EXTENSION SETS FOR REAL ANALYTIC FUNCTIONS AND APPLICATIONS TO RADON TRANSFORMS

The real analytic character of a function f(z, /) is determined from its behavior along radial directions fo(s) f(scos0, ssin0) for 0 E E, where E is a "small" set A support theorem for Radon transforms in the plane is proved In particular if fo extends to an entire function for 0 E E and f(z, /) is real analytic in IR then it also extends to an entire function in C


INTRODUCTION
The determination of the behavior of functions on small sets is an old problem that has been studied in many branches of analysis The purpose of this article is to study the real-analytic character of a function of two variables f(z,l) from its behavior along radial directions fo(s) f(s cos0, s sin0) for 0 E, where E is a "small" set We introduce a property of sets, the separation condition, that allow us to infer the behavior of f(x, /) from that of fo(s) for 0 E. In particular, if fo extends to an entire function for each 0 E and if f(z, /) is real analytic in ]R then it also extends to an entire function in C If fo is of exponential order for E then so is f The plan of the paper is as follows The second section gives the definition of the separation condition and studies some of its properties The third section gives the main results on the characterization of the real analytic character of f(z, /) from that of fo(s) for 0 E E, a set that satisfies the separation condition The last section applies these results to obtain a support theorem for Radon transforms in the plane 2. THE SEPARATION CONDITION Motivated by its prospective use in the theory of Radon transforms, we introduce the following definition DEFINITION.Let e > 0 We say that a set A c_ ]R satisfies the separation condltlon of order e, denoted as S C (e), if for each n 11 there are points 01, 0, A such that lot 031 > eli 3lln (2 1) THEOREM 1.Let A C ]R be a bounded set Then #(A) sup{e A satisfies S C (e)}, (22)   where # is the Lebesgue measure In particular, A satisfies S C if and only if #(A) > 0 PROOF.Let e < #(A) and let 6 be such that e < 6 < #(A) Let 0 E A with #(A F [01, oo D _> 6 and define 02, 0, E A recursively as Ok+l inf{0 -'#(ffl[Ok,O]) >_ 6In}. ( Then Ok+ O 6In, 1 k 72 1 Let now/1, ...,/n A be chosen so that IOk-Akl < (6-e)/2n.Then ,k.Ak >_ e/n It follows that A satisfies S C (e) and, consequently, #(A) < sup{e satisfies S C.(e)}.
Conversely, suppose A satisfies S C (e) Let I be the closed interval [infA, supA] Then I\A is open, therefore /\A= u U

3=1
where the U are disjoint open intervals For each q 1, 2, 3, the set I\ U U.
3=1 consists of q + 1 disjoint closed intervals A q) zl(q) ordered from left to right Since A satisfies "*q+l, S C (e), there exists/91, ...,/gn A, with/91 </92 < </9,,, such that IO, 01 _> I Jl/, (q) if Then there are integers, 1 kl _< ]2 <_ _< kq+l <_ kq+2 n q-2, such that /9 A Thus q+l q+l n If we first let n -oo we obtain 3=1 3=1 (2 4) and if we then let q we obtain e _< #(), ( as desired Let now E be a subset of the circle S {z E C'lz 1} We assume E symmetric, namely, 0 E if/9 E We denote by the set of arguments of elements of E, so that E {e * '/9 / } DEFINITION.We say that E satisfies the separation condition of order (for symmetric sets), denoted as S C (e), if for each n E 1 there are arguments 0, 0T E such that 10, 0j + kTr _> (2 6)   for each k Z Notice that the separation condition for not necessarily symmetric sets requires (2 6) to hold for k even only In our case, however, the points e and e '(+) will be identified and that is why we require (2 6) for all k Z It is not hard to see that E satisfies S C (e) if and only if E N [a, a + 7r] satisfies S C (e) for some, and hence for each, a It follows that sup{e" satisfies S C (e)} #() so that, in particular, E satisfies S C if and only if # () > 0.
mE U E,T {Sz "0 <_ S, Z e E}. ( Let XE. be the space of germs of functions f defined in E,,., that satisfy the following conditions a) f can be extended to a neighborhood of the origin as a smooth (i e C) function, b) fo(s) f(scosO, s sin0) is real-analytic as a function ofs for I1 < r for each 0 E E, c) the quantity n!
n=0 is finite Observe that if the functions fl, f2 extend to smooth functions F1, F2 that agree of infinite order at the origin, then they are identified The space X E is defined as the intersection of the X z, for r > 0 Observe that if f E X z then fo(s) can be extended to an entire function of s C for each 0 ' Let F C_ S be a superset of E, E c F Then each element of XF,,.can be restricted to X.T Our aim is to show that under suitable assumptions on E this process can be reversed, namely, each element of X, can be extended to X,.t for some < r In particular, XE and XF would result to be isomorphic and, by taking F S, each element of X would extend to an entire function in C Let us first show how to compute the directional derivatives f(r)(0) from the corresponding derivative values f) (0) for 0 E E Indeed, let us first suppose , 0 Then for each 0 E we have f' Oxk OYn-k If 00, 01,..., 0, E then evaluating (3 4) at 0 0 yields a linear system for the computation of the partial derivatives where we have denoted by/h, (0o, On) the determinant /Xn(00, On) det[coskOzsinn-kO]O<_k:<_n. ( These determinants are easily evaluated if we observe that ZX,(0, 01 ,0,) Gn-1 (01, 0,)sin 01... sin 0,, ( x(Oo, on) zx /XI(O0, 01) sin(Oo 01). ( This yields Gn(0o, 0n) H sin(0, 0s). ( :< Replacing 0 by 0, using (3 10) and doing some simplification in (3 5) give the formula fn) (0) E fo, (0) H sin(0 0s) =o 3#z sin(0, 0s)" ( Naturally (3 11) can be applied only if An (0o, 0n) 0, that is, only if 0, 03 7:: kr for 3 and k Z When E satisfies the S C then such points 0o, 01 can always be found, but, more than that, if they satisfy the separation condition 10, 0; + 1 > li 31e 0 < i, j < n, ( n+l then f') (0) can be estimated on S\E from estimates on E. Actually, if we suppose that ]f') -< b, for where M(O) sup{Isin(0-)1 q: /} -< 1. ( But 'n-> 2/r for 0 < Iz _< n-/2, so that if for each i, j we choose k k(i,3) Z with 10 0 + kr[ < 7r/2 then we obtain H Isin(O, O;)l _> H io, o + 2e (n -> r(n + 1) z., (3 15) so that Specializing the value of b, in (3 16) we obtain several extension theorems THEOREM 2. If E satisfies the S C (e) then each f E XE,r can be extended to XE,t for each { E whenever < er/Mre, where M sup M(O) 0 (S\E) PROOF.If f X,,-then according to (3 3) I:g'><>l-< 7' where K Ilfll, Thus for each 0 E R the series n=0 is majorized by the series which converges for s < er/Mrre i-I THEOREM 3. If E satisfies the S C (e) then X E is isomorphic to XF for each F _ E. If f XE then the Taylor series Okl +k2 kl z2k2 defines an entire function in C PROOF.That X E is isomorphic to XF if F 2)E follows by letting r oo in the previous theorem The convergence of the Taylor series for each (Zl, z2) C with IZll < r, Iz2l < r is equivalent to the convergence of the series k! z k=0 for each Izl < r and each 0 E , so that the Taylor series indeed defines an entire nction in C A paicularly usel result along these lines is the follong COROLLARY.Let f be real analytic in 2 If f E X F for some E that satisfies the S C then f extends to an entire function in C and its Taylor series converges to f in all ]R !-1 We say that an entire function of n variables F(zl,...,z,,) is of exponential order if there are constants K and A such that ]F(zl, Zn)] <_ Ke a(Izl+ +iz,l) (3 18)   for each (Zl, z,) E C In that case we say that F is of exponential order A TIIEOREM 4. Let f(z, t) be real analytic in ]R Let f E Xz, where E satisfies the S C (e) If fo (z) f(z cos 0, z sin 0) is of uniform exponential order in E in the sense that [fo(z)] < KeAlzl, zEC, 0, ( then f extends to an entire function of exponential order in C PROOF.If fo(z) satisfies (3 19) then it follows by Cauchy estimates that If')(O)l<Kn'(me') 0. (320, If we now use (3 16) we obtain the bound Up to now we have suppose 0 real, but as should be clear (3 21) also holds in 0 C If 0 ]R then [M(0)I < 1, an inequality that has to be replaced by IM(0)I < [cos0[ + Isin01 (3 22) if0C It follows that if z w cos 0, z2 w sin 0 for some w, 0 E C then 1 AreM(O)lw] KeeAeM(O)lwle-If(zl,z2)l --Ifo(w)l < Ke Z . ,< (3 23) or If(za,)l K ea+(Ae/')(lzl+l21) (3 24) Not every pair (Zl,Z2)C admits the representation Zl =-wcos0, z =wsin0 the extra condition z + z -0 has to be satisfied However, the set {(za,z) zx + z 0} is dense in C , so that by continuity (3.24) holds in all C S.
1) Notice that our deflation of Xz. requires f to have a smooth eension to a neighborhood of the origin.A nction like f(z, ) (z + )2 (z + 22)-1 shows that our theorems do not hold if condition (a) is oued in the deflation of X, 2) A nction like exp (x 2)-X2 < 2 f(X,y) 0 X2 y shows that Theorem 4 is false if f is not required to be real analic but just smooth

APPLICATION TO DON TNSFOS
In this section we show that our results have application in the suppo theorems for Radon transforms If f is a function defined in Rn, integrable over every hyperplane, then its Radon transform is defined as f(w, t) f(x)d#(x), for (w, t) E S R, where d# is the Lebesgue measure on the hyperplane x-w Clearly, if f has compact suppo contained in the closed ball B(0, A) then the suppo of its Radon transform is contained in sx I-A, A] The converse is not te in general [6,11], unless f satisfies some additional conditions For instance, the suppo theorem of Helgason [4,5,8] says that if f is continuous and rapidly decreasing (i e f(x) 0(ll k) as I ) then the suppo of is contained in -1 [_ A,A] if and only if the suppo of f is contained in B(0, A) In [10], Wiegerlnck introduced a simple but powerl method for the study of suppo theorems As we show, the use of this method and of our results of the previous sections allow us to obtain interesting suppo results for the

Radon transform
Instead of requiting f to be rapidly decreasing in the ordina sense, it would be enough to ask f to be dstnbutonally small in the sense that it satisfies the moment asymptotic expansion [2, Ik[ =0 For our purposes, it would be convenient to take f O The space Oc (see [7,9]) is defined as the space of smooth nctions (x) defined in that satis Dk( for each k and some q The space M is defined silarly but now Dk(x) 0(Ixl<k) for some q(k) that may va with k The Fourier transform interchanges these spaces f(O) , f(O) Oc Eve element of O is distributionally small [2,3] so obsee that if f E O then its Fourier transform {f(x), u} (f(x), e*x'u) can be obtained by evaluation at e'u, not just duality THEOM 5. Let f be a continuous nction in that satisfies three conditions (a) f E O (b) Its don transfo satisfies the estimate [?(w,t)[ Cwe -'wltl, ew>0, (w,t) ESx.
(43) (c) There is a set E C_ S that satisfies the S C (e) and a number R > 0 such that f(w,t) 0, w E E, Itl > R.
(44) Then f has compact support contained in the disc B(0, RTre/e) PROOF.If f (9, then its Fourier transform F '(f) belongs to OM, SO that F is smooth in If Iwl-1, the values of F along the line sw, s JR, can be expressed in terms of the Radon ]1 transform as F(sw) I(x)eSW'Xdx I(w, t)e-'dt. ( The condition (b) guarantees that F(s) can be extended to strip {s C :Ims < e} so that, in paicular, F(s) is real analic in s en E then (4 4) shows that F(s) extends to an entire nction of exponential order R If(w)l Ke1"1, s C, w E. defines an entire function of exponential order RTre/e in C But since F is smooth near the origin and since F(sw) is real analytic in s, it follows that the Taylor series converges to F in .oThus F extends to an entire function of exponential order Rrce/e Since F E OM, the Paley-Wiener theory shows that f .T'-IF is a distribution with compact support in the disc B(0, RTre/e) D

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.