Orthogonal polynomials are, along with Fourier series, one of the most widely used tools in the theory of approximation. Specific properties of trigonometric functions and polynomials assure effcient computation as well as stable and convergent numerical solutions. Spectral methods are, besides finite difference and finite element methods, an important tool for solving boundary value problems for ordinary as well as partial differential equaations. In the first part of the doctoral thesis, some basic properties of the Fourier series and orthogonal polynomials as well as some basic approaches for the construction of spectral methods with fundamental tools for convergence and error analysis are described. In the sequel, two non-classical families of orthogonal polynomials are presented, i.e., the half-range Chebyshev polynomials of the first and second kind as well as the corresponding half-range Chebyshev-Fourier series. Both families are constructed via the modified Chebyshev algorithm used for the computation of the recursive coefficients for the three-term recurrence relation. The approximation of square integrable functions with half-range Chebyshev-Fourierseries yields comparable results to the approximation with Fourier or Chebyshev series. In the central part of the doctoral thesis, a new class of Chebyshev-Fourier collocation spectral methods for solving linear two-point boundary value problems with Dirichlet boundary conditions is constructed. We seek for the numerical solution in the form of the truncated half-range Chebyshev-Fourier series, where spectral coefficients are computed using the collocation method. Convergence and error analysis shows that these methods are comparable with standard ones, where the solution is approximated with the Fourier series for periodic or with the Chebyshev series for non-periodic problems. We construct a new class of methods also for some evolutive boundary value problems, i.e., for generalized heat and wave equations. Numerical examples confirm theoretical results and show the comparability of the error of the numerical solution obtained with the new or the standard methods. Yet, computational costs are not comparable, because in the case of half-range Chebyshev-Fourier series there does not exist a tool for the computation of coefficients being comparable with fast Fourier transform.
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