This thesis consists of original contributions in the area of digital signal processing. The reconstruction of signals sparse (highly concentrated) in various transform domains is the primary problem analyzed in the thesis. The considered domains include Fourier, discrete Hermite, one-dimensional and two-dimensional discrete cosine transform, as well as various time-frequency representations. Sparse signals are reconstructed using sparsity measures, being, in fact, the measures of signal concentration in the considered domains. The thesis analyzes the compressive sensing reconstruction algorithms and introduces new approaches to the problem at hand. The missing samples influence on analyzed transform domains is studied in detail, establishing the relations with the general compressive sensing theory. This study provides new insights on phenomena arising due to the reduced number of signal samples. The theoretical contributions involve new exact mathematical expressions which describe performance and outcomes of reconstruction algorithms, also including the study of the influence of additive noise, sparsity level and the number of available measurements on the reconstruction performance, exact expressions for reconstruction errors and error probabilities. Parameter optimization of the discrete Hermite transform is also studied, as well as the additive noise influence on Hermite coefficients, resulting in new parameter optimization and denoising algorithms. Additionally, an algorithm for the decomposition of multivariate multicomponent signals is introduced, as well as an instantaneous frequency estimation algorithm based on the Wigner distribution. Extensive numerical examples and experiments with real and synthetic data validate the presented theory and shed a new light on practical applications of the results.