aa r X i v : . [ m a t h . C A ] A ug The Continuous Graph FFT
Robert CarlsonDepartment of MathematicsUniversity of Colorado at Colorado [email protected] 10, 2018
Abstract
The discrete Fourier transform and the FFT algorithm are ex-tended from the circle to continuous graphs with equal edge lengths.
Introduction
The discrete Fourier transform (DFT) and algorithms for its efficient compu-tation (FFT) enjoy an enormous range of applications. One might roughlydivide such applications into the analysis of data which is sampled in timeor in space. Applications involving spatially sampled data are basic in nu-merical analysis, since constant coefficient partial differential equations andsome of their discrete analogs, including the heat, wave, Schrodinger, andbeam equations in one space dimension, have exact solutions in Fourier se-ries. Other applications of the DFT involving the analysis of spatially sam-pled data include noise removal, data compression, data interpolation, andapproximation of functions. These applications often take advantage of thetight and explicit linkage binding the harmonic analysis of uniformly sampleddata with the harmonic analysis of periodic functions defined on R .Many aspects of Fourier analysis can be developed through the spec-tral theory of the second derivative operator − D = − d /dx acting on theHilbert space of square integrable functions with period 1. An orthogonalbasis of eigenfunctions is given by exp(2 πikx ). If N uniformly spaced sam-ples x n = n/N for n = 0 , . . . , N − πikn/N ) , k, n = 0 , . . . , N − , again provides an orthogonal basis for functions on the set { x n } . Thesesampled exponential functions are also eigenfunctions for ’local’ operatorssuch as ∆ N , which acts by∆ N f ( x n ) = f ( x n ) −
12 [ f ( x n − + f ( x n +1 )] , index arithmetic being carried out modulo N . The fact that efficient FFTalgorithms are available for sampled data is not a consequence of abstractspectral theory, but relies on the exponential form of the eigenfunctions andthe arithmetic progression of the frequencies k .Our discussion now shifts to graphs and their refinements, with the circleand its uniform sampling serving as an example. Discrete graphs with N V < ∞ vertices and N E edges have a well developed spectral theory [7] based onthe adjacency or Laplace operators acting on functions defined on the vertexset V . While these ideas play a role here, the lead actor is a continuous graph,2lso known as a metric graph or, when a differential operator is emphasized,a quantum graph. Here the edges of a graph are identified with intervals[ a n , b n ] ⊂ R , functions are elements of the Hilbert space ⊕ n L [ a n , b n ], anddifferential operators such as − D provide a basis for both physical modelingand harmonic analysis. (The terminology for such ’continuous graphs’ is notsettled, with the terms topological graphs or networks also used.)The natural and technological worlds offer numerous opportunities forgraph modeling, where applications such as those mentioned above can beconsidered with data sampled from continuous graphs. These include thesampling of populations along river systems, and the description of traf-fic density on road networks. Biological systems transport nutrients, waste,heat, and pressure waves through vascular networks, and electrochemical sig-nals through natural neural networks. Elaborate networks are manufacturedin microelectronics, and for microfluidic laboratories on chips.Despite a rapidly growing literature on ’quantum graphs’ [4, 8, 9], andsome work [2] generalizing the classical theory of Fourier series to functionsdefined on a continuous graph, there is a very limited literature consideringthe harmonic analysis of sampled data for these geometric objects. Thepapers [13, 14] consider sampling on continuous graphs with virtually nolength restrictions on the edges. In contrast, this work exploits the featuresappearing when graphs have equal length edges, and obtains the followingconclusion: every finite continuous graph with edges of equal length admits afamily of DFTs closely analogous to that of the circle, and an FFT algorithmfor their efficient computation.To unify the presentation a bit, we focus on simple discrete graphs, with-out loops or multiple edges between vertices, whose vertices have degree atleast two. Given a continuous version of a nonsimple graph without bound-ary vertices, one can insert additional vertices of degree 2 to reduce to thesimple case. This insertion of ’invisible’ vertices can also be used to reducegraphs whose edge lengths are integer multiples of a common value to theequal length case. Algorithmically, this reduction increases the size of thediscrete graph spectral problem whose solution is an important componentof the DFT.There are four subsequent sections in the paper. The second section startswith a review of differential operators on continuous graphs. This leads toa more detailed discussion of the spectral theory of the standard