WAVE FRONTS OF ULTRADISTRIBUTIONS VIA FOURIER SERIES COEFFICIENTS

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1 Matematiki Bilten ISSN X Vol.39 (LXV) No.2 UDC: : (5359) Skopje, Makedonija WAVE FRONTS OF ULTRADISTRIBUTIONS VIA FOURIER SERIES COEFFICIENTS DIJANA DOLI ANIN-DJEKI, SNJEZANA MAKSIMOVI, AND PETAR SOKOLOSKI Abstract. We shall use the properties of the product of periodic ultradistributions and give a new description of the wave front of an ultradistribution f D (R d ) in terms of Fourier series coecients. 1. Introduction We have analyzed in [14] the wave front and the Sobolev wave front for distributions through the Fourier coecients of their appropriate localizations. In this paper we extend the ideas of [14] to ultradistribution spaces. More precisely, we analyze the microlocal properties of an ultradistribution f at x 0 R d which are determined through the Fourier series expansion of periodizations of ϕf, where ϕ is a cut-o function near x 0. Spaces of periodic ultradistributions have been studied mostly in the last 30 years of the last century. We refer here just a few of the papers and books [5, 6, 7, 15] and also to the papers [18, 17] for the discrete wave fronts which provide another approach to the microlocal analysis. We refer to [3, 4, 10, 11, 16] for a new approach to the analysis of wave fronts and to [1, 2, 12, 19, 20, 22] for the theory of periodic distributions. The main dierence between the study of periodic distributions and periodic ultradistributions is in the weight functions we use and the conseuences. This will be explained in this paper Notation. We follow our notation from [14]. For x = (x 1,..., x d ) R d, we write x = (x x 2 d )1/2 and x = (1 + x 2 ) 1/2. Let 0 < η 1. We will use the notation d I η,x = (x j η/2, x j + η/2) and I η := I η,0. j=1 We refer to [13] for the spaces of ultradierentiable functions and their duals, spaces of ultradistributions. In our case, we consider spaces determined by the Gevrey seuence of positive numbers M p = p! s, p N 0 for some s > 1. The seuences of this type satisfy the conditions (M.1), (M.2), (M.3) of Komatsu [13] Mathematics Subject Classication. 35A18; 46F30, 46T30. Key words and phrases. Wave fronts, Fourier series, multiplication of ultradistributions. 53

2 54 D.DOLI ANIN-DJEKI, S. MAKSIMOVI, AND P. SOKOLOSKI Recall that functions or ultradistributions on R d are periodic of period 1 in each variable, if f(x + n) = f(x), for all x R d, n Z d. We put e y for e y (x) = e 2πiy x, y R d. We will consider periodic extensions of localizations of ultradistributions around a point x 0 R d, so if an ultradistribution g is ported by I η,x0, with 0 < η < 1, we shall write g p (x) := n Zd g(x + n) for its periodic extension Spaces of ultradierentiable functions and their duals. The space of periodic test functions P = P (R d ) where = (s), (respectively = {s}) (Beurling and Roumieu cases, respectively) consists of all the ultradierentiable periodic functions ϕ of Beurling and Roumieu classes, respectively. It is proved in [7] that ϕ is periodic ultradierentiable function of order s of Roumieu type P {s} (R d ), (resp., Beurling type P (s) (R d )) if and only if ϕ n 2 exp(2α 1/s n 1/s ) <, n Z d for some (resp., any) α > 0, where ϕ n = I 1 ϕ(x)e 2πin x dx = ϕ, e n, n Z d. The dual space of P, the space of periodic ultradistributions, is denoted by (P ). Recall [7]: f = n Z d f ne n is a periodic ultradistribution of order s of Roumieu (Beurling) type if and only if n Z d f n 2 exp( 2α 1/s n 1/s ) <, for any (some) α > 0. If f = n Z d f ne n (P ) and ϕ = n Z d ϕ ne n P, then f, ϕ = n Z d f nϕ n. In the seuel, we will consider only Roumieu type basic spaces and their duals. The corresponding results for the Beurling type spaces are similar and even easier to prove. Let, ν be positive functions over Z d. We call the function ν submultiplicative if ν(m + n) ν(m)ν(n), m, n Z d. (1.1) The function is a ν-moderate weight if there is a constant C such that (m + n) C(m)ν(n), m, n Z d. (1.2) For a xed submultiplicative function ν, the set of all ν-moderate weights is M ν. Following [11], for s > 1, M {s} (Z d ) = { M ν : ( C > 0)( k > 0)( n Z d )(ν(n) Ce k n 1/s )}. For M {s} (Z d ), we dene Pl = {f (P {s} ) : {f n (n)} n Z d l, where f n = f, e n } plied by the norm f Pl = {f n (n)} l. We consider from now on only values 1. Clearly, Pl 1 1 Pl 2 2, if 1 2 and 2 C 1. We will also consider the local space Pl,loc consisting of distributions f (D {s} ) (R d ) such that the periodic extensions (ϕf) p Pl, for all x 0 R d and ϕ D {s} (I 1,x0 ). Its topology is dened via the family of seminorms f x0,ϕ= (ϕf) p Pl, where x 0 R d and ϕ D {s} (I 1,x0 ). For the sake of completeness, we give the following elementary proposition (its proof also shows that the denition of Pl,loc is consistent).

3 WAVE FRONTS OF ULTRADISTRIBUTIONS VIA FOURIER SERIES COEFFICIENTS 55 Proposition 1.1. Pl Pl,loc. Proof. Let f Pl and ϕ D {s} (I 1,x0 ), then (ϕf) p = ϕ p f. Write f = n f ne n and ϕ p = n ϕ ne n P {s}. By (1.2) and the generalized Minkowski ineuality, we have 1/ ϕ p f Pl C ϕ j ν(j) f n j (n j) n j C ϕ p Pl 1 ν f Pl <. For xed s > 1, set k (n) = e k n 1/s, k R. For convenience, we write Pl k := Pl k and Pl k,loc := Pl k,loc. We clearly have P {s} = Pl k = Pl k 0 and Moreover, and (P {s} ) = k 0 E {s} = k 0 (D {s} ) F = k 0 Pl k = Pl k,loc = Pl k,loc = M {s} (Z d ) M {s} (Z d ) M {s} (Z d ) M {s} (Z d ) Pl. Pl,loc Pl,loc, where E {s} is the space of all ultradierentiable functions of order s of Roumieu type and (D {s} ) F is the space of ultradistributions of nite order of Roumieu type on R d, that is g (D {s} ) F if for some ultradierential operator P of class {s} and some G, continuous on R d, there holds g = P (D)G,[13] Multiplication within periodic test spaces. We x two weight functions M ν, ν M {s} (Z d ) (cf. (1.2)). Let f 1 = n Z f 1,ne d n Pl 1 and f 2 = n Z f 2,ne d n Plν 2. Then we dene their product as f := f 1 f 2 := n Z f ne d n, where f n = f 1,n j f 2,j, n Z d. j Z d This denition allows us to introduce multiplication in the local versions of these spaces. Let f 1 Pl 1,loc and f 2 Pl 2 ν,loc. Then the product f := f 1f 2 is dened as follows. Let x 0 R d, 0 < η < 1 and let φ D {s} (I 1,x0 ) be such that φ(x) = 1 for x I η,x0. We dene f Iη,x0 (D {s} ) (I η,x0 ) as the restriction of (φf 1 ) p (φf 2 ) p to I η,x0. Note that dierent choices of φ lead to dierent Fourier coecients but, by Proposition 1.1, we have f Iη,x0 = f Iη,x on I η,x0 I η,x. Thus, by {f 0 0 I η,x0 } we have a distribution f Pl,loc as the product of f 1f 2 := f.

4 56 D.DOLI ANIN-DJEKI, S. MAKSIMOVI, AND P. SOKOLOSKI Proposition 1.2. Let, 1, 2 [1, ] are such that mappings and are continuous. Pl = Then the Pl 2 ν (f 1, f 2 ) f 1 f 2 Pl (1.3) Pl 1,loc Pl2 ν,loc (f 1, f 2 ) f 1 f 2 Pl,loc (1.4) Proof. The continuity of (1.4) follows from (1.3), while (1.3), Young's ineuality and (1.2) imply f 1 f 2 Pl C f 1 Pl 1 f 2 Pl 2 ν. As a corollary, we have: Corollary 0.1. Let k, k 1, k 2 R, Then, the mappings Pl 1 k 1 Pl 2 k 2,loc (f 1, f 2 ) f 1 f 2 Pl k,loc k 1 + k 2 0, k min{k 1, k 2 }. (1.5) Pl 2 k 2 (f 1, f 2 ) f 1 f 2 Pl k are continuous. and Pl1 k 1,loc Proof. Assume k 1 0 and k = k 2 (it is not a restriction). Clearly k 1 k 2 has to hold in order to have k 1 + k 2 0. The continuity then follows from Proposition 1.2 with (n) = e k2 n 1/s and ν(n) = e k1 n 1/s because (1.2) holds in this case. 2. Wave Front As in the case of distributions, our aim is to describe the wave front of an f (D {s} ) (R d ) via the Fourier series coecients of the periodic extension of an appropriate localization of f around x 0 R d. Recall that (x 0, ξ 0 ) / W F (f) if there exist ψ (D {s} ) (R d ) with ψ 1 in a neighborhood of x 0 and an open cone Γ R d \ {0} containing ξ 0 such that ( N > 0)( C N > 0))( ξ Γ) ψf(ξ) C N e N ξ 1/s. (2.1) Theorem 1. Let f (D {s} ) (R d ) and (x 0, ξ 0 ) R d (R d \ {0}). The following conditions are euivalent: (i) There exist φ D {s} (I ε,x0 ), with ε (0, 1) and φ 1 in a neighborhood of x 0, and an open cone Γ containing ξ 0 such that ( N N)( C N > 0)( n Γ Z d ) ψf(n) C N e N n 1/s. (2.2) (ii) (x 0, ξ 0 ) / W F (f). Proof. The proof is similar to the one given in [14] for distributions, but for the sake of completeness we give all the details. By shrinking the conic neighborhood of ξ 0, we can choose ψ in (2.1) with arbitrarily small port around x 0. With this (ii) implies (i). So, now it is enough to show that (i) implies (ii). Assume (i). We will prove that there are ε and an open cone ξ 0 Γ 1 such that ( B bounded set in D {s} (I ε,x 0 ))( N > 0)( C N > 0)

5 WAVE FRONTS OF ULTRADISTRIBUTIONS VIA FOURIER SERIES COEFFICIENTS 57 ( n Γ 1 Z d ) ϕf(n) C N e N n 1/s. (2.3) ϕ B Let ε be such that φ(x) = 1 for every x I ε,x 0. Let Γ 1 be an open cone with ξ 0 Γ 1 and Γ 1 Γ {0}. Let 0 < c < 1 be a constant smaller than { the distance between Γ and the intersection of Γ 1 with the unit sphere. Then y R d : ( ξ Γ 1 )( ξ y c ξ ) } Γ. Let B D {s} (I ε,x 0 ) be a bounded set. Since φϕ = ϕ, ϕ B we have that fϕ(n) are the Fourier coecients of the periodic distribution (φf) p (ϕ) p. Thus, for ϕ B and n Γ 1 Z d, ϕf(n) = Now we estimate I 1 (n): I 1 (n) = φf(n j) ϕ(j) j Z ( d n j c n j c n + j >c n =: I 1 (n) + I 2 (n) φf(j) ϕ(n j) C n j c n ) φf(n j) ϕ(j) φf(j), where C only depends on B. Since n j c n implies j (1 c) n, we have e N n 1/s I 1 (n) C e N n 1/s φf(j) ϕ B, n Γ 1 n Γ 1 n j c n C (1 c) N e N j 1/s φf(j) = C(1 c) N C N. j Γ ξ0 (2.4) Next we estimate I 2. For this we use that n j (1 + c 1 ) j if j c n. By the Paley-Wiener theorem, there are M > 0, D > 0 such that φf(n j) De M n j 1/s, n, j Z d. The boundedness of B D {s} (R d ), implies 1/s ϕ(j) =: K N <. Moreover, we have e (M+N) j ϕ B j Z d e N n 1/s I 2 (n) D e N n 1/s n Γ 1 n Γ 1 j c n e M n j 1/s ϕ(j) DC N (1 + c 1 ) M K N, ϕ B. (2.5) By (2.4) and (2.5), we obtain (2.3). Now we show that (x 0, ξ 0 ) / W F (f) by (2.3). Let ψ D {s} (I ε,x 0 ) be eual to 1 in a neighborhood of x 0. Then, the set B = {ϕ t := ψe t : t [0, 1) d } is a bounded subset of D {s} (I ε,x 0 ). This implies t [0,1) d ψf(n + t) = t [0,1) d ϕ t f(n) C N e N n 1/s, n Γ 1 Z d, e N ξ 1/s ψf(ξ) (1 + 4d) N/2 C N. (2.6) ξ (Γ 1 Z d )+[0,1) d

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7 WAVE FRONTS OF ULTRADISTRIBUTIONS VIA FOURIER SERIES COEFFICIENTS 59 [21] J. Vindas, R. Estrada, Distributional point values and convergence of Fourier series and integrals, J. Fourier Anal. Appl. 13 (2007), [22] G. Walter, Pointwise convergence of distribution expansions, Studia Math. 26 (1966), Faculty of Technical Sciences in Kosovska Mitrovica, and, State University of Novi Pazar, Novi Pazar, Serbia address: Faculty of Electrical Engineering, University of Banja Luka, Patre 5, Banja Luka, Bosnia and Herzegovina address: Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Republic of Macedonia address: