Measure-geometric Laplacians for discrete distributions

TL;DR

This paper extends measure-geometric Laplacian theory to discrete distributions, providing matrix representations and explicit eigenvalues and eigenfunctions for uniform discrete cases.

Contribution

It generalizes Freiberg and Zähle's harmonic calculus to finite support distributions and derives explicit spectral properties for uniform discrete distributions.

Findings

Matrix representation of Laplacians for discrete distributions

Explicit eigenvalues and eigenfunctions for uniform discrete case

Extension of harmonic calculus to finite support distributions

Abstract

In 2002 Freiberg and Z\"ahle introduced and developed a harmonic calculus for measure-geometric Laplacians associated to continuous distributions. We show their theory can be extended to encompass distributions with finite support and give a matrix representation for the resulting operators. In the case of a uniform discrete distribution we make use of this matrix representation to explicitly determine the eigenvalues and the eigenfunctions of the associated Laplacian.

Equations

$$⟨f,g⟩=⟨f,g{⟩}_{\mu }:=\sum _{i=1}^N{\alpha }_{i}f(z_i)g(z_i).$$

$$\displaystyle\begin{aligned} \mathscr{D}_{\mu}^{1}=\mathscr{D}_{\mu}^{1}(\mathcal{I})\coloneqq\bigg{\{}f\in\mathfrak{L}^{2}({\mu})\colon\text{there exists}\,f^{\prime}\in L^{2}_{\mu}\;\text{such that}\;f(0)=f(1)\;\text{and}&\\ f(x)=f(0)+\int\mathds{1}_{[0,x)}f^{\prime}\,\mathrm{d}\mu\;\text{for all}\;x\in\mathcal{I}&\bigg{\}},\end{aligned}$$

$$\int \text{mathds}{1}_{[0,1)}{f}^{\prime }\mathrm{d}\mu =0.$$

$$f(x)=f(0)+\sum _{\begin{array}{c}i\in \{1,\dots ,N\}\\ z_i<x\end{array}}{\alpha }_i{\nabla }^{\mu }f(z_i),$$

$$A(f(z_1),\dots ,f(z_N){)}^{\top }=({\nabla }^{\mu }f(z_1),\dots ,{\nabla }^{\mu }f(z_N){)}^{\top }.$$

$${\nabla }^{\mu }f(z_N)$$

$$A=\left(\begin{array}{ccccccc}-{\alpha }_1^{-1}& {\alpha }_1^{-1}& 0& \cdots & 0& 0& 0\\ 0& -{\alpha }_2^{-1}& {\alpha }_2^{-1}& \cdots & 0& 0& 0\\ 0& 0& -{\alpha }_3^{-1}& \cdots & 0& 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & -{\alpha }_{N-2}^{-1}& {\alpha }_{N-2}^{-1}& 0\\ 0& 0& 0& \cdots & 0& -{\alpha }_{N-1}^{-1}& {\alpha }_{N-1}^{-1}\\ {\alpha }_N^{-1}& 0& 0& \cdots & 0& 0& -{\alpha }_N^{-1}\end{array}\right).$$

$$\mathcal{E}(f,g)={\mathcal{E}}^{\mu }(f,g):=⟨{\nabla }^{\mu }f,{\nabla }^{\mu }g⟩,$$

$$|\hat{f}(z_{i+1})-\hat{f}(z_i)|\le |f(z_{i+1})-f(z_i)|\text{and}|\hat{f}(z_1)-\hat{f}(z_N)|\le |f(z_1)-f(z_N)|,$$

$$⟨{\nabla }^{\mu }f,{\nabla }^{\mu }g⟩=-⟨{\Delta }^{\mu }f,g⟩,\text{and thus,}{\Delta }^{\mu }=-{\left({\nabla }^{\mu }\right)}^{\ast }\circ {\nabla }^{\mu }.$$

$$B=\left(\begin{array}{cc}-{\alpha }_1^{-2}-{\alpha }_2^{-2}& {\alpha }_1^{-2}+{\alpha }_2^{-2}\\ {\alpha }_1^{-2}+{\alpha }_2^{-2}& -{\alpha }_1^{-2}-{\alpha }_2^{-2}\end{array}\right),$$

