Explore the advanced theory of function approximation with “Uniform Approximations by Trigonometric Polynomials,” a seminal work for mathematicians. This monograph delves into the approximation of periodic functions using trigonometric polynomials via linear summation methods. It meticulously investigates classical methods by Fourier, Fejér, Riesz, and Roginski, focusing on the Kolmogorov-Nikol’skii problem. Discover exact and asymptotic bounds for polynomial deviations on specific function classes, with a significant focus on multidimensional cases. This updated edition presents deep results and new methods, offering crucial insights for researchers in mathematical analysis. The problems, solved with asymptotic equalities, ensure lasting relevance and value.