Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 27 (2011), ISSN

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1 Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 27 20, ISSN CERTAIN ESTIMATES FOR DOUBLE SINE SERIES WITH MULTIPLE-MONOTONE COEFFICIENTS XHEVAT Z. KRASNIQI Abstract. In this paper we obtain estimates of the sum of double sine series near the origin, with multiple-monotone coefficients tending to zero. These estimates extend some results of Telyakovski [] and Popov [7] from single to multidimensional case.. Introduction and Preliminaries Many authors considered the sine series gx : a n sin nx with monotone coefficients tending to zero. Young [3] was the first to consider the problem of estimates of gx for x 0 expressed in terms of the coefficients a n. Then Salem [8], [9], Shogunbekov [0], Aljančić, Bojanić and Tomić [] considered this problem, as well. Later Telyakovski in [] has proved the following fact: Theorem T. Assume that a n 0. Then for x π, π m m] Im, m, 2,... the following estimate is valid: gx m na n x O m 3 Likewise, among others Popov [7] has proved m n 3 a n. Theorem P. For any nonincreasing sequence of positive numbers a k tending to zero, the following inequality holds: 2 a sin x 2 gx, x 0, π]. 200 Mathematics Subject Classification. 42A20, 42A6. Key words and phrases. sine series, asymptotic behaviour, multiple-monotone coefficients. 233

2 234 XHEVAT Z. KRASNIQI The problem shown above is considered by the present author [4] [6] as well. The object of this work is to extend these two theorems, formulated above, from single to multidimensional case. In fact, we shall investigate the behavior near the origin of the sum of double sine series with multiple-monotone coefficients. Let us now consider the following double sine series. a m,n sin mx sin ny whose coefficients satisfy conditions a m,n 0 for m and all n fixed, and for n and all m fixed. For k 0, k 2 0 denote k,0a m,n i k a mi,n, i i0 0,k2 a m,n j0 i0 j k2 j j0 a m,nj, k,k 2 a m,n ij k i k2 j a mi,nj. Parallel with series. we consider the series of the form.2 k,k 2 a m,n B k m xb k 2 n y, where Let.3 B rx D r x sin x sin 2x sin rx, r ; B ν rx B ν µ x, ν 2, 3,.... µ be double numerical series and S m,n ν µ n c µ,ν m ν µ its rectangular partial sums. If there exists a number S such that for all ε > 0 there exist natural numbers k and l such that S m,n S < ε c µ,ν

3 CERTAIN ESTIMATES FOR DOUBLE SINE SERIES for all n > k and m > l, then series.3 converges in Pringsheim s sense to the number S see [2], page 27. It s well-known see [3] that if sequence of numbers {a k,l } satisfies conditions A and k,k 2 a k,l 0 for all k and l, then: Lemma.. Series. converges almost everywhere in Pringsheim s sense, in other words there exists a function gx, y such that the sum of series. is gx, y. 2 Series.2 converges almost everywhere in Pringsheim s sense to gx, y. Throughout this paper the O expressions contain positive constants and they may depend only on k and k Main Results To prove our main results we need first the following lemma. Lemma 2.. For ν, 2, 3,... and x 0, π] the following inequality is true B ν rx rν 2 sin x 2. Proof. For the proof we shall use mathematical induction. Namely, for ν we have B rx D r x cos x cos r 2 2 x 2 sin x cos x 2 2 sin x 2 2 Assuming that 2 tan x 4 2 sin x 2. we obtain B ν r x rν 2 2 sin x 2 B ν rx µ B ν µ x 2 sin x 2 which completes the proof of the Lemma 2.. µ µ ν 2 rν 2 sin x 2, We shall prove two theorems. The first theorem gives an estimate of the sum gx, y near the origin from above, while the second one gives the estimate from below. Theorem 2.2. Assume that {a m,n } satisfies conditions A and k,k 2 a m,n 0, for k, k 2. Then for x I r and y I l, r, l, 2,... the following

4 236 XHEVAT Z. KRASNIQI estimate is valid 2. gx, y O mnxya m,n rn 2 mn 2 lm 2 j rl 3 l 2 n 2 m 3 n r 4 i l 3 i,0 a m,n r 2 m 2 mn 3 r 3 l 4 j 0,j a m,n j mn 3 r 4 i l 4 j i,j a m,n mna m,n. Proof. By the Lemma. we have 2.2 gx, y nl k,k 2 a m,n B k m xb k 2 n y mr nl mr k,k 2 a m,n B k m xb k 2 n y k,k 2 a m,n B k m xb k 2 n y k,k 2 a m,n B k m xb k 2 n y A k,k 2 l,r x, y A k,k 2,r x, y A k,k 2 l, x, y Ak,k 2, x, y. The expression A k,k 2 l,r x, y can be written as follows 2.3 A k,k 2 l,r x, y [ k,k 2 a m,n B k m x where H k,k 2 r,n x r k,k 2 a m,n B k m x. Since ] B k 2 n y B s mx B s m x B s m x, s 2, 3,... k, H k,k 2 r,n xb k 2 n y,

5 then H k,k 2 r,n x. CERTAIN ESTIMATES FOR DOUBLE SINE SERIES [ k,k 2 a m,n k,k 2 a m,n ] B k m x [ ] k,k 2 a m,n B k m x B k m x k,k 2 a r,n B k r x k,k 2 a m,n B k m x k,k 2 a r,n B k r x k 2,k 2 a m,n B k 2 m x ik i,k2 a r,n B i rx 0,k2 a m,n sin mx i,k2 a r,n B i rx. Applying the relation sin u uou 3, as u 0, i,k2 a r,n i,k2 a r,n by our assumption, the well-known estimate B i rx O r i for i, x [0, π], and x I r we obtain 2.4 H k,k 2 r,n x m 0,k2 a m,n x O r 3 m 3 0,k2 a m,n r i i,k2 a r,n. In a similar way using the same arguments as for H k,k 2 r,n x we arrive at the following estimates y I l : and 0,k2 a m,n B k 2 n y 0,0 a m,n sin ny 0,j a m,l B j ly na m,n y O l 3 i,k2 a r,n B k 2 n y n i,0 a r,n y O l 3 j n 3 a m,n l j 0,j a m,l, j n 3 i,0 a r,n l j i,j a r,l. j

6 238 XHEVAT Z. KRASNIQI Using 2.3, 2.4, and last estimates we get 2.5 A k,k 2 l,r x, y mx O r 3 r i r 0,k2 a m,n B k 2 n y m 3 0,k2 a m,n B k 2 n y i,k2 a r,n B k 2 n y mnxya m,n O j rl 3 l rl 3 ml j 0,j a m,l r 3 l mn 3 a m,n r 3 nr i i,0 a r,n l 3 r i l j i,j a r,l. j mn 3 a m,n m 3 na m,n m 3 l j 0,j a m,l j n 3 r i i,0 a r,n But r ml j 0,j a m,l r j r ml j l 0,j a m,l j ml j 4 l 3 j j 4 rl 4 j n 3 0,j a m,l mn 3 0,j a m,n,

7 CERTAIN ESTIMATES FOR DOUBLE SINE SERIES and in a similar manner we can find m 3 l j 4 r 3 0,j a m,l r 3 l 4 j j l 3 l nr i i,0 a r,n n 3 r i i,0 a r,n Therefore, 2.6 j j 4 r 4 i l 4 r 4 i l 3 r i l j i,j a r,l A k,k 2 l,r x, y O j mnxya m,n 6 r 4 i l 4 j mn 3 0,j a m,n, m 3 n i,0 a m,n, mn 3 i,0 a m,n, rn 2 mn 2 lm 2 j rl 3 l 2 n 2 m 3 n r 4 i l 3 i,0 a m,n r 2 m 2 mn 3 r 3 l 4 j 0,j a m,n j mn 3 r 4 i l 4 j i,j a m,n mn 3 i,j a m,n. mna m,n Now, we estimate A k,k 2,r x, y. Indeed, since B k 2 n y O y 2 k for y 0, π], 2.4, and x I r, y I l, then 2.7 A k,k 2,r x, y O y k 2 r O l k 2 r O r k,k 2 a m,l B k m x k,k 2 a m,l B k m x ml k 2 0,k2 a m,l m 3 l k 2 r 3 0,k2 a m,l r i l k 2 i,k2 a r,l..

8 240 XHEVAT Z. KRASNIQI and and It is not difficult to prove that r 3 r ml k 2 0,k2 a m,l m 3 l k 2 0,k2 a m,l From last two estimates we have r ml k 2 0,k2 a m,l r 3 j j 4 rl 4 j 4 r 3 l 4 j m 3 l k 2 0,k2 a m,l j r i l k 2 i,k2 a r,l r i rl k 2 i,k2 a r,l r i l k 4 2 r 3 mn 3 0,j a m,n. mn 3 0,j a m,n. 4 r 2 m 2 mn 3 r 3 l 4 j 0,j a m,n, m 3 i,k2 a m,l r i l k 2 6 rl 3 j mn 3 i,k2 a m,l 6mn 3 r 4 i l 4 j i,j a m,n. Therefore, from these estimates and 2.7 we obtain 2.8 A k,k 2,r x, y O k2 j r 2 m 2 mn 3 r 3 l 4 j 0,j a m,n j mn 3 r 4 i l i,j a 4 j m,n.

9 CERTAIN ESTIMATES FOR DOUBLE SINE SERIES In a similar manner we can find the following estimate k 2.9 A k,k 2 l, x, y O n 2 l 2 m 3 n r 4 i l 3 i,0 a m,n j mn 3 r 4 i l i,j a 4 j m,n. In the end from B k m x O x k k 2, B n y O y 2 k for x, y 0, π], and x I r, y I l we obtain 2.0 A k,k 2, x, y O x k y k 2 nl mr k,k 2 a m,n O r k l k 2 k,k2 a r,l. By virtue of the monotonicity of k,k 2 a r,l, for k, k 2, we get 2. r k l k 2 k,k 2 a r,l r k l k2 rl k,k 2 a r,l 6 r 4 k l 4 k 2 mn 3 k,k 2 a r,l 6 r 4 k l 4 k 2 j mn 3 k,k 2 a m,n 6 mn 3 r 4 i l 4 j i,j a m,n. Thus, from 2.0 and 2. we get 2.2 A k,k 2, x, y O k j mn 3 r 4 i l i,j a 4 j m,n. Inserting 2.6, 2.8, 2.9 and 2.2 into 2.2 we obtain 2.. As a direct consequence of the theorem 2.2 is the following corollary:

10 242 XHEVAT Z. KRASNIQI Corollary 2.3. [4] Assume that {a m,n } satisfies conditions A and, a m,n 0. Then for x I r and y I l, r, l, 2,... the following estimate is valid 2.3 gx, y mnxya m,n O mn 3 a rl 3 m,n r 3 l m 3 na m,n rl 3 mn 3 a m,n. Now we prove the statement that gives the estimate of gx, y from below. Theorem 2.4. Assume that a k,l satisfy conditions A and k,k 2 a k,l 0. Then the following estimate is valid sin x 2 sin y m k n k2 k,k 2 2 a m,n gx, y. Proof. Based on Lemma. and Lemma 2. we immediately obtain 2.4. Theorem 2.4, for k, k 2, implies the following statement proved in [4]. Corollary 2.5. Assume that a k,l satisfy conditions A and, a k,l 0. Then the following estimate is valid a, 4 sin x 2 sin y gx, y. 2 References [] S. Aljančić, R. Bojanić, and M. Tomić. Sur le comportement asymptotique au voisinage de zéro des séries trigonométriques de sinus à coefficients monotones. Acad. Serbe Sci. Publ. Inst. Math., 0:0 20, 956. [2] N. K. Bari. Trigonometricheskie ryady. With the editorial collaboration of P. L. Ul janov. Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 96. [3] H. G. Hardy. On double fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters. Quarterly J. Math., 37:53 79, 906. [4] Xh. Z. Krasniqi. On the behavior near the origin of double sine series with monotone coefficients. Math. Bohem., 343: , [5] Xh. Z. Krasniqi. Some estimates of r-th derivative of the sums of sine series with monotone coefficients of higher order near the origin. Int. J. Math. Anal. Ruse, 3-4:59 69, [6] Xh. Z. Krasniqi and N. L. Braha. On the behavior of r-derivative near the origin of sine series with convex coefficients. JIPAM. J. Inequal. Pure Appl. Math., 8:Article 22, 6 pp. electronic, [7] A. Y. Popov. Estimates for the sums of sine series with monotone coefficients of certain classes. Mat. Zametki, 746: , [8] R. Salem. Détermination de l ordre de grandeur à l origine de certains séries trigonométriques. C. R. Acad. Sci. Paris, 86: , 928. [9] R. Salem. Essais sur les séries trigonométriques. Hermann et Cie., Paris, 940. Thèse presentée à la Faculté des Sciences de l Université de Paris.

11 CERTAIN ESTIMATES FOR DOUBLE SINE SERIES [0] S. S. Shogunbekov. Certain estimates for sine series with convex coefficients. In Application of functional analysis in approximation theory Russian, pages [] S. A. Teliakovsky. On the behavior near the origin of the sine series with convex coefficients. Publ. Inst. Math. Beograd N.S., 5872:43 50, 995. [2] E. T. Whittaker and G. N. Watson. A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions. Fourth edition. Reprinted. Cambridge University Press, New York, 962. [3] W. H. Young. On the mode of oscillation of a fourier series and of its allied series. Lond. Math. Soc. Proc., 22: , 93. Received October 3, Department of Mathematics and Computer Sciences, University of Prishtina, Ave. Mother Teresa 5, Prishtinë, 0000, Republic of Kosova address: xhevat-z-krasniqi@hotmail.com