$$|\lambda |\le 2\max \{|B_{i,i}|:i\in \{1,\dots ,N\}\}.\text{qed}$$

$$f_0(x)=1,\text{for all x∈[0,1],}\text{and}{f}_1(x)=\{\begin{array}{ll}1& \text{for}x\in [0,z_1]\cup (z_2,1],\\ -1& \text{otherwise.}\end{array}$$

$$f_{\kappa }^{\mu }(x)=\sin (\pi \kappa F_{\mu }(x))\text{or}{g}_{\kappa }^{\mu }(x)=\cos (\pi \kappa F_{\mu }(x)),$$

$${\lambda }_0$$

$${\lambda }_3$$

$$v^{(0)}$$

$$v^{(1)}$$

$$v^{(2)}$$

$$v^{(3)}$$

$$v^{(4)}$$

$$v^{(5)}$$

$$S_{1,5}:=\left\{\left(v_1^{(1)},v_1^{(5)}\right),\dots ,\left(v_6^{(1)},v_6^{(5)}\right)\right\}\text{and}{S}_{2,4}:=\left\{\left(v_1^{(2)},v_1^{(4)}\right),\dots ,\left(v_6^{(2)},v_6^{(4)}\right)\right\}$$

$$\sqrt{1+r^{-4}-r^{-2}}\text{lower}0.4pt\text{vrule}height=6.44444pt,depth=-5.15558pt\left(x^2+y^2-1\right)=\left(2-r^{-2}\right)xy,$$

$$w^{(\kappa )}:={\left(\sin (\pi \kappa F_{\mu }(z_1)),\dots ,\sin (\pi \kappa F_{\mu }(z_6))\right)}^{\top },u^{(\kappa )}:={\left(\cos (\pi \kappa F_{\mu }(z_1)),\dots ,\cos (\pi \kappa F_{\mu }(z_6))\right)}^{\top },$$

$$B=\left(\begin{array}{ccccccc}-2N^2& N^2& 0& \cdots & 0& 0& N^2\\ N^2& -2N^2& N^2& \cdots & 0& 0& 0\\ 0& N^2& -2N^2& \cdots & 0& 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & -2N^2& N^2& 0\\ 0& 0& 0& \cdots & N^2& -2N^2& N^2\\ N^2& 0& 0& \cdots & 0& N^2& -2N^2\end{array}\right).$$

$${\lambda }_l=-2N^2+2N^2\cos \left(2\pi l/N\right)$$

$$\displaystyle v^{(l)}=\bigg{(}1,\exp\left(2\pi\mathbf{i}\,l/N\right),\exp\left(2\pi\mathbf{i}\,2l/N\right),\dots,\exp\left(2\pi\mathbf{i}\,(N-1)l/N\right)\bigg{)}^{\top},$$

$$\sum _{k=1}^{N-j}{m}_k{v}_{k+j}+\sum _{k=N-j+1}^N{m}_k{v}_{k-N+j}=\lambda v_{j+1},$$

$${\lambda }_l$$

$$=-2N^2+2N^2\cos \left(2\pi l/N\right).\text{qed}$$

Full text

1. Introduction

Motivated by the fundamental theorem of calculus, and based on the works of Feller [4] and Kac and Kreĭn [8], given an atomless Borel probability measure $\mu$ supported on a compact subset of $\mathbb{R}$, Freiberg and Zähle [6] introduced a measure-geometric approach to define a first order differential operator $\nabla^{\mu}$ and a second order differential operator $\Delta^{\mu,\mu}\coloneqq\nabla^{\mu}\circ\nabla^{\mu}$, with respect to $\mu$. In the case that $\mu$ is the Lebesgue measure, it was shown that $\nabla^{\mu}$ coincides with the weak derivative. Moreover, a harmonic calculus for $\Delta^{\mu,\mu}$ was developed and, when $\mu$ is a self-similar measure supported on a Cantor set, the authors proved the eigenvalue counting function of $\Delta^{\mu,\mu}$ is comparable to the square-root function. In [10] for continuous measures the exact eigenvalues and eigenfunctions were obtained and it was shown the eigenvalues do not depend on the given measure. Arzt [1] has also considered the Kreĭn-Feller operator $\Delta^{\mu,\Lambda}\coloneqq\nabla^{\mu}\circ\nabla^{\Lambda}$, where $\mu$ denotes a continuous Borel probability measure and $\Lambda$ denotes the Lebesgue measure, see [5, 7] for further results in this direction.

Here, we show this framework can be extended to included purely atomic measures $\mu$. Unlike in the case when one has a measure with a continuous distribution function (see for instance [6, 10]), we prove the operators $\nabla^{\mu}$ and $\Delta^{\mu,\mu}$ are no longer symmetric. To circumvent this problem, we consider the operator $\nabla^{\mu}$, its adjoint $(\nabla^{\mu})^{*}$ and define the $\mu$-Laplacian to be $\Delta^{\mu}=-(\nabla^{\mu})^{*}\circ\nabla^{\mu}$. We give matrix representations for these operators, noting they coincide with the normalised Laplacian matrix of a cycle graph [2] and resemble a discretisation of a one-dimensional Laplacian on a non-uniform grid [15]. Further, we discuss properties of the eigenvalues and eigenfunctions of the operator $\Delta^{\mu}$. In particular, we show the eigenfunctions for distributions with finite support are not necessarily of the form $f_{\kappa}^{\mu}(\cdot)\coloneqq\sin(\pi\kappa F_{\mu}(\cdot))$ or $g_{\kappa}^{\mu}(\cdot)\coloneqq\cos(\pi\kappa F_{\mu}(\cdot))$, for $\kappa\in\mathbb{R}\setminus\{0\}$ and where $F_{\mu}$ denotes the distribution function of $\mu$. This differs from the case of continuous distributions, see [6, 10, 17]. Additionally, in the case that $\mu$ is a uniform discrete probability distribution we explicitly determine the eigenvalues and eigenfunctions of $\Delta^{\mu}$.

Outline

In Section 2 we present necessary definitions and basic properties of $\nabla^{\mu}$, $(\nabla^{\mu})^{*}$ and $\Delta^{\mu}$ and give matrix representations for these operators. In Section 3 we prove general results concerning the spectral properties of $\Delta^{\mu}$. We conclude with Section 4, where explicit computations are carried out when $\mu$ is a uniform discrete probability distribution.

2. Definitions and analytic properties of $\Delta^{\mu}$

Set $\mathcal{I}\coloneqq[0,1]$ and let $\delta_{z}$ denote the Dirac-measure at $z$, for some fixed $z\in\mathcal{I}$. Let $\mu$ denote the probability measure $\mu\coloneqq\sum_{i=1}^{N}\alpha_{i}\delta_{z_{i}}$, where $N\in\mathbb{N}$, $0\leq z_{1}<z_{2}<\dots<z_{N}<1$ and $\alpha_{i}>0$, for $i\in\{1,\dots,N\}$. We denote the set of real-valued square-integrable functions on $\mathcal{I}$ by $\mathfrak{L}^{2}_{\mu}=\mathfrak{L}^{2}_{\mu}(\mathcal{I})$, we define $\mathcal{N}_{\mu}(\mathcal{I})$ to be set of $\mathfrak{L}^{2}_{\mu}$-functions which are constant zero $\mu$-almost everywhere, and we let $L^{2}_{\mu}=L^{2}_{\mu}(\mathcal{I})\coloneqq\mathfrak{L}^{2}_{\mu}(\mathcal{I})\setminus\mathcal{N}_{\mu}(\mathcal{I})$. The latter space is a finite-dimensional inner product space with inner product $\langle\cdot,\cdot\rangle$ given by

[TABLE]

We define the set of $\mu$-differentiable functions on $\mathcal{I}$ with periodic boundary conditions by

[TABLE]

where we understand $[0,0)=\emptyset$. Note, the function $f^{\prime}$ defined in (1) is unique in $L^{2}_{\mu}$. Since $f(0)=f(1)=f(0)+\int\mathds{1}_{[0,1)}f^{\prime}\,\mathrm{d}\mu$, it follows that

[TABLE]

For $f\in\mathscr{D}_{\mu}^{1}$ and $f^{\prime}$ as in (1), the operator $\nabla^{\mu}\colon\mathscr{D}_{\mu}^{1}\to L^{2}_{\mu}$ defined by $\nabla^{\mu}f\coloneqq f^{\prime}$ is called the ${\mu}$-derivative. Linearity of the integral yields $\nabla^{\mu}$ is linear on $\mathscr{D}_{\mu}^{1}$. As $\mu$ is a linear combination of Dirac measures, we can reformulate the defining equation of $\nabla^{\mu}f$ given in (1) by

[TABLE]

where $f\in\mathscr{D}_{\mu}^{1}$ and $x\in\mathcal{I}$. Thus, if $\mu$ is a Dirac measure, that is $N=1$, then (2) becomes $\alpha_{1}\nabla^{\mu}f(z_{1})=0$. Hence, from (3), it follows that $\mathscr{D}_{\mu}^{1}$ is the set of constant functions. In other words, the operator $\nabla^{\mu}$ is the null-operator, and so, from here on we assume $N\geq 2$.

