STRONG BOUNDEDNESS OF ANALYTIC FUNCTIONS IN TUBES

Certain classes of analytic functions in tube domains T in n-dimensional complex space, where C is an open connected cone in n, are studied. We show that the functions have a boundedness property in the strong topology of the space of tempered distributions Ig. We further give a direct proof that each analytic function attains the Fourier transform of its spectral function as distributional boundary value in the strong (and weak) topology of 8'.

V the space of distributions of L. Schwartz  [2], which possesses the t following properties e -yt V g 8' for all y B; <V eiZt> for all z TB.
(1.2) t , iz Here is the space of tempered distributions of Schwartz [2] and <V t e is the Fourier-Laplace transform of the spectral function V t.
In [3] Vladimirov defined certain classes of analytic functions in tubular T C 1R n + iC, where C is an open cone, and analyzed the spectral cone s functions of these analytic functions corresponding to C being an open connected cone The results of [3] have been incorporated into the book [i] of Vladlmirov [i, section 26.4].
In this paper we add information to the main results of [3] and [i, section 26.4] which are [i, pp. 238-239,Theorems i and 2].We show that the analytic functions considered by Vladimirov in these results have boundedness properties in the strong topology of the space of tempered distributions Further we give a direct proof by elementary means that each analytic function attains the Fourier transform of its spectral function as distributional boundary value in the strong (and weak) topology of ', a fact which has been recognized by Vladlmirov [i, p. 238] and which is obtained by him as a special case of a more general result.

NOTATION AND DEFINITIONS.
Our n-dimensional notation is that of Vladlmirov [i, p. i].x, y, and t will be points in n in this paper and z n, n-dimensional complex space.Note the inner products zt ZltI +...+ Zntn and yt Yltl +...+ Yntn for t and y in I n and z n.Note also the differential operator D in [I, p. i], and we shall write D D ( or to indicate that the differentiation is with respect z t to z or t, respectively.Here c is an n-tuple of nonnegative integers.The definitions of cone C in n, compact subcone of a cone, indlcatrlx Uc(t) of a cone, and of the number 0C which characterizes the nonconvexlty of a cone C, can all be found in [i, section 25.1].Note that 0C > 1 [i, p. 220] for , any cone C. The cone C {t e An :Yt > 0, y e C} is the dual cone of C and C, will denote C, n \ C 0(C) will denote the convex envelope (hull) of the cone C, and we define the tubes T C and T 0(C) by T C A n + iC and T0(C) n + i0(C) respectively.
Let C be a cone in A n We make the convention throughout this paper T C T0( C T C that by z e (e and y e C(E 0(C)) we mean that z and y e C' for an arbitrary compact subcone C 'C C (C'c 0(C)).
The space of functions of rapid decrease g g(An) and the space of tempered distributions ' ' (An) are defined and discussed in Schwartz L 1 [2 Chapter 7] The Fourier (inverse Fourier) transform of an (An) function (t), denoted [(t);x] (-l[(t);x]), will be as defined in Vladlmlrov [i, p. 21].The Fourier transform of a tempered distribution Vt, denoted [V], is defined in Schwartz [2, p. 250, (VII 6; 6)].All terminology and definitions concerning distributions in this paper, such as support of a dlstrl- bution, will be that of Schwartz [2].Uc(t) < a}.Conversely, if V t e ' and has support in {t Uc(t) _< a} for some a _> 0 and some open connected cone C, then all the derivatives DS(f(z)) of the Fourier-Laplace transform f(z) of V t z belong to the class HI(aOc;0(C)).

LEMMAS.
As noted in the introduction, we shall add information to Theorems i and 2.
We shall show that the analytic functions in these theorems have a strong boundedness property in 8 In addition we give a direct proof that the analytic functions attain the Fourier transform of their spectral functions as distributional boundary values in the strong (and weak) topology of 8 The following lemma is the basis of the boundary value result, and its proof in turn is useful in obtaining our strong boundedness properties.Through- out this section C is an open connected cone.
LEMMA i.Let f(z) e Hp(a + 6;C), p > i and a > 0. The spectral function V of f(z) is in 8' as is (e -yt V ), y e 0(C) and in the strong (and weak) topology of 8 PROOF.Let C be an arbitrary compact subcone of 0(C).By the sufficiency of Theorem 1, the spectral function V of f(z) has the repre- sentation (3.2).Since each ga(t) in (3.2) is continuous and of power increase over n we immediately have V t e ' The fact that (e -yt V e 8' t y e C c 0(C), follows by the proof of Theorem i given in [i, section 26.5].
Let # be an arbitrary element of 8. Using the notion of distributional differentiation and the generalized Leibnitz rule, we have for y e C c 0(C) that <V (e -yt -l)(t)> t' a! DB(e -yt i) Y((t)) dt where , 8, and y are n-tuples of nonnegative integers and I (,B,Y) f g(t) ((-i) IBI yB e-Yt-DB(1)) DY(@(t)) dr. (4.3) For the arbitrary C c 0(C) we apply [i, p. 223, Lemma  II('B'Y)y (C*)' ga(t) ((-i) [BI yB e-Yt-DE(1))t  Recall that we desire a convergence result in this lemma as y + 0, y e 0(C).Hence to obtain (4.1) it suffices to consider y e 0(C) such that IYl < Q for Q > 0 fixed.Now consider the integrand of the integral ll(,8,y) in (4.6)  and y g C c 0(C) such that IYl < Q" Since g , the right side of the last inequality in (4.10) is an L I function over n which is independent of y g C 0(C) such that IYl < Q" Using this fact, (  (3.3).Using (3.3), the relations (3.4), the facts -yt < IYl U0(c)(t) U0(c)(t) < PC Uc(t) t g C, y E 0(C) (4.12) contained in [i, section 25.1], and analysis as in [I, p. 244], we have for t g C a C a and y g C 0(C) such that Yl < Q that < Me(Ca exp[-(a-E)(Uc(t))P (i + IYl II e-Yt) ID ((t)) < ME(C , exp[-(a-E)(Uc(t))P (i + IYl III exp[ly 0 C Uc(t) ]) IDJ((t))l (4.13) < ME(C , (i + IYl IBI) exp[-(a -E) (Uc(t) )P / lyl c Uc(C)] ID: 3) holds for all E > 0. In particular (3.3), and hence (4.13), holds for E >0 fixed such that (a 2E) > 0 for the fixed a in (3.4).For E > 0 fixed in this way in obtaining (4.13), we now conclude from (4.13) that B B(1)) Y((t)) < g(t)((-l)ISl Y e-Yt Dt Ot  The next lemma is the basis of our strong boundedness results concerning the analytic functions H (a + E; C), p > I and a > 0. P LEMMA 2. Let p > i and a > 0. Let C be an open connected cone.Let V t be any generalized function of the form (3.2) where the g(t) satisfy the 8 conditions stated in Theorem i Then V t ' (e -yt V t) E for all y e 0(C), and {[e -yt V t] ' y E 0(C) IYl < Q} is a strongly bounded set in ' for Q > 0 being arbitrary but fixed.
PROOF.Let C be an arbitrary compact subcone of 0(C).The facts that , -yt ' V t and (e V t) for all y e C c 0(C) follow as at the beginning of the proof of Lemma i.The locally convex topology of 8 is defined by the norms sup lal<_k ( + Itl) k ID<,(t))l Using (4.4), (4.18), and the fact that each go(t) satisfies (4.9) for some polynomial P (t), we have where R o is a constant and k s is a positive integer with both depending on o; and (4.21) holds for each o and y, e 8 + Y, in (4.19).Also recall that each go(t) satisfies (3.3).Using (3.3), (4.12), and analysis as in (4.21), (4.13), and (4.14) we have for y e C 0(C) that M' II2(s,)l < (c',) f, exp[-(a-6)(Uc(t))P exp[lyl o c Uc(t)] IDY((t))l dt Y 6 t C, < ME(C,) If*Ilk' /, exp[-(a'-S)(Uc(t))P + lyl Pc Uc(t)] (i + Itl) -n-I dt (4.22) S C, i i )p/p' p <_ M e(C,) llllk exp[(p, (a'-26) PC fn (i + Itl) -n-1 dt where ME(C , is a constant and k s is a positive integer depending on . Because of (3.3), we can assume that E > 0 in (4.22)  (5.3)

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: 2] to obtain a number (C > 0 and an open cone (C) both depending on C such that (C contains the cone C {t n:yt > 0, y C}, the dual cone of C, and yt > lyl ItI y g c t (c*) c*) c' * Put C R n\ is a compact subcone of C IR n\ C and wehave C, N (C*) = and C, U (C*) n.We now write the integral ly(a,B,y) in(4 IBI yB e-Yt_ Dt U_I) IBI yB e-Yt B # (0,...,0), ln for all y C c 0(C) and in fact for all y E hence for any a in the last sum in (4.2) and any subsequent B and Y, B + Y a,(4.7) of (4.14) is an function over ]R n and is independent of y C c 0(C) such that IYl < Q.Thus by (4.14), (4.8), and the Lebesgue dominated convergence theorem we have lira bounded set in 8.For the arbitrary C c 0(C) we apply [i, p. 223, Lemma 2] as in the proof of Lemma i and obtain a number (C') > 0 and an open cone (Ca compact subcone of C, \ C as in the proof of Lemma i.Using the form of V t in (3.2) and the generalized Leibnitz rule we obtain for any $ $ and y C c 0(C) that 24 R. D. CARMICHAEL<e -yt V t (t)> .(-i) II I (_1) 161 yB (ll(,y) + 12(s,y)) y) , go(t) e -yt DYt((t)) dt.ly C,

First
Round of Reviews March 1, 2009 2. [i, p. 239] Let f(z) e H I(a + E..'C) where C is an open connected cone and a > 0. Then its spectral function V e 8 THEOREM I. [I, p. 238] Let f(z) H (a + E;C), where C is an open P connected cone, p > i, and a > 0. The spectral function V of f(z) can t be represented in the form of a finite sum of distributional derivatives of continuous functions g(t) of power increase, V Da(g_ (t)) (3.2) t t which, for all t C, where C, is an arbitrary compact subcone of C, n \ C and for all E > O, satisfy Ig( t) < ME(C,) exp[-(a -E)(Uc(t))P (3.3) .,ere the numbers p nd a are connected with p and a by the relations i +--i,_i (p' t p > i and the cone C,, then all derivatives DB(f(z)) of its Fourier- Z Laplace transform f(z) belong to the class H (a0 + E;0(C)).P Notice that the C as printed in [1, p. 239, line 8] should be C, instead as we have written in Theorem I. THEOREM for t e (C*)'.Since each g(t) in (3.2) is of power t n.Using (4.9) and (4.4) we get ig(t) ((_i) 181 y8 e-Yt.DE(l)) D((t))I <