Sampling with Bessel Functions

Transcription

1 amping with Besse Functions K.I. Kou, T. Qian and F. ommen Abstract. The paper deas with samping of σ-bandimited functions in R m with Cifford-vaued, where bandimitedness means that the spectrum is contained in the ba B, σ that is centered at the origin and has radius σ. By comparing with the genera setting, what is new in the samping is using the expicit Bochner-type reations invoving spherica harmonics and monogenics in the Cifford agebra setting. Convergence of the samping formuas in the L sense and in the uniform and absoute pointwise sense are studied. Mathematics ubect Cassification. Primary 4A38; econdary 3G35. Keywords. samping, spherica harmonics, spherica monogenics, Besse functions. 1. Introduction The Besse function of arbitrary compex order ν is defined by [16] z/ ν+ J ν z =! Γ + ν + 1, 1.1 = where 1/Γ + ν + 1 are assigned to be of zero vaue for non-positive integers + ν + 1 due to anaytic continuation of the function 1/Γz. It is shown that for a ν the Besse function defined through 1.1 satisfies the corresponding Besse s differentia equation [16] 3.11 z d y dz + z dy dz + z ν y =. For compex numbers ν with Re ν > 1, there hods the integra formua [16] z/ ν π J ν z = cosz cos θ sin ν θdθ. π Γν + 1/ The study was supported by Research Grant of the University of Macau No. RG59/5-6/QT/FT.

2 K.I. Kou, T. Qian and F. ommen Besse s equation arises when finding separabe soutions to Lapace s equation and the Hemhotz equation in cyindrica or spherica coordinates, and Besse functions are therefore especiay important for many probems of wave propagation, static potentias, and so on. For cyindrica probems, one obtains Besse functions of integer order ν = n; for spherica probems, one obtains haf integer orders ν = n + 1, incuding eectromagnetic waves in a cyindrica waveguide; heat conduction in a cyindrica obect and modes of vibration of a thin circuar or annuar membrane. Besse functions aso have usefu properties for other probems, such as signa processing e.g., Frequency moduation synthesis and Kaiser window, see [16] and []. The function J ν z has a countaby infinite number of positive rea zeros, and a finite number of conugate compex zeros. Moreover, a the zeros are simpe, except possiby the zero at the point z = [14] 5.13 Theorem. Denote these rea zeros in the ascending order < u ν 1 < u ν < < u ν caed the th positive zero of J ν. <, where u ν To treat a bandimited function see Definition 1.1 we wi encounter a positive number σ as, or arger than the spectrum radius. Throughout the foowing discussion we wi dea with an arbitrary but fixed positive parameter σ. To simpify the notation, we et is o λ ν λ ν = uν, = 1,, 3, σ is the vaue of the th positive zero of J ν scaed by a fixed factor 1/σ. Accordingy, we have the orthogona property J ν sλ ν J ν sλ ν sds = σ δ, J ν+1u ν, 1.3 where Re ν > 1, δ, is the Kronecer deta [14] , If F is square-integrabe with respect to the weight sds on, σ, then F has a Besse series expansion of order ν on the interva, σ given by F s = A J ν =1 sλ ν, 1.4 where A can be determined by the orthogonaity property of J ν in 1.3 A = σ J ν+1 uν F sj ν sλ ν sds. 1.5

3 amping with Besse Functions 3 The number A is caed the th Besse coefficient of the function F [14] , Parseva s identity for Besse series 1.4 reads F s sds = F σ = J ν+1u ν A. 1.6 Fourier transform of functions in L 1 R m is defined by fξ = e i<x,ξ> fxdx. R m The inverse Fourier transform is formay defined by ǧx = 1 π m e i<x,ξ> gξdξ. R m We now define a cass of functions in terms of their Fourier transforms. =1 Definition 1.1. If f L R m and supp f B, σ the ba in R m centered at with radius σ, then f is caed a bandimited function with bandwidth σ. The paper is organized as foows. In the second section we review the genera samping theory. In section 3 by using the decomposition of square integrabe functions into spherica harmonics, we formuate a corresponding samping theorem. In section 4 we estabish the samping resut in reation to homogeneous eft-monogenic functions. The symbos Z and N denote the sets of the integers and the natura numbers, respectivey.. The Genera Theory of amping Theorems We wi be woring with the weighted L space L I, ρ I R and ρs a.e. in I. The inner product is given by < F, G >= F sgsρsds. The norm of F with respect to the weighted Lebesgue measure is 1/ F =< F, F > 1/ = F s ρsds. In the rest of the note we wi fix the weight function ρs = s. I Our question of samping and reconstruction can be stated as foows: For a set R of functions on a domain Ω, can we identify a discrete set {r } Ω such I

