Constructie van Ingebedde Roostergebaseerde Algoritmen voor het Benaderen van Multivariate Periodieke Functies met of zonder Bekende Gladheid

This thesis focuses on approximating $d$-variate periodic functions in weighted Korobov spaces with general weight parameters, using function evaluations at a rank-$1$ lattice of size~$n$. <br />The approximation methods under consideration include truncated trigonometric polynomials and kernel interpolations. We utilise the component-by-component (CBC) construction to obtain a ``good'' generating vector for a given~$n$ or even a range of $n$ being powers of primes. Notably, in the search of generating vectors, there is no computation or storage of frequency index sets, which are typically used to truncate the Fourier series. Unlike currently available literature, our approach does not restrict the number of point evaluations~$n$ to prime numbers, but allows any integer~$n \ge 2$ in both implementation and theoretical analysis.This non-trivial generalisation requires new proof techniques, and serves as a theoretical foundation for constructing embedded lattice sequences. The approximation error is analysed in the worst-case setting under both the $L_2$ and $L_{\infty}$ norms. For $L_2$-approximation, our error bound, denoted as $\calS_n^*$, converges at the best possible rate for a full rank-$1$ lattice. For $L_{\infty}$-approximation, the CBC construction with adapted parameters also works, extending previous results of product-type weight parameters to general weights. We further demonstrate that embedded lattice sequences are only a logarithmic factor worse than lattice rules with a fixed~$n$, while offering the advantages of reusing generating vectors and existing function evaluations. Beyond approximation, the constructive strategy to generate embedded lattice sequences applies to numerical integration in weighted Korobov spaces and unanchored Sobolev spaces, and further simplifies the corresponding proof of CBC error bounds. Interpolating the kernel method at rank-$1$ lattices completely bypasses the construction of frequency index sets, leading to the lattice-based kernel method. The approximation error bound~$S_n^*$ also applies to these methods, due to the optimality of kernel interpolants in the worst-case sense using the same function evaluations. An alternative error bound, denoted as $P_n^*$, is based on the power function commonly used in machine learning. Via numerical comparisons, we observe that the empirical difference in approximation error between lattices constructed by minimising $P_n^*$ and $S_n^*$ is negligible. The criterion $S_n^*$ is preferred owing to its computational efficiency and numerical stability. Furthermore, we propose a smoothness-independent lattice-based algorithm that approximates continuous functions within a variant of weighted Korobov spaces. This method combines a smoothness-independent hyperbolic index set with an efficient CBC construction of rank-1 lattices, requiring no prior knowledge of the smoothness parameter. We provide a theoretical analysis of the worst-case error measured under the $L_2$ norm. Our algorithm adaptively captures the true smoothness of the underlying function space and achieves the best-possible rate of error convergence for using a full rank-$1$ lattice as a sample set. We present summability conditions for general weights under which the approximation errors become dimension-independent across the proposed algorithms.