## 6 Expansions and Interpolations by Polyharmonic-Neumann Eigenfunctions  
In this section, we outline the derivation of eigenfunctions (see the system \(\mathcal{H}_{q}\) in (6)) and eigenvalues of the operator \(\mathcal{L}_{0}\) for realization of the corresponding expansions \(\mathcal{F}_{N}^{q}[f](x)\), interpolations \(\mathcal{I}_{N}^{q}[f](x)\), and QP interpolations \(\mathcal{I}_{N,m}^{q}[f](x)\). We follow [58] and briefly describe well-known results for completeness. The eigenfunctions of the problem (3) and (4) corresponding to \(\alpha\
eq 0\) can be derived explicitly. They are linear combinations of the products of trigonometric and hyperbolic functions. The corresponding coefficients can be derived as a solution of a \(q\times q\) algebraic eigenproblem. The explicit calculation of the eigenfunctions requires the separation of cases corresponding to even and odd \(q\). When \(q\) is even, the eigenfunction takes one of two possible forms \(\phi^{[i]}\), which is even if \(i=0\  
The eigenfunctions \(\phi_{n}\) are exponentially close to trigonometric functions  
\[\phi_{n}(x)=\cos\left[\frac{1}{4}(2n+q-1)\pi x+\frac{1}{2}(n+q-1) \pi\right]\\ +O(e^{-\frac{1}{2}\gamma_{q}(1-|x|)n\pi}),\ n>>1,\ x\in(-1,1). \tag{162}\]  
The latter can be applied to accelerate the computation of the corresponding expansions. The \(n\)-th eigenvalue \(\alpha_{n}\) has the following asymptotic:  
\[\alpha_{n}=\frac{1}{4}(2n+q-1)\pi+O(e^{-\gamma_{q}n\pi}),\ n>>1, \tag{163}\]  
for some \(\gamma_{q}>0\) depending only on \(q\). This asymptotic can be used to calculate eigenvalues with any predefined precision. In general, the eigenvalues can be calculated as the roots of the determinants of the mentioned systems. Computation can be carried out extremely easily using iterations. The values \(\alpha_{n}\approx 1/4(2n+q-1)\pi\)can be used as the starting points of iterations. Having calculated the values \(\alpha_{n}\) and functions \(\varphi_{n}(x)\) and \(\varphi_{0,n}(x)\), we can experiment with the expansions \(\mathcal{F}_{N}^{q}[f]\). The traditional interpolation for the system \(\mathcal{H}_{q}\) and for the grid \(x_{k}=2k/(2N+1)\), \(|k|\leq N\) we seek in the following form:  
\[\mathcal{I}_{N}^{q}[f](x)=\sum_{k=-N}^{N}a_{k}(x)f(x_{k}), \tag{164}\]  
where the unknown- functions are \(a_{k}\), and we determine requiring that the interpolation is exact for the eigenfunctions of \(\mathcal{H}_{q}\):  
\[\sum_{k=-N}^{N}\varphi_{0,n}(x_{k})a_{k}(x)\equiv\varphi_{0,n}(x),\ n=0,\ldots,q-1, \tag{165}\]  
and  
\[\sum_{k=-N}^{N}\varphi_{n}(x_{k})a_{k}(x)\equiv\varphi_{n}(x),\ n=1,\ldots,2N- q+1. \tag{166}\]  
Denoting the matrix of the system \(\{M_{n,k}\}\), \(n=0,\ldots,2N\), \(k=-N,\ldots,N\), we can write \(\mathcal{I}_{N}^{q}[f](x)\) as follows:  
\[a_{k}(x)=\sum_{n=0}^{q-1}M_{k,n}^{-1}\varphi_{0,n}(x)+\sum_{n=q}^{2N}M_{k,n}^ {-1}\varphi_{n-q+1}(x), \tag{167}\]  
and  
\[\mathcal{I}_{N}^{q}[f](x)=\sum_{n=0}^{q-1}\varphi_{0,n}(x)\sum_{k=-N}^{N}f(x _{k})M_{k,n}^{-1}+\sum_{n=q}^{2N}\varphi_{n-q+1}(x)\sum_{k=-N}^{N}f(x_{k})M_{ k,n}^{-1}. \tag{168}\]  
Again, by denoting  
\[\tilde{f}_{0,n}^{q}=\sum_{k=-N}^{N}f(x_{k})M_{k,n}^{-1},\ n=0,\ldots,q-1, \tag{169}\]  
and  
\[\tilde{f}_{n}^{q}=\sum_{k=-N}^{N}f(x_{k})M_{k,n}^{-1},\ n=q,\ldots,2N, \tag{170}\]we can write  
\[\mathcal{I}_{N}^{q}[f](x)=\sum_{n=0}^{q-1}\int_{0,n}^{q}\varphi_{0,n}(x)+\sum_{n= q}^{2N}\int_{n}^{q}\varphi_{n-q+1}(x). \tag{171}\]  
By the same procedure, we derive the the corresponding QP interpolations for the system \(\mathcal{H}_{q}\) and for the grid \(x_{k}=k/N\), \(|k|\leq N\) that includes also the endpoints.