4 4 K.I. Kou, T. Qian and F. ommen that every f R is uniquey determined by its vaues on {r }? If such a discrete set {r } exists, we, specificay, expect to have a samping formua of the form fr = fr r.1 vaid for a functions f in R where are some nown functions, and the convergence is in a certain norm or point-wise sense. amping in Besse functions fits into the foowing frame wor but with some variations. Assume that we have an integra erne K, r L I, ρ, and there exists a sequence {r } =1 Ω such that Ks, r = a φ s, where a > are normaizing constants so that {φ } =1 forms an orthonorma basis for L I, ρ. Now define on L I, ρ the inear integra transformation with the erne Ks, r mapping F L I, ρ to fr := F sks, rρsds.. I The integra transformation. is we defined because F and K, r both are assumed to be in L I, ρ. ince {φ } =1 is a basis the transformation is one to one. where Next we expand the function Ks, r into Ks, r = rφ s, s, r I Ω,.3 =1 r =< K, r, φ >..4 The inner product is one associated with the Hibert space L I, ρ. ubstituting.3 into. and interchanging the order of summation and integration due to the L -convergence, we obtain fr = r F sφ sρsds = r < F, φ >. I =1 Recaing the reation φ s = Ks,r a, and., we have < F, φ >= fr a fr = =1 =1 and fr a r..5

5 amping with Besse Functions 5 This hods in the point-wise convergence sense for a functions f in the range of the integra transformation., viz. { } R = f : Ω C such that fr = F sks, rρsds, F L I, ρ. To study uniform convergence, we first have fr f N r fr r a =N+1 fr where f N r = N =1 =N+1 I a 1/ fr a r. In the expansion F s = =1 A φ s, the coefficients A =< F, φ >= fr. a The Panchere identity gives F fr = <. =1 o, if N is arge enough, then the tai =N+1 a fr a =N+1 r 1/,.6 is sma. Based on this, if there exists M < and B Ω such that for arge enough N r < M, r B,.7 =N+1 then the point-wise convergence in.5 is strengthened to becomes uniform and absoute in B. Note that the condition impies.7. K, r = r M, r B, =1

6 6 K.I. Kou, T. Qian and F. ommen Now we estabish L convergence of.5. Based on the inective mapping. we may equip a Hibert space structure onto the set R by etting f H = F, where H denotes the formuated Hibert space. The convergence of.5 then may be seen to be aso in the norm H. In fact, due to the poarization identity, the transformation. under the defined norm induces an isometrica isomorphism from H to L I, ρ. Under the isomorphism orthonorma bases in L I, ρ are mapped to orthonorma bases in H. ince {φ i } i=1 is an orthonorma basis in L I, ρ, then the reation.4 exhibits that { } =1 is an orthonorma basis in H. This concudes that the convergence of.5 is in the norm H sense. For more comprehensive expanations of this genera samping theory, the reader may refer to [5], [6], [9], [1] and [13].The underying idea of the procedure is borrowed from Hardy [11] who first noticed that samping formua.1 is an orthogona expansion. 3. amping in reation to decomposition into spherica harmonics The wor wi be based on the direct sum decomposition L R m = N, 3.1 = where the dimension m, N is the set of finite inear combinations of scaar vaued functions of the form gry x, where r = x, g is a function defined on [, satisfying gr r m+ 1 dr < and Y a soid spherica harmonics of degree. Each space N is invariant under Fourier transformation [15] p.151. The space of soid spherica harmonics of degree, being isomorphic to the space of surface spherica harmonics of degree, denoted by Y, has finite dimension γ = m+ 1!!m 1!. When =, the space N is of dimension 1 generated by the constant function 1, and functions gry x are reduced to ust radia functions gr. The foowing theorem gives the samping expansion for bandimited functions in N. Theorem 3.1. Let f be a radia function fx := f r r = x in L R m and f be bandimited with bandwidth σ. Then for a x R m we have fx = f =1 λ c λ c c c;λ r cr 3.