Laplacedifferential operator on continuous graphs with equal length edges. Most ofthe material in this section was previously known [3, 11], although Theorem3.9 appears to be new. The third section explores the linked spectral theoryof Laplacians for continuous graphs and their uniformly sampled subgraphs.The fourth section shows that efficient algorithms for Fourier analysis areavailable. The final section presents a simple example where many of thecomputations can be done ’by hand’. In this work a discrete graph G will be finite and simple, with a vertexset V having N V points, and a set E of N E edges. Each edge e has a positive length l e . It is convenient to number the edges. For n = 1 , . . . , N E the edge e n isidentified with a real interval [ a n , b n ] of length l n . The resulting topologicalgraph is also denoted G . In an obvious fashion one may extend the standardmetric and Lebesgue measure from the edges to G .The identification of graph edges and intervals allows us to define theHilbert space L ( G ) = ⊕ N E n =1 L [ a n , b n ]with the inner product h f, g i = Z G f g = X n Z b n a n f n ( x ) g n ( x ) dx, f n : [ a n , b n ] → C . We will employ the standard vertex conditions, requiring continuity at thevertices, lim x ∈ e ( i ) → v f ( x ) = lim x ∈ e ( j ) → v f ( x ) , e ( i ) , e ( j ) ∼ v, (2.1)and the derivative condition X e n ∼ v ∂ ν f n ( v ) = 0 . (2.2)Here the derivative ∂ ν f n ( v ) is f ′ n ( a n ), respectively − f ′ n ( b n ) if a n , respectively b n , is identified with v .The standard vertex conditions are used to define the Laplacian L f = − f ′′ of the continuous graph G . Let D max denote the set of functions f ∈ L ( G ) with f ′ absolutely continuous on each e n , and f ′′ ∈ L ( G ). The domain4 of L is then the set of f ∈ D max satisfying the standard conditions (2.1)and (2.2). From the classical theory of ordinary differential operators [12, p.S123] one knows that L is self adjoint with compact resolvent. This operatoris nonnegative, with 0 being a simple eigenvalue if G is connected. Writingeigenvalues with multiplicity, the spectrum is thus a sequence 0 = λ < λ ≤ λ ≤ . . . , and there is a orthonormal basis of eigenfunctions.We now impose the additional requirement that all edges of G have equallength. In this situation the operator L will be denoted ∆ ∞ . Initially theedge lengths are taken to be 1, but the rescaling ξ = x/L converts the system D x Y = λY to D ξ Y = µY, ≤ ξ ≤ , µ = L λ. (2.3)Values of eigenfunctions at graph vertices are not effected, and derivativesare scaled by L . Thus eigenfunctions with eigenvalue λ for the standardLaplacian on the graph with edge lengths L are taken to eigenfunctions witheigenvalue L λ for the standard Laplacian on the graph with edge lengths 1.The spectrum of ∆ ∞ is tightly linked to the discrete graph G and thespectrum of the discrete Laplacian of combinatorial graph theory. Let E ( λ )denote the eigenspace of ∆ ∞ for the eigenvalue λ . The connections between E ( λ ) and the discrete graph G arise in two ways, depending on whether ornot λ ∈ { n π | n = 1 , , , . . . } . Before exploring this dichotomy, we notethe following result. Proposition 2.1. If ω > , dim E ( ω ) = dim E ([ ω + 2 nπ ] ) , n = 1 , , , . . . . Proof.
Suppose n ≥ Y is an eigenfunction with eigenvalue ω . On eachedge y ( x ) = A cos( ωx ) + B sin( ωx ) . The function y ( x ) = A cos([ ω + 2 nπ ] x ) + B sin([ ω + 2 nπ ] x )then satisfies the equation, − y ′′ = [ ω + 2 nπ ] y , and at the endpoints of the edge we have y (0) = y (0) , y (1) = y (1) . y ′ (0) = ωB, y ′ (1) = ω [ B cos( ω ) − A sin( ω )]and y ′ (0) = [ ω + 2 nπ ] B, y ′ (1) = [ ω + 2 nπ ][ B cos( ω ) − A sin( ω )] , so y ′ (0) = ω + 2 nπω y ′ (0) , y ′ (1) = ω + 2 nπω y ′ (1) . Thus y satisfies the same interior vertex conditions that y does. More-over, the linear map taking y → y is one-to-one, as is the analogous mapfrom E ([ ω + 2 nπ ] ) to E ( ω ). Given a vertex v ∈ G , let u , . . . , u deg ( v ) be the vertices adjacent to v . Adiscrete graph carries a number of linear operators acting on the vertex space H of functions f : V → C , including the adjacency operator Af ( v ) = deg ( v ) X i =1 f ( u i ) , and the degree operator T f ( v ) = deg ( v ) f ( v ) . Define the operator ∆ by∆ f ( v ) = f ( v ) − T − Af ( v ) . ∆ is similar to the much studied Laplacian [7, p. 3], I − T − / AT − / . The distinction of the cases λ ∈ { n π } is related to the following fact. Lemma 2.2.
Fix λ ∈ C , and consider the vector space of solutions of − y ′′ = λy on the interval [0 , . The linear function taking y ( x ) to ( y (0) , y (1)) is anisomorphism if and only if λ / ∈ { n π | n = 1 , , , . . . } . roof. For λ / ∈ { n π } , the formula y ( x, λ ) = y (0) cos( ωx ) + [ y (1) − y (0) cos( ω )] sin( ωx )sin( ω ) (2.4)shows that the map is surjective. On the other hand, if λ ∈ { n π } , then y (0) = 0 implies y (1) = 0, since y ( x ) = B sin( nπx ).As an immediate consequence we have the following result for graphs. Lemma 2.3.
Suppose the edges of G have length , and λ / ∈ { n π | n =1 , , , . . . } . Let y : G → C be continuous, and satisfy − y ′′ = λy on the edges.If y ( v ) = 0 at all vertices of G , then y ( x ) = 0 for all x ∈ G . Theorem 2.4.
Suppose λ / ∈ { n π | n = 1 , , , . . . } and y is an eigenfunc-tion for ∆ ∞ . If v has adjacent vertices u , . . . , u deg( v ) , then cos( ω ) y ( v ) = 1deg( v ) deg( v ) X i =1 y ( u i ) . (2.5) Proof.
In local coordinates identifying each u i with 0, (2.4) for y i on theedges e i = ( u i , v ) gives y ′ i ( v ) = − ω sin( ω ) y i ( u i ) + [ y i ( v ) − y i ( u i ) cos( ω )] ω cos( ω )sin( ω ) . (2.6)Summing over i , the derivative condition at v then gives0 = X i y ′ i ( v ) = − ω sin( ω ) X i y i ( u i ) + ω cos( ω )sin( ω ) X i [ y i ( v ) − y i ( u i ) cos( ω )] . Using the continuity of y at v and elementary manipulations gives (2.5).(2.5) is clearly an eigenvalue equation, with eigenvalue cos( ω ), for the linearoperator T − A acting on the space of (real or complex valued) functions onthe vertex set. With this background established, we are ready to relate thespectra of ∆ and ∆ ∞ . Theorem 2.5. If λ / ∈ { n π | n = 0 , , , . . . } , then λ is an eigenvalue of ∆ ∞ if and only if − cos( ω ) = 1 − cos( √ λ ) is an eigenvalue of ∆ , with thesame geometric multiplicity. roof. Since ∆ = I − T − A , we may work with T − A . Suppose first that y ( x, λ ) is an eigenfunction of ∆ ∞ satisfying the given vertex conditions. The-orem 2.4 shows that the (linear) evaluation map taking y : G → C to y : V → C takes eigenfunctions to solutions of (2.5). By Lemma 2.3 thekernel of this map is the zero function, so the map is injective.Suppose conversely that y : V → C satisfies T − Ay ( v ) = µy ( v ) , | µ | < . Pick λ ∈ cos − ( µ ). By Lemma 2.2 the function y : V → C extends to aunique continuous function y ( x, λ ) : G → C satisfying − y ′′ = λy on eachedge.In local coordinates identifying v with 0 for each edge e i = ( v, u i ) incidenton v , this extended function satisfies (2.6). Summing gives X i y ′ i ( v ) = ω sin( ω ) [ − sin ( ω ) − cos ( ω )] X i y i ( u i ) + X i y i ( v ) ω cos( ω )sin( ω )= − ω sin( ω ) X i y i ( u i ) + deg( v ) y ( v ) ω cos( ω )sin( ω ) . The vertex values satisfy (2.5), so X i y ′ i ( v ) = − ω sin( ω ) deg( v ) cos( ω ) y ( v ) + deg( v ) y ( v ) ω cos( ω )sin( ω ) = 0 . Thus the extended functions are eigenfunctions of ∆ ∞ satisfying the standardvertex conditions. Since the extension map is linear, and the kernel is thezero function, this map is also injective. n π Now we turn to eigenvalues λ ∈ { n π } . First recall [7, p. 7] that for both ∆ and ∆ ∞ G .For n ≥ ∞ also have a combinatorial interpreta-tion, closely related to the cycles in G . If C is a cycle, and x is distance alongthe cycle starting at some selected vertex, then the function sin(2 nπx ) is aneigenfunction of ∆ ∞ . Similarly, if C is an even cycle, then sin( nπx ) gives asimilar eigenfunction. We can make the following observation.8 emma 2.6. Suppose G is connected, and ψ is an eigenfunction of ∆ ∞ witheigenvalue λ = n π for n ≥ . If ψ vanishes at any vertex, then ψ vanishesat all vertices.Proof. Suppose ψ ( v ) = 0 for some vertex v . On any edge incident on v , theeigenfunction is a linear combination ψ ( x ) = A cos( nπx ) + B sin( nπx ) , v ≃ , and clearly A = 0. Since all edge lengths are 1, at all adjacent vertices w ,we then have ψ ( w ) = B sin( nπ ) = 0. By continuity of ψ and connectivity ofthe graph, ψ vanishes at all vertices.The next result explores the existence of eigenfunctions vanishing at novertices. Lemma 2.7. If λ = (2 nπ ) , for n = 1 , , , . . . , then ∆ ∞ has an eigenfunc-tion vanishing at no vertices.If λ = (2 n − π , n = 1 , , , . . . , then ∆ ∞ has an eigenfunction van-ishing at no vertices if and only if G is bipartite.Proof. If λ = (2 nπ ) , then the desired eigenfunction is simply cos(2 nπx ) inlocal coordinates on each edge.Suppose G is bipartite, with the two classes of vertices labelled 0 and 1.Pick local coordinates on each edge consistent with the vertex class labels,and define the eigenfunction to be cos([2 n − πx ).Suppose conversely that for some λ = (2 n − π , there is an eigenfunc-tion ψ vanishing at no vertex. Label the vertices v according to the sign of ψ ( v ). In local coordinates for an edge, ψ ( x ) = a cos([2 n − πx ) + b sin([2 n − πx ) , a = 0 . Then if w is a vertex adjacent to v we see that ψ ( w ) = − ψ ( v ), showing thatvertices are only adjacent to vertices of opposite sign, and G is bipartite.In addition to the vertex space mentioned above, an edge space maybe constructed using the edges of a graph as a basis (we assume the fieldis R or C ). The edge space has the cycle subspace Z ( G ) generated bycycles, with dimension N E − N V + 1 [5, pp. 51–58] or [10, pp. 23–28]. Let E ( n π ) ⊂ E ( n π ) be those eigenfunctions of ∆ ∞ vanishing at the vertices.9 heorem 2.8. dim Z ( G ) = dim E (4 n π ) . Proof.
It suffices to prove the result for a connected graph. Suppose Z ( G )has dimension M . Picking a spanning tree T for G , there is [5, p. 53]a basis of cycles C , . . . , C M such that each C j contains an edge e j / ∈ T ,with e j not contained in any other C i . Fix n ∈ { , , , . . . } and constructeigenfunctions f j = sin(2 πnx ) on the edges of C j , and 0 on all other edges.Here x denotes distance along the cycle starting at some selected vertex. Ifa linear combination P a i f i is 0, then for x ∈ e j M X i =1 a i f i ( x ) = a j f j ( x ) , j = 1 , . . . , M. There are x ∈ e j where f j ( x ) = 0, so a j = 0 and the functions f i areindependent. This shows dim E (4 n π ) ≥ dim Z ( G ).Now suppose that ψ ∈ E (4 n π ). After subtracting a linear combination P a i f i we may assume that ψ vanishes on all edges not in the spanning tree T . For every boundary vertex v of T , ψ vanishes identically on all but oneedge of G incident on v , and by the vertex conditions it then vanishes on alledges incident on v . Continuing away from the boundary of the spanningtree, we see that ψ is the 0 function.Figure 2.1: Bowtie graphThe proof of Theorem 2.8 provides a combinatorial basis construction for E (4 n π ). A similar construction using even cycles will provide independentelements of E ((2 n − π ), but these may not form a complete set. Considerthe bowtie graph in Figure 2.1, whose cycles have length 3. An eigenfunction ψ with eigenvalue π can be constructed by letting ψ ( x ) = 2 sin( πx ) on the10iddle edge. The function ψ then continues as − sin( πx ) on the adjacentfour edges, and sin( πx ) on the remaining two edges. Notice that on eachedge the function ψ is an integer multiple of sin( πx ).A generalization of the even cycles will be used for a combinatorial con-struction of E ((2 n − π ). Let Z denote the set of functions f : E → C such that X e ≃ v f ( e ) = 0 , v ∈ V . Theorem 2.9.
The linear map taking f ∈ Z to g ( x ) defined by g e ( x ) = f ( e ) sin((2 n − πx ) , is an isomorphism from Z onto E ((2 n − π ) . The subspace Z has anintegral basis.Proof. Since sin((2 n − πx ) = sin((2 n − π (1 − x )), the edge orientation doesnot affect the definition, so g is well defined. Clearly g ( x ) satisfies the eigen-value equation and vanishes at each vertex. The condition P e ≃ v f ( e ) = 0 forall v ∈ V gives the derivative condition, so g ∈ E ((2 n − π ). Moreoverthe map is one to one.Suppose ψ ( x ) ∈ E ((2 n − π ). Then ψ e ( x ) = a e sin((2 n − πx ) oneach edge, and because ψ satisfies the derivative conditions we have X e ≃ v a e = 0 , v ∈ V . Define f ( e ) = a e to get a linear map from E ((2 n − π ) to Z , which isalso one to one. This establishes the isomorphism.To see that Z has an integral basis, let e n be a numbering of the edgesof G , and let x n = f ( e n ). The set of functions Z is then given by the set of x , . . . , x N E satisfying the N V equations X e n ≃ v x n = 0 . This is a system of linear homogeneous equations whose coefficient matrixconsists of ones and zeros. Reduction by Gaussian elimination shows thatthe set of solutions has a rational basis, and so an integral basis.11
Graph refinements
Now we introduce the notion of graph refinement for graphs whose edgelengths are 1. Let the original combinatorial graph be denoted G , with theoperator ∆ acting on the vertex space H . For each integer N > G N with vertex space H N and operator ∆ N : H N → H N bysubdividing each edge e ∈ G into N edges. Pick local coordinates identifying e with [0 , x and 1 = x N . For n = 1 , . . . , N − x n = n/N in the local coordinates. The new graph G N with I = N V + ( N − N E vertices will have the vertex set consisting of the verticesof G , together with the new vertices x n , for each edge e of G . The vertex x n is adjacent to x n − and x n +1 for n = 1 , . . . , N −
1. If v is a vertex in G ,and the local coordinates for edges incident on v are chosen so v is identifiedwith 0 on each edge, then v is adjacent in G N with the vertices x for eachof the incident edges. An example is illustrated in Figure 3.1. G G G Figure 3.1: Refinement of a graphAn inner product on the vertex space H N is defined by h f, g i N = 1 W X v deg ( v ) f ( v ) g ( v ) , W = X v ∈G N deg ( v ) = 2 N N E . Using the above identifications of edges of G N with subintervals of [0 , /N . These continuous graphs may all be identified with the contin-uous graph G ∞ corresponding to G . In particular they will share the innerproduct h f, g i ∞ = 1 N E Z G ∞ f g, N E is the number of edges in G . As a continuous graph, the standardvertex conditions hold at all vertices of G N . Although appearing different indefinition, the continuous graph Laplacian ∆ ∞ has not changed as we passfrom G to G N .Let ∆ N denote the following operator on the vertex space of G N ,∆ N = N ( I − T − A ) . The eigenspaces of these operators with eigenvalue λ will be denoted E N ( λ ),while E ∞ ( λ ) will denote an eigenspace for ∆ ∞ . By (2.3) and Theorem 2.5,the mapping N (1 − cos( √ λ/N )) carries eigenvalues of ∆ ∞ to eigenvaluesof ∆ N if λ/N / ∈ { n π } . That is, the eigenvalues of the normalized adja-cency operator are cos( √ λ/N ) whenever λ / ∈ { ( N nπ ) } is an eigenvalue for∆ ∞ . Notice the eigenvalues of ∆ N are approximately λ/ λ is smallcompared to N . Lemma 3.1.
For ≤ N < ∞ the operator ∆ N is self adjoint on the vertexspace with the inner product h f, g i N .Proof. It suffices to check the normalized adjacency operator T − A .Let E p ( n π ) denote the subspace spanned by eigenfunctions of ∆ ∞ hav-ing the form cos( nπx ) on each edge (so not vanishing at the vertices). Let S N ⊂ L ( G ∞ ) denote the subspace S N = span { E p ( N π ) , E ∞ ( λ ) , ≤ λ < N π } . Here is preliminary result describing the restriction of eigenfunctions of ∆ ∞ to the vertices of G N . Proposition 3.2.
The restriction map R N : S N → H N is an bijection. For ≤ λ < N π this map takes distinct orthogonal eigenspaces E ∞ ( λ ) of ∆ ∞ onto distinct orthogonal eigenspaces E N ( N (1 − cos( √ λ/N ))) of ∆ N , and R N takes E p ( N π ) onto E N (2 N ) .Proof. By Theorem 2.5 and (2.3), R N is a bijection from the eigenspace E ∞ ( λ ) of ∆ ∞ to the eigenspace E N ( N (1 − cos( √ λ/N ))) of ∆ N . The 0eigenspaces for ∆ ∞ and ∆ N are just the functions which are constant on theconnected components of the respective graphs, so R N is a bijection from E ∞ (0) to E N (0). From the proof of Lemma 2.7 we also see that R N is abijection from E p ( N π ) to E N (2 N ).13ince ∆ ∞ and ∆ N are self adjoint on their respective function spaces,and since cos( t ) is strictly decreasing on (0 , π ), distinct eigenspaces of ∆ ∞ ,which are orthogonal, are mapped to distinct eigenspaces of ∆ N , which areorthogonal and span H N .The restriction map R N : S N → H N also has noteworthy features on theindividual eigenspaces E ∞ ( λ ) of ∆ ∞ . Before stating the results, we start witha simple observation relating the H N inner product and the sums appearing inthe trapezoidal rule for integration. Recalling that W = P v deg ( v ) = 2 N N E ,the inner product for G N satisfies the identity W h f, g i N = X v deg ( v ) f ( v ) g ( v )= X e ∈G [ f e ( x ) g e ( x ) + f e ( x N ) g e ( x N ) + 2 N − X n =1 f e ( x n ) g e ( x n )] . These last sums are just the trapezoidal rule sums used for integrals overthe edges of G ∞ . With this motivation, if the continuous linear functional T N : C [0 , → C is defined by T N ( f ) = 12 N [ f ( x ) + f ( x N ) + 2 N − X n =1 f ( x n )] , then h R N f, R N g i N = 1 N E X e ∈E T N ( f e g e ) . (3.1)The following identities are also useful. First Z e iωx = exp(2 iω ) − iω = e iω sin( ω ) ω . Then, for 0 < ω < N , a geometric series computation gives2
N T N (exp( iωx )) = exp( iω ) − N − X n =0 exp( iωn/N )= exp( iω ) − − exp( iω )1 − exp( iω/N ) = (1 − exp( iω )) 1 + exp( iω/N )1 − exp( iω/N ) , T N ( e iωx ) = M ( ω N ) Z e iωx , M ( z ) = M ( − z ) = z cot( z ) . (3.2)These identities will be useful for comparing the inner products on L ( G ∞ )and H N . The cases λ = k π with 0 ≤ k < N are considered first. Theorem 3.3.
