (non) stationary behavior of the channel and (2) estimate and track channel parameters. This class of applications is finite-dimensional signals. Compressed sensing demonstrates great capability in sampling finite-dimensional signals. Nevertheless, it does not show the same performance sampling the semi-infinite and infinite-dimensional signals. The second part of the thesis is more theoretical works on compressed sensing toward application. In chapter 4, we leverage the group Fourier theory and the stochastical nature of the directional communication to introduce families of the linear and quadratic family of displacement operators that track the join-distribution signals by mapping the old coordinates to the predicted new coordinates. We have shown that the continuous linear time-variant millimeter-wave channel can be represented as the product of channel Wigner distribution and doubly directional channel. We notice that the localization operators in the given model are non-associative structures. The structure of the linear and quadratic localization operator considering group and quasi-group are studied thoroughly. In the last two chapters, we propose continuous compressed sensing to address infinite-dimensional signals and apply the developed methods to a variety of applications. In chapter 5, we extend Hilbert-Schmidt integral operator to the Compressed Sensing Hilbert-Schmidt integral operator through the Kolmogorov conditional extension theorem. Two solutions for the Compressed Sensing Hilbert Schmidt integral operator have been proposed, (1) through Mercer's theorem and (2) through Green's theorem. We call the solution space the Compressed Sensing Karhunen-Loéve Expansion (CS-KLE) because of its deep relation to the conventional Karhunen-Loéve Expansion (KLE). The closed relation between CS-KLE and KLE is studied in the Hilbert space, with some additional structures inherited from the Banach space. We examine CS-KLE through a variety of finite-dimensional and infinite-dimensional compressible vector spaces. Chapter 6 proposes a theoretical framework to study the uniform convergence of a compressible vector space by formulating the compressed sensing in locally convex Hausdorff space, also known as Fréchet space. We examine the existence of an optimum subspace comprehensively and propose a method to compute the optimum subspace of both finite-dimensional and infinite-dimensional compressible topological vector spaces. To the author's best knowledge, we are the first group that proposes continuous compressed sensing that does not require any information about the local infinite-dimensional fluctuations of the signal.