7 amping with Besse Functions 7 where c c;λ r = λ c σj c+1 σλ c J cσr λ c r, r λ c ; 1, r = λ c, 3.3 and c = m, λc are the scaed positive zeros of the Besse function J c given by 1., and the series 3. being convergent in L -norm, and absoutey, uniformy, point-wisey convergent on the whoe space R m. Proof. The Fourier transform of a radia function is sti radia. With an abuse of notation but informativey, we write ft := f s s = t for a t R m. By the assumption f L R m, being bandimited with bandwidth σ, the inverse Fourier formua [15] Theorem 3.3 in Chapter 4 gives f r = π m [ f ss m ] [r m J m ] rs sds. 3.4 This corresponds to the integra transformation. with I = [, σ], ρs = s, c = m, F s = π m f ss c and the integra erne Ks, r = r c J c rs. Recaing 1.3, the set of functions J c sλ c Jc sλ c φ s = J c λ c = 3.5 σj c+1 u c forms an orthonorma basis for L I, ρ. For r = λ c with J c λ c σj c+1 a = λ c c = λ c we have Ks, r = a φ s u c c. Next expand Ks, r into.3 with φ s given in 3.5, as Besse series expansion of order c = m, we have r c J c rs = r c c c;λ rj c sλ c. 3.6 =1 Using specia function formua see [1], , for order c we have c c;λ r = λ c σj c+1 σλ c J cσr λ c r, r λ c ; 1, r = λ c. Hence the corresponding coefficients in.3 are r = J c λ c r c c;λ r = c σj c+1 r c u c c c;λ r. 3.7

8 8 K.I. Kou, T. Qian and F. ommen ubstituting 3.6 into 3.4, we have fx = π m r c f s c c;λ rj c = r c =1 =1 [ c c;λ r π m sλ c s c+1 ds ] f sj c sλ c s c+1 ds, 3.8 where interchanging the order of summation and integration is due to the L - convergence. Using 3.4, c π m f sj c sλ c s c+1 ds = λ c f λ c. 3.9 This, together with 3.8, gives 3.. Now we study uniform convergence. To this end, based on the genera theory, it is sufficient to estimate K, r = r c J c r = r c Jc rssds. using the modified expressions in [3] p.1, for Re ν > 1, we have γ β J ν σ s J ν σ s sds { σ γβ = νβ γ [J ν 1γJ ν+1 β J ν+1 γj ν 1 β], γ β; [ J ν γ J ν 1 γj ν+1 γ ], γ = β. σ For γ = β = rσ and ν = c = m, [ r c Jc rssds = σ J c rσ r c J ] c 1rσJ c+1 rσ r c. 3.1 The estimate of the bounds of 3.1 for the cases m > 3 is based on the foowing asymptotic resuts. For the Besse functions of order Re ν > there hods r ν J ν r ν Γ1 + ν ; 3.11 whie for arge arguments r >> 1, J ν r r πr cos 1 νπ 14 π, 3.1 [14] For m =, J m 4 1 m J m r m Z [16].1. For m = 3, J m 4 = J 1 r = J 1 r because of the property J m r = = J 1 r = πr 1/ cosr

