Abstract:
We introduce two new heuristic ideas concerning the spectrum of a Laplacian, and we give theorems and conjectures from the realms of manifolds, graphs and fractals that validate these heuristics. The first heuristic concerns Laplacians that do not have discrete spectra: here we introduce a notion of "spectral mass", an average of the diagonal of the kernel of the spectral projection operator, and show that this can serve as a substitute for the eigenvalue counting function. The second heuristic is an "asymptotic Schur's lemma" that describes the proportions of the spectrum that transforms according to the irreducible representations of a finite group that acts as a symmetric group of the Laplacian. For this to be valid we require the existence of a fundamental domain with relatively small boundary. We also give a version in the case that the symmetry groups is a compact Lie group. Many of our results are reformulations of known results, and some are merely conjectures, but there is something to be gained by looking at them together with a new perspective.

Abstract:
One of the ways that analysis on fractals is more complicated than analysis on manifolds is that the asymptotic behavior of the spectral counting function $N(t)$ has a power law modulated by a nonconstant multiplicatively periodic function. Nevertheless, we show that for the Sierpinski gasket it is possible to write an exact formula, with no remainder term, valid for almost every $t$. This is a stronger result than is valid on manifolds.

Abstract:
Let $N(t)$ denote the eigenvalue counting funtion of the Laplacian on a compact surface of constant nonnegative curvature, with or without boundary. We define a refined asymptotic formula $\tilde{N}(t)=At+Bt^{1/2}+C$, where the constants are expressed in terms of the geometry of the surface and its boundary, and consider the average error $A(t) = \frac{1}{t} \int_0^{t}D(s)ds$ for $D(t) = N(t) - \tilde{N}(t)$. We present a conjecture for the asymptotic behavior of $A(t)$, and study some examples that support the conjecture.

Abstract:
We characterize functions of ?nite energy in the plane in terms of their traces on the lines that make up "graph paper" with squares of side length $mn$ for all $n$, and certain $\12-$order Sobolev norms on the graph paper lines. We also obtain analogous results for functions of ?nite energy on two classical fractals: the Sierpinski gasket and the Sierpinski carpet.

Abstract:
We study boundary value problems for the Laplacian on a domain $\Omega$ consisting of the left half of the Sierpinski Gasket ($SG$), whose boundary is essentially a countable set of points $X$. For harmonic functions we give an explicit Poisson integral formula to recover the function from its boundary values, and characterize those that correspond to functions of finite energy. We give an explicit Dirichlet to Neumann map and show that it is invertible. We give an explicit description of the Dirichlet to Neumann spectra of the Laplacian with an exact count of the dimensions of eigenspaces. We compute the exact trace spaces on $X$ of the $L^2$ and $L^\infty$ domains of the Laplacian on $SG$. In terms of the these trace spaces, we characterize the functions in the $L^2$ and $L^\infty$ domains of the Laplacian on $\Omega$ that extend to the corresponding domains on $SG$, and give an explicit linear extension operator in terms of piecewise biharmonic functions.

Abstract:
We study the spectral asymptotics of wave equations on certain compact spacetimes where some variant of the Weyl asymptotic law is valid. The simplest example is the spacetime $S^1 \times S^2$. For the Laplacian on $S^1 \times S^2$ the Weyl asymptotic law gives a growth rate $O(s^{3/2})$ for the eigenvalue counting function $N(s) = \#\{\lambda _j: 0 \leq \lambda _j \leq s\}$. For the wave operator there are two corresponding eigenvalue counting functions $N^{\pm}(s) = \#\{\lambda _j: 0 < \pm \lambda _j \leq s\}$ and they both have a growth rate of $O(s^2)$. More precisely there is a leading term $\frac{\pi^2}{4}s^2$ and a correction term of $as^{3/2}$ where the constant $a$ is different for $N^{\pm}$. These results are not robust, in that if we include a speed of propagation constant to the wave operator the result depends on number theoretic properties of the constant, and generalizations to $S^1 \times S^q$ are valid for $q$ even but not $q$ odd. We also examine some related examples.

Abstract:
The difference between the number of lattice points in a disk of radius $\sqrt{t}/2\pi$ and the area of the disk $t/4\pi$ is equal to the error in the Weyl asymptotic estimate for the eigenvalue counting function of the Laplacian on the standard flat torus. We give a sharp asymptotic expression for the average value of the difference over the interval $0 \leq t \leq R$. We obtain similar results for families of ellipses. We also obtain relations to the eigenvalue counting function for the Klein bottle and projective plane.

Abstract:
In this paper, we establish an analogue of the classical mean value property for both the harmonic functions and some general functions in the domain of the Laplacian on the Sierpinski gasket. Furthermore, we extend the result to some other p.c.f. fractals with Dihedral-3 symmetry.

Abstract:
In the case of some fractals, sampling with average values on cells is more natural than sampling on points. In this paper we investigate this method of sampling on $SG$ and $SG_{3}$. In the former, we show that the cell graph approximations have the spectral decimation property and prove an analog of the Shannon sampling theorem.. We also investigate the numerical properties of these sampling functions and make conjectures which allow us to look at sampling on infinite blowups of $SG$. In the case of $SG_{3}$, we show that the cell graphs have the spectral decimation property, but show that it is not useful for proving an analogous sampling theorem.

Abstract:
We study the analog of power series expansions on the Sierpinski gasket, for analysis based on the Kigami Laplacian. The analog of polynomials are multiharmonic functions, which have previously been studied in connection with Taylor approximations and splines. Here the main technical result is an estimate of the size of the monomials analogous to x^n/n!. We propose a definition of entire analytic functions as functions represented by power series whose coefficients satisfy exponential growth conditions that are stronger than what is required to guarantee uniform convergence. We present a characterization of these functions in terms of exponential growth conditions on powers of the Laplacian of the function. These entire analytic functions enjoy properties, such as rearrangement and unique determination by infinite jets, that one would expect. However, not all exponential functions (eigenfunctions of the Laplacian) are entire analytic, and also many other natural candidates, such as the heat kernel, do not belong to this class. Nevertheless, we are able to use spectral decimation to study exponentials, and in particular to create exponentially decaying functions for negative eigenvalues.