###### Abstract  
Fourier expansions employing polyharmonic-Neumann eigenfunctions have demonstrated improved convergence over those using the classical trigonometric system, due to the rapid decay of their Fourier coefficients. Building on this insight, we investigate interpolations on a finite interval that are exact for polyharmonic-Neumann eigenfunctions and exhibit similar benefits. Furthermore, we enhance the convergence of these interpolations by incorporating the concept of quasi-periodicity, wherein the basis functions are periodic over a slightly extended interval. We demonstrate that those interpolations achieve significantly better convergence rates away from the endpoints of the approximation interval and offer increased accuracy over the entire interval. We establish these properties for a specific case of polyharmonic-Neumann eigenfunctions known as the modified Fourier system. For other basis functions, we provide supporting evidence through numerical experiments. While the latter methods display superior convergence rates, we demonstrate that interpolations using the modified Fourier basis offer distinct advantages. Firstly, they permit explicit representations via the inverses of certain Vandermonde matrices, whereas other interpolation methods require approximate computations of the eigenvalues and eigenfunctions involved. Secondly, these matrix inverses can be efficiently computed for numerical applications. Thirdly, the introduction of quasi-periodicity improves the convergence rates, making them comparable to those of other interpolation techniques. t +
Footnote †: journal: _Algorithms_  
**Citation:** [PERSON]; [PERSON]; [PERSON] On an Exact Convergence of Quasi-Periodic Interpolations for the Polyharmonic-Neumann Eigenfunctions. **2024, 17**, 497. [[https://doi.org/10.3390/a17110497](https://doi.org/10.3390/a17110497)]([https://doi.org/10.3390/a17110497](https://doi.org/10.3390/a17110497))  
**1. Introduction**  
It is well known that trigonometric expansions and interpolations using the classical system  
\[\mathcal{H}_{\text{class}}=\{\cos\pi nx:n\in\mathbb{Z}_{+}\}\cup\{\sin\pi nx:n \in\mathbb{N}\},\;x\in[-1,1], \tag{1}\]  
exhibit degraded convergence when approximating non-periodic functions on a finite interval \([-1,1]\), even when the functions are sufficiently smooth. A substantial body of literature is devoted to the methods that improve convergence or mitigate the Gibbs phenomenon. Given the extensive amount of work on this subject, providing a comprehensive summary is impractical. Numerous individual articles explore topics such as [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. In this paper, we focus exclusively on ideas relevant to our study: quasi-periodic (QP) interpolations and expansions using polyharmonic-Neumann eigenfunctions. The main purpose of this paper is to combine these ideas and explore QP interpolations that are exact for polyharmonic-Neumann eigenfunctions. The main idea of the QP approach is to extend basis functions from 2-periodic to \((2+\epsilon)\)-periodic, where \(\epsilon\) tends to zero as the number of terms in the expansions tends to infinity. It generally does not completely remove the Gibbs phenomenon but reduces its impact. Meanwhile, it accelerates the convergence rate away from the endpoints. This idea was introduced in [65]. Papers [66; 67; 68] investigate the pointwise convergence of the QP interpolation and its convergence in the \(L_{2}\)-norm. It can be combined with the polynomial correction method with improved convergence ([69]) for complete elimination of the Gibbs phenomenon and additional convergence acceleration. Another efficient idea is the utilization of the polyharmonic-Neumann eigenfunctions.