The periodic boundary conditions and (3) together imply that a function $f\in\mathscr{D}_{\mu}^{1}$ is piecewise constant; namely, $f|_{[0,z_{1}]\cup(z_{N},1]}$ and $f|_{(z_{i},z_{i+1}]}$ are constant, for $i\in\{1,\dots,N-1\}$. Therefore, $f$ is uniquely determined by the vector $(f(z_{1}),\dots,f(z_{N}))^{\top}$, and thus, there exists an $N\times N$-matrix $A$ with

[TABLE]

From (3) and the fact that $f(1)=f(0)=f(z_{1})$, we have

[TABLE]

for $n\in\{1,\dots,N-1\}$, and hence,

[TABLE]

Since $\sum_{i=1}^{N}\alpha_{i}g^{2}(z_{i})<\infty$, for all $g\in\mathscr{D}_{\mu}^{1}$, there exists a natural embedding $\pi\colon\mathscr{D}_{\mu}^{1}\to L_{\mu}^{2}$. In fact, from the matrix representation given above it follows that $\pi(\mathscr{D}_{\mu}^{1})=L_{\mu}^{2}$. In other words, every equivalence class of $L_{\mu}^{2}$ has a $\mu$-differentiable representative, and so, from here on we will not distinguish between $\mathscr{D}_{\mu}^{1}$ and $\pi(\mathscr{D}_{\mu}^{1})$.

Notice, $A$ is not self-adjoint, and $A^{2}$ is self-adjoint if and only if $N=2$ and $\alpha_{1}=\alpha_{2}$. Hence, the operators $\nabla^{\mu}$ and $\Delta^{\mu}\coloneqq\nabla^{\mu}\circ\nabla^{\mu}$ are not in general self-adjoint. To obtain a self-adjoint operator we follow the program of Kigami [11, 12, 13], and Kigami and Lapidus [14], and use the bilinear form $\mathcal{E}$ defined by

[TABLE]

for $f,g\in\mathscr{D}_{\mu}^{1}$. We refer to $\mathcal{E}$ as the $\mu$-energy form.

Theorem 2.1**.**

The $\mu$-energy form $\mathcal{E}$ is a Dirichlet form.

Proof.

The $\mu$-energy form is bilinear since the inner product is bilinear, $\nabla^{\mu}$ is linear and every equivalence class of $L_{\mu}^{2}$ has a $\mu$-differentiable representative. The symmetry and the non-negativity of $\mathcal{E}$ follow from the properties of the inner product. For every $f\in\mathscr{D}_{\mu}^{1}$, the function $\hat{f}\colon\mathcal{I}\to\mathbb{R}$, defined by $\hat{f}(x)\coloneqq\min(\max(f(x),0),1)$, belongs to $\mathscr{D}_{\mu}^{1}$, and as

[TABLE]

it follows that $\mathcal{E}(\hat{f},\hat{f})\leq\mathcal{E}(f,f)$. The properties of $\langle\cdot,\cdot\rangle$ and $\mathcal{E}$ yield $\mathscr{D}_{\mu}^{1}$ equipped with $\langle\cdot,\cdot\rangle_{\mathcal{E}}\coloneqq\langle\cdot,\cdot\rangle+\mathcal{E}(\cdot,\cdot)$ is an inner product space. Now, for every Cauchy sequence $(f_{n})_{n\in\mathbb{N}}$ in $(\mathscr{D}_{\mu}^{1},\langle\cdot,\cdot\rangle_{\mathcal{E}})$, we have both $(f_{n})_{n\in\mathbb{N}}$ and $(\nabla^{\mu}f_{n})_{n\in\mathbb{N}}$ are Cauchy-sequences in $L_{\mu}^{2}$. Hence, there exist $\tilde{f_{0}},\tilde{f_{1}}\in L_{\mu}^{2}$ with $\lim_{n\to\infty}\lVert f_{n}-\tilde{f_{0}}\rVert=0$ and $\lim_{n\to\infty}\lVert\nabla^{\mu}f_{n}-\tilde{f_{1}}\rVert=0$, where $\lVert f\rVert^{2}\coloneqq\langle f,f\rangle$, for $f\in L_{\mu}^{2}$. Since $\nabla^{\mu}$ is linear, it is continuous, and so $\nabla^{\mu}\tilde{f_{0}}=\tilde{f_{1}}$. This implies that $\lim_{n\to\infty}f_{n}=\tilde{f_{0}}\in\mathscr{D}_{\mu}^{1}$, with respect to $\langle\cdot,\cdot\rangle_{\mathcal{E}}$. ∎

Notice, since the analysis reduces to a finite dimensional vector space, the $\mu$-energy form is also a graph energy form as treated in [9].

We say that $f\in\mathscr{D}_{\mu}^{1}$ belongs to $\mathscr{D}_{\mu}^{2}=\mathscr{D}_{\mu}^{2}(\mathcal{I})$, if there exists a $h\in L^{2}_{\mu}$, necessarily unique, such that $\mathcal{E}(f,g)=-\langle h,g\rangle$, for all $g\in\mathscr{D}_{\mu}^{1}$. We define the $\mu$-Laplacian to be the operator $\Delta^{\mu}\colon\mathscr{D}_{\mu}^{2}\to L^{2}_{\mu}$ given by $\Delta^{\mu}f\coloneqq h$. Indeed, for an arbitrary $g\in\mathscr{D}_{\mu}^{1}$, we observe

[TABLE]

With this, we conclude $B\coloneqq-A^{\top}A$ is a matrix representation of $\Delta^{\mu}$; in fact, for $N=2$,

[TABLE]

and, for $N\geq 3$, we have $B$ is the $N\times N$-matrix

$\begin{pmatrix}-\alpha_{N}^{-2}-\alpha_{1}^{-2}&\alpha_{1}^{-2}&0&\cdots&0&0&\alpha_{N}^{-2}\\[2.5pt] \alpha_{1}^{-2}&-\alpha_{1}^{-2}-\alpha_{2}^{-2}&\alpha_{2}^{-2}&\cdots&0&0&0\\[2.5pt] 0&\alpha_{2}^{-2}&-\alpha_{2}^{-2}-\alpha_{3}^{-2}&\cdots&0&0&0\\[2.5pt] \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\[2.5pt] 0&0&0&\cdots&-\alpha_{N-3}^{-2}-\alpha_{N-2}^{-2}&\alpha_{N-2}^{-2}&0\\[2.5pt] 0&0&0&\cdots&\alpha_{N-2}^{-2}&-\alpha_{N-2}^{-2}-\alpha_{N-1}^{-2}&\alpha_{N-1}^{-2}\\[2.5pt] \alpha_{N}^{-2}&0&0&\cdots&0&\alpha_{N-1}^{-2}&-\alpha_{N-1}^{-2}-\alpha_{N}^{-2}\end{pmatrix}$.

Theorem 2.2**.**

The operator $\Delta^{\mu}$ is linear, self-adjoint and non-positive.

Proof.

Linearity follows from linearity of $\nabla^{\mu}$ and bilinearity of $\mathcal{E}$. Self-adjointness is a consequence of symmetry of $\mathcal{E}$. An application of (4) yields $\langle\Delta^{\mu}f,f\rangle=-\langle\nabla^{\mu}f,\nabla^{\mu}f\rangle\leq 0$, and hence, $\Delta^{\mu}$ is non-positive. ∎

The above demonstrates one can view the operator $\Delta^{\mu}$ as a Laplacian matrix of a weighted cycle graph, see [2, 3]. Indeed, the analytic results from spectral graph theory may be carried over to our setting, which would certainly be interesting to investigate and could lead to further results concerning $\Delta^{\mu}$. Moreover, the matrix $B$ resembles the matrices appearing in the finite difference methods used in numerical analysis of ordinary differential equations111 

We would like to thank Paul Choboter, Maik Gröger, Jens Rademacher and Alfred Schmidt for bringing this to our attention. , see [15]. We also observe the matrix representation of the operator $\Delta^{\mu}$ only depends on the order and the weighting of the atoms of $\mu$ and is independent on the distances between $z_{i}$ and $z_{j}$, for all $i,j\in\{1,\dots,N\}$.

3. Spectral properties of $\Delta^{\mu}$

By definition of the matrix $B$, to find the eigenvalues and eigenfunctions of $\Delta^{\mu}$, it suffices to compute the eigenvalues and eigenvectors of $B$.

Proposition 3.1**.**

If $\lambda$ is an eigenvalue of $\Delta^{\mu}$, then $\lambda\in\mathbb{R}$ and $\displaystyle 2\min_{i\in\{1,\dots,N\}}B_{i,i}\leq\lambda\leq 0$.

Proof.

Theorem 2.2 together with the fact that $B$ is self-adjoint with entries in $\mathbb{R}$ yields all eigenvalues are non-positive real numbers. Since the spectral norm is bounded above by the column-sum norm, given an eigenvalue $\lambda$ of $B$, it follows that

[TABLE]

Proposition 3.2**.**

The operator $\Delta^{\mu}$ has a simple eigenvalue at $\lambda=0$ where the corresponding eigenfunction is the constant function with value $1$.

Proof.

A direct calculation reveals $B(1,\dots,1)^{\top}=(0,\dots,0)^{\top}$, and hence, we have that $\lambda=0$ is an eigenvalue of $\Delta^{\mu}$ where the constant function with value $1$ is the corresponding eigenfunction. A row reduced echelon form of $A$ is an upper triangular matrix with a single zero on the diagonal, and so $\operatorname{rank}(A)=N-1$. Combining this with the fact that $\operatorname{rank}(B)=\operatorname{rank}(A^{\top}A)=\operatorname{rank}(A)$ it follows that the eigenvalue $\lambda=0$ of $B$ is simple. ∎

We observe, for $N=2$, that the eigenvalues of $\Delta^{\mu}$ are $\lambda_{0}=0$ and $\lambda_{1}=-2(\alpha_{1}^{-2}+\alpha_{2}^{-2})$ with corresponding eigenfunctions,

[TABLE]

This follows since $\lambda_{0}$ and $\lambda_{1}$ are eigenvalues of the matrix $B$, given in (5), with corresponding eigenvectors $v^{(0)}=(1,1)^{\top}$ and $v^{(1)}=(1,-1)^{\top}$, respectively. In particular, we see that, in this case, the lower bound in Theorem 3.1 is sharp.

Different to the case of continuous distributions, the eigenfunctions for distributions with finite support are not necessarily of the form

[TABLE]

for $x\in[0,1]$ and $\kappa\in\mathbb{R}\setminus\{0\}$, where $F_{\mu}$ denotes the distribution function of $\mu$, which we now address in the following paragraph.

Let $m_{1}$ and $m_{2}$ denote two positive real numbers with $3m_{1}+3m_{2}=1$, and set $r=m_{2}/m_{1}$. Consider the discrete distribution $\mu=\sum_{i=1}^{6}\alpha_{i}\delta_{z_{i}}$, where $0<z_{1}<z_{2}<\cdots<z_{6}<1$, $\alpha_{1}=\alpha_{3}=\alpha_{5}=m_{1}$ and $\alpha_{2}=\alpha_{4}=\alpha_{6}=m_{2}$. A direct calculation shows the eigenvalues for the matrix representation of $\Delta^{\mu}$ are

[TABLE]

with corresponding eigenvectors

[TABLE]

For the eigenvalues with multiplicity two, namely $\lambda_{1}=\lambda_{5}$ and $\lambda_{2}=\lambda_{4}$, notice that the sets of tuples

[TABLE]

determine the same ellipse, namely,

[TABLE]

which is non-axisymmetric; see Figure 2 for an example and compare with Theorem 4.1 and Corollary 4.2, where the analogous set of tuples lie on the unit circle. This latter property demonstrates that the eigenspace for $\lambda_{1}=\lambda_{5}$ (or $\lambda_{2}=\lambda_{4}$) is not spanned by $\{f_{\kappa_{1}}^{\mu},g_{\kappa_{2}}^{\mu}\}$ for any $\kappa_{1},\kappa_{2}\in\mathbb{R}$. Moreover, in this explicit case, setting

[TABLE]

a direct calculation shows $Bw^{(\kappa)}\neq\lambda_{i}w^{(\kappa)}$ and $Bu^{(\kappa)}\neq\lambda_{i}u^{(\kappa)}$, for all $\kappa\in\mathbb{R}\setminus\{0\}$ and all $i\in\{0,\dots,5\}$. This latter result also holds when replacing $F_{\mu}(z_{i})$ by $\widetilde{F}_{\mu}(z_{i})\coloneqq F_{\mu}(z_{i}-\varepsilon)$, for a fixed $\varepsilon\in(0,\min\{z_{1},\min\{z_{i+1}-z_{i}\colon i\in\{1,\dots,5\}\}\})$. For further details and examples, we refer the reader to [16].

For the case $m_{1}=1/4$ and $m_{2}=1/12$, in Figure 1 the corresponding eigenfunctions of the operator $\Delta^{\mu}$ are sketched and Figure 2 illustrates the point plots of $S_{1,5}$ and $S_{2,4}$ together with the corresponding ellipse.

4. Uniform discrete probability distributions

Here, we consider the case when $\mu$ is a uniform discrete probability distribution with $N\geq 3$, namely $\alpha_{i}=N^{-1}$ for all $i\in\{1,\dots,N\}$, in which case,

[TABLE]

The following theorem reveals the spectrum of this matrix, and in the case that $N$ is even, the result gives a second example for which the lower bound in Theorem 3.1 is sharp.

Theorem 4.1**.**

The eigenvalues of the matrix $B$ given directly above are of the form

[TABLE]

with corresponding eigenvectors

[TABLE]

for $l\in\{0,\dots,N-1\}$.

Proof.

Set $m_{1}=-2N^{2},m_{2}=m_{N}=N^{2}$ and $m_{i}=0$, for $i\in\{3,\dots,N-1\}$. The identity $Bv=\lambda v$ can be formulated as a system of $N$ difference equations of the form

[TABLE]

where $j\in\{0,\dots,N-1\}$ and $v=(v_{1},\dots,v_{N})$. To obtain the eigenvalue $\lambda_{l}$, we follow the ansatz $v_{k}^{(l)}=\varphi_{l}^{k-1}$, where $\varphi_{l}^{k}\coloneqq\exp(2\pi\mathbf{i}\,kl/N)$, for $k,l\in\{0,\dots,N-1\}$. Substituting this into (6), and using the facts that $\varphi_{l}^{-N}=\varphi_{l}^{0}=1$ and $\varphi_{l}^{j}\neq 0$, for all $j,l\in\{0,\dots,N-1\}$, we obtain $\lambda_{l}=\sum_{k=1}^{N}m_{k}\varphi_{l}^{k-1}$. Hence, for $l\in\{0,\dots,N-1\}$, we have $Bv^{(l)}=\lambda_{l}v^{(l)}$ and

[TABLE]

Corollary 4.2**.**

The eigenvalues of the operator $\Delta^{\mu}$ are $\lambda_{l}=-2N^{2}+2N^{2}\cos\left(2\pi l/N\right)$, for $l\in\{0,\dots,N-1\}$, with corresponding eigenfunctions $f_{l}\in\mathscr{D}_{\mu}^{2}$, where

1. (1)

$f_{0}$* is the constant function with value $1$,*

and, for $j\in\{1,\dots,N-1\}$,

1. (2)

$\displaystyle f_{l}|_{[0,z_{1}]\cup(z_{N},1]}=0$* and $\displaystyle f_{l}|_{(z_{j},z_{j+1}]}=\operatorname{Im}\left(\exp\left(2\pi\mathbf{i}\,jl/N\right)\right)$, for $0<l<N/2$, and* 2. (3)

$\displaystyle f_{l}|_{[0,z_{1}]\cup(z_{N},1]}=1$* and $\displaystyle f_{l}|_{(z_{j},z_{j+1}]}=\operatorname{Re}\left(\exp\left(2\pi\mathbf{i}\,jl/N\right)\right)$, for $N/2\leq l\leq N-1$.*

(See Figures 3 and 4.)

In the situation of Corollary 4.2, we have for $N$ tending to infinity, that the pure point measures converge weakly to the Lebesgue measure, and indeed, the eigenfunctions of the discrete Laplacians converge uniformly to (appropriately re-scaled) cosine and sine functions. Moreover, the eigenvalue approach $-(2\pi k)^{2}$; the corresponding eigenvalues of the classical weak Laplacian.