FOURIER TRANSFORMS OF DINI-LIPSCHITZ FUNCTIONS ON VlLENKIN GROUPS

In (4) we proved some theorems on the Fourier Transforms of functions satisfying conditions related to the Dini-Lipschitz conditions on the n-dimensional Euclidean space R n and the torus group T. In this paper we extend those theorems for functions with Fourier series on Vilenkin groups.

Here we intruce me dtions d notations that H u Ist on.Ts is by done since we sh my follow Oewr [I] d Quek d Yap [2] in ts rt.V n= I 0 if all V, are finite, the inclusion is proper.We introduce the numbers mo, ml, m2,...,m k such that mo=l,m k+l=pkmk; keN, pk being a prime >_ 2. Then evetT V, has m, as its measure and the quotient subgroup Vn/V,_ has P, for its measuxe.DEFINITION 2.1.For zG, let (n,z) denote the continuous character of z, i.e. (,,z)G.The Fourier transform (,) of f(z)L'(G) is defined by 2(") I/()(.,)dG where (n,z) is the complex conjugate of (n,z).
DEFINITION 2.3 if evexT P/ is finite as k --) oo we say that G has the boundedness property (P).
REMARK 3.2.We remark here that for special choice of a, 7, and P like.a I, 7 1, P 2, the previous theorem gives special interesting cases.This is quite obvious and we shall not deal with it any further.However, the special case P 2 and o < c < 1 is particularly important and deserves special consideration.
4. FUNCTIONS IN L2(G).The origin of this section is a theorem proved in Titchmarsh ([3] Theorem 85, p. 117) for functions belonging to Lip(a,2) on the real line R.For further reference we state it as.
This theorem was studied rather extensively in [5] and [6] for functions in L2(R2), and L2(T2) respectively, where several conditions of the order of magnitude for the Fourier transforms j of f proved to be equivalent to one another.
In [4] (Theorems 5.1, 5.2) we proved an analogue of Theorem 4.1 for the Dini-Lipschitz functions in L2(R).In this section we shall prove Theorem 5.2 in [4] for functions in L2(G).THEOREM 4.2.Let f(z) belong to L2(G).Then the conditions w2(J',k 0(hO/(Log h)7), hG k ( are equivalent.Here h m -1. PROOF.That the first implication is tree follows from Theorem 3.1 where it is proved that I](,,)I o(,(f,))q n=m k Taking p q 2 and substituting for h rn -1 we obtain (4.2).We also hint that an argument based on the Parseval's identity similar to that of Titchmarsh's leads independently to the same result.To prove the converse let (4.2) hold.Then  Since G has the boundedness property (P); hence every Pk =mt + 1link is finite for all keN, the same is true of Log P k.Thus the right hand sides of (4.2) and (4.3) are the same.This applies to estimates of the form This is equivalent to (4.1) upon substituting for h m "1, heG t and the proof is complete.REMARK 4.2.We conclude by indicating that Theorem 5.1 in [4] is true for Vilenkin Fourier series, since, it can be deduced as a special case of Theorem 4.1.We also add that for 0 < c < 1, Theorems 3.1 and 4.1 of the present paper can be proved for higher differences of f(z)eLP(G).The statements and the proofs are almost straightforward and will not be given.
ACKNOWLEDGEMENT: This research was supported by a grant from Yarmouk University.

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To sum up, by the Parseval's identity one obtains[[f(z+h)-f(z)[2dz= _ [(n)[2[(n,h)-l G a Vilen oup.Then its du G is a &screte co.table to.ion oup.It is w o that one c intru on G a table bic t of n&ghurhs {Gn} of the identity element {e} of G such that t G=GoDG1, DG2,..., d G={e}.=o On the other hd, let V dote the lator in G of the suboup Gn in