Suppose f, g ∈ E ∞ ( λ ) , with ≤ λ ≤ N π . If λ = k π foran integer k with ≤ k < N , then h f, g i ∞ = h R N f, R N g i N . (3.3) If f, g ∈ E p ( N π ) , then h f, g i ∞ = 12 h R N f, R N g i N . (3.4) Proof.
Suppose f, g ∈ E ∞ ( λ ), with 0 ≤ λ < N π . On each edge e we have f e = α e e iωx + β e e − iωx , g e = γ e e iωx + δ e e − iωx , ω = λ, (3.5) f e g e = α e γ e + β e δ e + α e δ e e iωx + β e γ e e − iωx . If λ = k π for an integer k with 0 ≤ k < N , then T N ( e πikx ) = Z e πikx = 0 . Using (3.1) and T N (1) = 1, it follows that h f, g i ∞ = 1 N E Z G ∞ f g = 1 N E X e [ α e γ e + β e δ e ]= 1 N E X e T N ( α e γ e + β e δ e ) = h R N f, R N g i N , which is (3.3). Similar calculations handle the case λ = 0.Suppose f, g ∈ E p ( N π ), a one dimensional space. For f e = g e =cos( N πx ), we have 1 N E Z G ∞ cos ( N πx ) = 1 / , h R N f, R N g i N = 1 N E X e ∈E T N ( f e g e ) = 1 N E X e ∈E , giving (3.4).For general ω a variation of a classical trapezoidal rule estimate appears. Theorem 3.4.
Suppose f, g ∈ E ∞ ( λ ) , with < λ < N π . Then for ω ≥ ω > , (cid:12)(cid:12)(cid:12) h R N f, R N g i N − h f, g i (cid:12)(cid:12)(cid:12) = O ( | ω | /N ) k f k k g k . (3.6) Proof.
The argument starts with a variation of standard error estimates [1, p.285], [p. 358-369][6] for the trapezoidal rule for integration. If φ = exp( iωx )then φ ′ (1) − φ ′ (0)12 N = 112 N Z φ ′′ = − ω N Z e iωx . Putting this together with (3.2) yields Z e iωx − T N ( e iωx ) = − φ ′ (1) − φ ′ (0)12 N + M ( ω N ) Z e iωx , with M ( z ) = 1 − z cot( z ) − z . A Taylor expansion gives M ( z ) = O ( z ) , and the function M ( z ) = M ( − z ) is analytic for | z | < π .On each edge we have the representation (3.5). Since f and g are in thedomain of ∆ ∞ , the function ψ = f g satisfies the vertex conditions (2.1) and(2.2). Thus X e [ ψ ′ e (1) − ψ ′ e (0)] = − X v ∈V X e ∼ v ∂ ν ψ e ( v ) = 0 , and N E [ h f, g i − h R N f, R N g i N ] = X e hZ ψ e − T N ( ψ e ) i = − X e h ψ ′ e (1) − ψ ′ e (0)12 N i + M ( ωN ) hZ e iωx X e α e δ e + Z e − iωx X e β e γ e i M ( ωN ) hZ e iωx X e α e δ e + Z e − iωx X e β e γ e i Using 2 | α e || β e | ≤ | α e | + | β e | , we find Z | f e | = | α e | + | β e | + αβe iω sin( ω ) ω + βαe − iω sin( ω ) ω ≥ (1 − sin( ω ) ω )( | α e | + | β e | ) . This gives the desired result N E (cid:12)(cid:12)(cid:12) h R N f, R N g i N − h f, g i (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) M ( ωN )(1 − sin( ω ) ω ) − X e ( Z | f e | ) / Z | g e | ) / (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) M ( ωN )(1 − sin( ω ) ω ) − Z G ∞ | f | ) / ( Z G ∞ | g | ) / (cid:12)(cid:12)(cid:12) . In this section we turn to an algorithm for the efficient computation of theFourier transform and its inverse transform on the spaces H N . ... π π π π π √ λ Figure 4.1: Square root of the spectrum of ∆ ∞ .Recall some of the structures associated with the spectrum of ∆ ∞ . Thesquare roots √ λ k of the positive eigenvalues of ∆ ∞ and the eigenspaces E ( λ k )exhibit the ’2 π -periodicity’ discussed in Proposition 2.1. and illustrated inFigure 4.1. Let 0 < ω , < ω , < · · · ≤ π denote the square roots ofthe K distinct eigenvalues λ k of ∆ ∞ with 0 < λ k ≤ (2 π ) . For each ω ,k ω ,k ) for E ( ω ,k ), with basis functions Ψ j ( ω ,k ), j = 1 , . . . , dim E ( ω ,k ). ω ,k ω ,k ω ,k ω ,k π • < − − − > • < − − − > • < − − − > • < −− > • < − − − > • < − − − > • < − − − > • < −− >π • < − − − > • < − − − > • < − − − > • < −− > • < − − − > • < − − − > • < − − − > • < −− > • < − − − > • < − − − > • < − − − > • < −− > • m Figure 4.2: Eigenfunction maps for ∆ ∞ .For m = 1 , , , . . . , let ω m,k = ω ,k + 2 mπ . From each basis Ψ( ω ,k ) wemay produce bases Ψ( ω m,k ) for E ( ω m,k ) by the method of Proposition 2.1.Recall that on each edge e a function Ψ j ∈ Ψ( ω ,k ) has the formΨ j = A e exp( iω ,k x ) + B e exp( − iω ,k x ) , ≤ x ≤ . The set of functions Ψ j ( ω m,k ) whose values on e areΨ j ( ω m,k ) = A e exp( i [ ω ,k + 2 mπ ] x ) + B e exp( − i [ ω ,k + 2 mπ ] x ) , (4.1)then give a basis Ψ( ω m,k ) for E ([ ω ,k + 2 mπ ] ). This basis need not beorthonormal. Figure 4.2 indicates the relations among the bases Ψ( ω m,k ). Proposition 4.1. If ω ,k ∈ { π, π } , then the functions Ψ j ( ω m,k ) are or-thonormal, as are the functions R N Ψ j ( ω m,k ) ∈ H N for ≤ ω m,k < N π .Proof. Suppose Ψ i ( ω m.k ) ∈ Ψ( ω m,k ) is represented on the edge e byΨ i ( ω m,k ) = C e exp( i [ ω ,k + 2 mπ ] x ) + D e exp( − i [ ω ,k + 2 mπ ] x ) , ≤ x ≤ , and (4.1) describes Ψ j . With ω ,k = π, π, Z exp( ± i [ ω ,k + 2 mπ ] x ) = 0 , so X e Z Ψ j ( ω m,k )Ψ i ( ω m,k ) = X e [ A e C e + B e D e ] = X e Z Ψ j ( ω ,k )Ψ i ( ω ,k ) . The extension to H N follows from Theorem 3.3.18et I = dim E (0), I k = dim E ( ω ,k ), and letΦ j ( ω m,k ) = R N Ψ j ( ω m,k )be the restriction of Ψ j ( ω m,k ) to the vertices of G N . The bases Φ( ω k,m ) willbe used for efficient Fourier transform algorithms [1, p. 182] [6, p. 383].Define the discrete Fourier transforms (DFT) F N : H N → C I ⊕ m,k C I k forthe continuous graph G ∞ by taking the inner product of a function f ∈ H N with the bases Φ( ω m,k ), F N ( f ) = {h f, Φ j ( ω m,k ) i N } , ω m,k ≤ N π. (4.2)The condition ω m,k ≤ N π amounts to m = 0 , . . . , N/ − N is even,which will hold in the cases of interest below. Except for the cases notedin Proposition 4.1 the functions of Φ( ω m,k ) may not be orthonormal, so theFourier transform will not be an isometry from Φ( ω m,k ) to C I k if the usualinner product is used on the range. We consider a modified inner product.Suppose B ( ω m,k ) = ( b ij ) is a matrix taking the basis Φ( ω m,k ) to an or-thonormal basis η i = X j b ij Φ j ( ω m,k ) (4.3)for the same eigenspace in H N . (Such a matrix may be obtained by the Gram-Schmidt process.) For f in the span of Φ( ω m,k ), the map f → {h f, η i i N } isan isometry to C I k with the usual inner product X • Y = P x j y j . Expressingthis in the original basis, k f k = X i |h f, η i i N | = X i |h f, X j b i,j Φ j i N | = X i | X j b i,j h f, Φ j i N | = BX • BX, X = ( h f, Φ i N , . . . , h f, Φ I k i N ) T . For f = P j c j Φ j ( ω m,k ) in the span of Φ( ω m,k ), the coefficients c j may berecovered from the DFT values h f, Φ j i N . Starting from f = X i a i η i , a i = h f, η i i N , and using (4.3) we find f = X i ( X j b ij h f, Φ j i N )( X l b il Φ l ) = X l X j ( X i b ij b il h f, Φ j i N )Φ l X l ( X j ( B ∗ B ) jl h f, Φ j i N )Φ l , or c l = X j ( B ∗ B ) jl h f, Φ j i N . (4.4)Calling BX • BX the B ( ω m,k ) inner product on C I k , and noting Φ( ω m,k )is a basis for the µ m,k eigenspace of ∆ N , we obtain the first part of the nextresult. Theorem 4.2.
The Fourier transform F N : H N → C I ⊕ m,k C I k satisfies F N (∆ N f ) = { µ m,k F ( f ) m,k } , and is an isometry if C I k m has the B ( ω m,k ) inner product.If N is a power of , then F N ( f ) can be computed in time O ( N log ( N )) .Proof. The inner products of (4.2) may be split into trapezoidal sums for theedges as in (3.1), F N ( f ) = 1 N E X e T N ( f e Φ j ( ω m,k )) . Using the representation (4.1) we have T N ( f e [ A e exp( i [ ω ,k + 2 mπ ] x ) + B e exp( − i [ ω ,k + 2 mπ ] x )])= A e T N (exp( − iω ,k x ) f e exp( − mπix ) + B e T N (exp( iω ,k x ) f e exp(2 mπix )) . If N is a power of 2, e and k are fixed, and m = 0 , . . . , N −
1, thenthe sequence T N (exp( ± iω ,k x ) f e ( x ) exp( ± mπix )) differs trivially from theconventional discrete Fourier transform of the sampled data f e ( x n ), whichmay be computed [1, p. 182] [6, p. 383] in time O ( N log ( N )). Since thenumber of edges e and indices k is fixed, we obtain the result.It remains to consider the inverse transform. Since the matrix B ∗ B of(4.4) depends only on the graph and functions Φ j ( ω m,k ), these matrices areassumed to be precomputed. Suppose the DFT sequence (4.2) of a function f is given. Since the computations in (4.4) are limited to sequence blocks20 I k , the coefficients c j ( ω m,k ) for all ω m,k may be computed in time O ( N ). Itremains to compute f ( x n ) = X m,k c j ( ω m,k )Φ j ( ω m,k )( x n ) . Fixing the edge e , the primitive frequency index k , and the basis index j ,and taking advantage of (4.1), the remaining sum has the form f e,k,j ( x n ) = A e exp( iω ,k n/N ) X m c j ( ω m,k ) exp( i mπn/N )+ B e exp( − iω ,k n/N ) X m c j ( ω m,k ) exp( − i mπn/N ) . These sums may be computed with a conventional FFT. Since the numberof edges e and primitive frequency indices k are constant, and the range ofbasis indices j is bounded independent of m , the next result is obtained. Theorem 4.3. If N is a power of , then the inverse DFT can be computedin time O ( N log ( N )) . A relatively simple family of examples is provided by the complete bipartitegraphs K ( m,
2) on m, m = 4 case is portrayed in Figure5.1. Discussion of these graphs is facilitated if the vertices are colored, the 2vertices of degree m being red and the remaining m vertices blue.Suppose v is a vertex, and { u i } are the vertices adjacent to v . Then theoperator ∆ on the m + 2 dimensional vertex space is given by∆ f ( v ) = f ( v ) − deg ( v ) deg ( v ) X i =1 f ( u i ) . Eigenvalues µ for ∆ are 0 , ,
2, with 1 having multiplicity m . The eigen-functions ψ for µ = 0 , µ = 0 , ψ ( v ) = 1 ,µ m +1 = 2 , ψ m +1 ( v ) = n , v red − , v blue o K , Next, consider the eigenfunctions for µ = 1. A function f ( v ) will satisfy∆ f ( v ) = f ( v ) if and only if P deg ( v ) i =1 f ( u i ) = 0 for every vertex v . Numberthe red vertices r and r , and the two blue vertices b , . . . , b m . For j =1 , . . . , m −
1, independent eigenfunctions are ψ j ( v ) = n , v = b j − , v = b j +1 , , otherwise o . There is one additional independent eigenfunction ψ m ( v ) = n , v = r , − , v = r , , otherwise o . To establish the independence of ψ , . . . , ψ m , note that if m X j =1 c j ψ j ( v ) = 0 , then c m = 0 by evaluation at r , and evaluation at the points b j leads toequations c = 0 , c = c , . . . , c m − = c m . Moving to the continuous graph, identify each edge with [0 ,
1] so the bluevertex is identified with 0. Consider the eigenvalues λ and correspondingeigenfunctions Ψ for ∆ ∞ , focusing on √ λ ∈ [0 , π ]. Of course we have22 = 0 with the constant eigenfunction Ψ(0) = 1. From µ j = 1 we obtaineigenvalues √ λ = cos − (1 − µ j ) = cos − (0) = π/ , π/ . That is, λ = ( π/ and λ = (3 π/ are eigenvalues, each with multiplicity m . The corresponding eigenfunctions Ψ j ( π/
2) and Ψ j (3 π/
2) are obtained byextrapolation as in Lemma 2.2 from the vertex values of ψ j .This graph [5, p. 53] has m − C i is given by the vertex sequence r , b i , r , b i +1 , r .Each C i supports eigenfunctions Ψ i ( π ) and Ψ i (2 π ) which are respectively ± sin( πx ) and ± sin(2 πx ) on the edges of the cycle.Finally, there are two additional independent eigenfunctions, Ψ m ( π ) hav-ing values cos( πx ) and Ψ m (2 π ) with values cos(2 πx ) on the edges of G . Thisgives a total of 2 N E = 4 m = 2 m + 2( m −
1) + 2independent eigenfunctions for ∆ ∞ with eigenvalues 0 < λ ≤ π .This graph has an obvious involution in which r and r are interchanged,and we may split the eigenspaces into even and odd subspaces with respect tothe involution. Notice that the eigenfunctions Ψ m ( π ), Ψ m (2 π ), Ψ i ( π/
2) andΨ i (3 π/
2) for i = 1 , . . . , m − m ( π/ m (3 π/ i ( π ) andΨ i (2 π ) for i = 1 , . . . , m − m edges andboundary vertices with the conditions f ′ e (0) = 0, respectively f e (0) = 0.23 eferences [1] K. Atkinson. An Introduction to Numerical Analysis.
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