9 amping with Besse Functions 9 [16] o the integra in 3.1 is bounded for m by a positive number M. This impies K, r < M for a r B = [,. Hence the series in 3. is uniformy convergent in the whoe space. Now we show the L -norm convergence of 3.. Based on the genera theory, by comparing 3.4 with., it is sufficient to prove that the norm H of f defined through the reation f H = F is a constant mutipe of the norm f L R m. This is to show that, for a constant c, we have f L R n = c F. But this is ust the Panchere theorem of a radia square-integra function in R m. This theorem shows that bandimited radia functions can be constructed competey from the specified samping. We now turn our attention to bandimited square integrabe functions not necessariy be radia. We begin by first examining the samping formuas for functions in functions casses N, 1. Theorem 3.. Let be a fixed non-negative integer and f be of the form fx := f ry x r = x, Y a soid harmonics of degree, in L R m, and f be bandimited with bandwidth σ. Then ft = f sy t s = t with and Moreover f s = i π m f r = i π m fx = f =1 where c = m+, c ;λ c of the Besse function J c λ c f rs c J c srr c +1 dr. f sr c J c rss c +1 ds λ c c r is given by 3.3, λ c c ;λ c ry x, 3.14 are the scaed positive zeros given by 1., and the series 3.14 being convergent in L -norm, absoutey and uniformy convergent point-wisey on whoe space R m. Proof. The Fourier transform of the function fx := f ry x r = x can be expressed as ft = f sy t s = t, where Y is a soid harmonics of degree [15] Theorem 3.1 and, f s = i π m ] [f rr m+ [s m+ J m+ ] sr rdr.

10 1 K.I. Kou, T. Qian and F. ommen With supp f [, σ] the radia function f can be obtained via the formua [ ] ] f r = i π m f ss m+ [r m+ rs sds J m+ Denote c = m+. The integra transformation 3.15 maps F s = i π m f ss c to f r, where the corresponding integra erne is Ks, r = r c J c rs. Aso based on 1.3, the set of functions Ks, λ c = a φ s gives rise to an orthonorma basis {φ } =1 of L I, ρ, where J c λ c u c a = λ c c = σj c +1 λ c c and J c sλ c Jc sλ c φ s = J c λ c =. σj c +1 u c The expansion of the function Ks, r into the series in φ s is where c ;λ c r c J c sr = =1 r c c ;λ c rj c sλ c is given by 3.3. The corresponding coefficient r = σj c +1 u c r c c ;λ c The proof of the point-wise convergence of 3.14 now foows from the genera theory in ection. The absoute, uniform and point-wise convergence in R m foows from the reation K, r = r c r. J c rssds, using formua 3.1 γ = β = rσ and ν = c, and J c rssds = 1, [ J c rσ J c 1rσJ c +1rσ ]. This is nothing but an increased dimension to m +. The same reasoning as in the proof of Theorem 3.1 based on the asymptotic formuas gives K, r < M for a r B = [,. Hence the series in 3.14 is absoutey and uniformy convergent in the whoe space.

11 amping with Besse Functions 11 To show the L -convergence of 3.14 we ony need to point out that the L R m norm of f is a constant mutipe of the Hibert space norm of f given by the reation f H = F. We finay remar that the mentioned constant depends ony on the space dimension m but not on. This fact wi be used in the proof of Theorem 3.3. Let {Y 1, Y,, Y γ } be an orthonorma basis of Y, the space of surface spherica harmonics of degree in R m. For any f N, f x = γ =1 f rr Y x, 3.16 where r = x, x = rx and f are functions defined on [,, = 1,,, γ, satisfying f r r m+ 1 dr <. If f and g Y, the inner product in L Σ m 1 is defined by f, g = fx gx dx, Σ m 1 where Σ m 1 is the m 1-dimensiona unit sphere in R m and dx is the m 1- dimensiona Lebesgue area measure on Σ m 1. The samping formua of f N is a summation of γ formuas of the ones obtained in Theorem 3.. Next we derive the samping expansions of square integrabe bandimited functions in R m. Theorem 3.3. Let f L R m be bandimited with bandwidth σ. Then γ fx = f x = = =1 = = =1 f where r = x, and f can be reconstructed competey via c λ c [ γ fx = r c ;λ c r f =1 ry x 3.17 λ c Y x ] 3.18 where c = m+, λ c are the scaed positive zeros of the Besse function J c given by 1., c ;λ c is given by 3.3, and the series 3.18 is convergent in the L -norm sense. In particuar, c N λ c f N x = r = =1 c ;λ c r [ γ =1 f λ c Y x ], 3.19

12 1 K.I. Kou, T. Qian and F. ommen converges to f in the L sense, and for every fixed N, the series 3.19 point-wisey, absoutey and uniformy convergent in R m. Proof. ince f L R m, it foows from 3.1 and the expansion 3.16 that f has the decomposition γ fx = f x = f ry x = γ f rr Y, 3. x = = =1 = =1 where x = rx, r = x and {Y x } γ =1 is an orthonorma basis of Y. The convergence is in the L sense. By taing Fourier transform on both sides of 3., we have γ ft = f t = f sy t = γ f ss Y t. 3.1 = = =1 = =1 ince f is bandimited with bandwidth σ, for s > σ, 3.1 becomes γ = f ss Y t. 3. = =1 Mutipying the both sides by Y t N and = 1,, γ, integrating over the unit sphere Σ m 1 term-by-term, and using the orthogonaity of the spherica harmonics Y t Y t dt = if or Σ m 1 and we obtain = γ f = =1 ss Σ m 1 Y t dt = 1, Therefore, f [, σ], N and = 1,,..., γ and f bandimited. Using 3.14 in Theorem 3., we get f ry x = f λ c λ c c r Σ m 1 Y t Y t dt = f ss. 3.3 f s = for N, = 1,..., γ and s > σ. This shows that supp =1 Adding up those terms in,, we obtain ry x L R m are a c ;λ c ry x. 3.4

13 amping with Besse Functions 13 The L convergence of 3.18 is based on the L convergence of 3.4 and that of The point-wise convergence property of 3.19 foows from the corresponding property of the series discussed in Theorem 3.. The theorem asserts that by ignoring an error in the L sense, the principa part 3.19 of the samping formua 3.18 may be uniformy and absoutey in the point-wise sense approximated by the corresponding interpoation series. A discussion foowing the idea of.6 regarding any reminder of the series is possibe but compicated. 4. amping in Reation to Decomposition into Homogeneous Left-Monogenic Functions In this section we estabish the corresponding samping resuts in reation to homogeneous eft-monogenic functions that refines what are obtained for homogeneous harmonics in the ast section. We first introduce basic concepts in reation to eftmonogenic functions in Cifford anaysis [4] and [1]. We wi be woring with R m 1, the rea-inear span of e, e 1,, e m, where e is identica with 1 and e i e + e e i = δ i i, = 1,,..., m. The rea- m + 1-dimensiona inear space R m 1 is embedded into the rea-cifford agebra R m and the compex-cifford agebra C m generated by e 1,, e m over the rea and compex number fieds, respectivey. A typica eement in R m 1 is denoted by x = x + x, where x R and x = x 1 e x m e m R m, the atter being identica with the Eucidean space R m. We usuay write x = rx, where r = x. A typica eement in the compex-cifford agebra C m is x = x e, = or 1,, where 1 1 < m, 1 m, x C, e = e 1 e, e = e. Functions to be studied in this note are R m 1 -variabe and compex-cifford agebra-vaued. A genera function is of the form fx = f xe, and the component functions f are compex-vaued. Left- and right-monogenic functions are introduced via the generaized Cauchy-Riemann operator D = x e + x 1 e x m e m : A function f with continuous first order derivatives is said to be eft-monogenic, or right-monogenic, if m f Df = e i e =, x i= i or m f fd = e e i =, x i i= in its domain, respectivey. In this note we ony study eft-monogenic functions. The theory for right-monogenic functions is parae. Note that there exist Cauchy s

14 14 K.I. Kou, T. Qian and F. ommen Theorem and Cauchy s formua in this setting [4] or [7]. There exists aso a Tayor and Laurent series theory for eft-monogenic functions. For m = 1 we have R 1 1 = C = R 1, the space of compex numbers, where eft- and right- monogenic functions both coincide with hoomorphic functions. Now we can estabish the samping formua via the decomposition of the spaces of spherica harmonics [7] p.16 into the spaces of spherica eft-monigenics. For any N, there hods Y = M + M 1, 4.1 where M + and M are the spaces consisting of the restrictions to the unit sphere Σ m 1 of, respectivey, the functions in M + and M. The notations M + and M are the spaces of eft-monogenic homogeneous functions in R m 1 of degree and in R m 1 \ {} of degree m + 1, respectivey. The eements of M + and M are caed spherica monogenics, or surface spherica monogenics, of degree. For any f N, N, the space invariant under Fourier transformation discussed in ection 3, combining 3.16 and 4.1, we have γ f x = f rr g + + g x, x =1 where r = x, g + M + and g M 1. Therefore, f x = γ =1 where P M +, Q M 1, h so h We have f rp x + γ =1 h rq x, 4. r = f rr m+ and f satisfy f r r m+ 1 dr <, r = f rr m+ satisfies the inequaity h r r m+ 3 dr <. N = Ω, N = Ω Ω, N, where Ω,, is the right-cifford modue of finite inear combinations of functions of the form frp x, where r = x, P M + and f is a function satisfying fr r m+ 1 dr < ; Ω, >, is the right-cifford modue of finite inear combinations of functions of the form hrqx, where r = x, Q M 1 and h

15 amping with Besse Functions 15 is a function satisfying hr r m+ 3 dr <. Note that for N, the space Ω corresponds to the space M 1. The direct sum decomposition L R m = = Ω 4.3 hods in the sense that: the subspaces Ω are cosed and mutuay orthogona, Z. Every function of L R m is a imit of finite inear combinations of functions in = Ω. Fourier transformation maps each subspace Ω into itsef. This resut is an extension of the cassica m = 1 case [15] 1.4 Chapter IV to any m N [8]. For any g Ω, Z, where r = x, R M + g x = γ =1 g rr x, 4.4, if ; and R M 1, if <. In the foowing, assume the dimension m, we derive the samping expansion for bandimited square integrabe functions in reation to the decomposition 4.3. Theorem 4.1. Let f Ω, Z, with the form fx := grrx r = x, where Rx M +, if ; and Rx M 1, if <, and g be bandimited with bandwidth σ. Then f can be constructed competey via fx = g λ c λc sgnc r c ;λ c rrx 4.5 =1 where λ c are the scaed positive zeros of the Besse function J c given by 1., c ;λ c is given by 3.3, c = m+ Z, sgn is the signum function that taes the vaue +1, 1 or, respectivey, for >, < or = and the series 4.5 being convergent in L -norm, absoutey and uniformy convergent point-wisey on whoe space R m. Proof. The Fourier transform of the function fx := grrx r = x where Rx M +, if ; and Rx M 1, if <, with an abuse of notation but informativey, can be expressed as ft = ĝsrt s = t

16 16 K.I. Kou, T. Qian and F. ommen and ĝs = i π m [ ] [ ] grr sgnc s sgnc J c sr rdr, where c = m+ Z and sgn is the signum function that taes the vaue +1, 1 or for >, < or = [8] Theorem. By the assumption supp ĝ [, σ] and g L [, σ], the radia function g can be obtained via the integra formua [ ] [ ] gr = i π m ĝss sgnc r sgnc J c rs sds. 4.6 The inear integra transformation 4.6 maps F s = i π m ĝss sgnc to gr, with weight function ρs = s, interva I = [, σ] and erne Ks, r = r sgnc Jc rs. and Let r = λ c such that Ks, r = a φ s, where a = φ s = σj c +1 u c λ c sgnc Jc sλ c σj c +1 u c forms a compete orthonoma basis in L I, ρ. When expand Ks, r into the series of φ, the corresponding coefficient =1 r = σj c +1 u c r sgnc c ;λ c r. Therefore by the genera theory described in ection, we have gr = g λ c λc sgnc r c ;λ c r where r = x. The desired reation 4.5 foows. The L and point-wise convergence properties may be proved by using the same methods as in Theorem 3.. Theorem 4.. If f L R m and f is bandimited with bandwidth σ, then f can be written as γ fx = g x = g rr x, 4.7 = = =1

17 where R M + constructed competey via fx = γ = =1 =1 amping with Besse Functions 17, if ; and R g λ c M 1, if <. Moreover, f can be λc sgnc r c ;λ c r R x, 4.8 where λ c are the scaed positive zeros of the Besse function J c given by 1., c ;λ c is given by 3.3, c = m+ Z, sgn is the signum function that taes the vaue +1, 1 or, for >, < or =. The series 4.8 converges to f in the L sense. In particuar, N γ f N,M x = g λ c λc sgnc r c ;λ c rr x,4.9 = M =1 =1 converges to f in the L sense as N, M, and for every fixed pair of positive integers N, M the series 4.9 point-wisey, absoutey and uniformy convergent in R m. Proof. As consequence of 4.3 and 4.4, f can be written as γ fx = g x = where R L sense. M + =, if ; and R = =1 g rr x r = x, 4.1 M 1, if <. The convergence is in the Appying Fourier transform on both sides of 4.1, we have γ ft = ĝ t = g sr t s = t = = =1 ince f vanishes outsides B, σ, for any s > σ, 4.11 becomes γ = g sr t. 4.1 = =1 Owing to the orthogonaity property of R, the same reasoning as proving Theorem 3.3 gives that supp ĝ is, g Theorem 4.1, reation 4.5 gives [, σ] =..., 1,, 1,..., = 1,,..., γ. That is bandimited with bandwidth σ. ince g rr x L R m, appying g rr x = =1 g λ c λc r sgnc c ;λ c rr x, 4.13

18 18 K.I. Kou, T. Qian and F. ommen where c ;λ c is given by 3.3, c = m+ Z and λ c are the scaed positive zeros of the Besse function J c given by 1.. ubstitute 4.13 into the right hand side of 4.1. We get the desired expansion 4.8. The convergence properties may be proved simiary as in the proof of Theorem 3.3. References [1] G. Abramowitz and I. tegun, Handboo of Mathematica Functions, Dover, New Yor, 197. [] George B. Arfen and Hans J. Weber, Mathematica Methods for Physicists, Harcourt, an Diego, 1. [3] F. Bowman, Introduction to Besse Functions, New Yor: Dover, [4] F. Bracx, R. Deanghe and F. ommen, Cifford Anaysis, Research Notes in Mathematics, Vo. 76, Pitman Advanced Pubishing Company, Boston, London, Mebourne, 198. [5] P. L. Butzer, A survey of the Whittaer-hannon samping theorem and some of its extensions, J. Math. Res. Expos., , [6] P. L. Butzer and G. Nasri-Roudsari, Kramer s samping theorem in signa anaysis and its roe in mathematics, in amping Theorey and igna Anaysis II, J. R. Higgins and R. tens, eds., Oxford University Press, Oxford, UK, 1999, [7] R. Deanghe, F. ommen and V. ouce, Cifford agebra and spinor vaued functions, A function theory for Dirac operator. Kuwer, Dordrecht, 199. [8] M. Fei and T. Qian, Direct sum decomposition of L R m 1 into subspaces invariant under Fourier transformation, J. Fourier Ana. App., 1, 6, [9] A. G. Garcia, Orthogona amping Formuas: A Unified Approach, IAM Review Vo. 43,, [1] J. E. Gibert and M. A. M. Murray, Cifford Agebras and Dirac operators in Harmonic Anaysis, Cambridge tudies in Advanced Mathematics, 6, Cambridge University Press, [11] G. H. Hardy, Notes on specia systems of orthogona functions, IV: The Whittaer s cardina series, Proc. Camb. Phi. oc., , [1] J. R. Higgins, Five short stories about the cardina series, Bu. Amer. Math. oc., , [13] A. J. Jerri, The hannon samping theorem-its various extensions and appications: A tutoria review, Proc. IEEE, , [14] N.N. Lebedev, pecia Functions and Their Appications, Prentice-Ha, London, [15] E. tein and G. Weiss, Introduction to Fourier anaysis on Eucidean spaces, Princeton university press, [16] G. N. Watson, A Treatise on the Theory of Besse Functions, econd Edition, Cambridge University Press, 1966.

19 amping with Besse Functions 19 K.I. Kou Facuty of cience and Technoogy University of Macau Av. Padre Tomas Pereira,. J., Taipa, Macao e-mai: T. Qian Facuty of cience and Technoogy University of Macau Av. Padre Tomas Pereira,. J., Taipa, Macao e-mai: F. ommen Department of Mathematica Anaysis University of Ghent Gagaan, B-9 Gent, Begium e-mai: