Int. Journal of Math. Analysis, Vol. 6, 0, no. 0, 493-50 On Absolute Matrix Summability of Orthogonal Series Xhevat Z. Krasniqi, Hüseyin Bor, Naim L. Braha 3 and Marjan Dema 4,3 Department of Mathematics and Computer Sciences University of Prishtina Avenue Mother Theresa 5 Prishtinë 0000, Kosovë xheki00@hotmail.com nbraha@yahoo.com P.O. Box, 0650 Bahçelievler Ankara, Turkey hbor33@gmail.com 4 Faculty of Electrical and Computer Engineering University of Prishtina Bregu i Diellit, p.n., Prishtinë 0000, Kosovë marjan.dema@uni-pr.edu Abstract. In this paper, we prove two theorems on A k, k, summability of orthogonal series. The first one gives a sufficient condition under which an orthogonal series is absolutely summable almost everywhere, and the second one, is a general theorem, which also gives a sufficient condition so that an orthogonal series is absolutely summable almost everywhere, but it involves a positive numerical sequence that satisfies certain additional conditions. Besides, several known and new results are deduced as corollaries of the main results. Mathematics Subject Classification: 4C5, 40F05, 40D5, 40G99 Keywords: Orthogonal series, matrix summability. Introduction Let a n be a given infinite series with its partial sums {s n } and let A := a nv ) be a normal matrix, i.e. a lower triangular matrix of non-zero diagonal entries. Then A defines the sequence-to-sequence transformation,
494 Xhevat Z. Krasniqi et al mapping the sequence s := {s n } to As := {A n s)}, where A n s) := a nv s v, n = 0,,,... v=0 The series a n is said to be summable A k, k, if see [5]) n k ΔA n s) k converges, where and we write in brief ΔA n s) = A n s) A n s), a n A k. Then, let p denotes the sequence {p n }. For two given sequences p and q, the convolution p q) n is defined by p q) n = p m q n m = p n m q m. m=0 When p q) n = 0 for all n, the generalized Nörlund transform of the sequence {s n } is the sequence {t p,q n } obtained by putting t p,q n = p n m q m s m. p q) n m=0 m=0 The series a n is absolutely summable N, p, q) if the series t p,q n t p,q n converges, and is written in brief a n N, p, q. We note that N, p, q summability is introduced by Tanaka [3]. Let {φ n x)} be an orthonormal system defined in the interval a, b). We assume that fx) belongs to L a, b) and.) fx) c n φ n x), where c n = b fx)φ a nx)dx, n = 0,,,... ). Also is written see [4]) R n := p q) n, Rn j := p n m q m m=j
On absolute matrix summability 495 where Rn n+ = 0, Rn 0 = R n, and the following two theorems are proved. Theorem. [4]). If the series { R j n R n Rj n R n c n φ n x) is summable N, p, q almost everywhere. ) c j } Theorem. [4]). Let {Ωn)} be a positive sequence such that {Ωn)/n} is a non-increasing sequence and the series converges. Let {p nωn) n} and {q n } be non-negative. If the series c n Ωn)w ) n) converges, then the orthogonal series c nφ n x) N, p, q almost everywhere, where w ) n) is defined by w ) j) := j n=j n Rn j R n Rj n R n ). The main purpose of the present paper is to generalize Theorem. and Theorem. for A k summability of the orthogonal series.), where k. Before starting the main results first introduce some further notations. Given a normal matrix A := a nv ), we associate two lower semi matrices A := a nv ) and ˆA := ˆa nv ) as follows: a nv := a ni, n, i = 0,,,... and i=v ˆa 00 = a 00 = a 00, ˆa nv = a nv a n,v, n =,,... It may be noted that A and ˆA are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. The following lemma due to Beppo Levi see [6]) is often used in the theory of functions.we will need it to prove the main results. Lemma.3. If f n t) LE) are non-negative functions and.) f n t)dt <, then the series E f n t)
496 Xhevat Z. Krasniqi et al converges almost everywhere on E to a function ft) LE). Moreover, the series.) is also convergent to f in the norm of LE). Throughout this paper K denotes a positive constant that it may depends only on k, and be different in different relations. We prove the following theorems. Theorem.. If the series { n k. Main Results ˆa n,j c j } k converges for k, then the orthogonal series c n φ n x) is summable A k almost everywhere. Proof. For the matrix transform A n s)x) of the partial sums of the orthogonal series c nφ n x) we have v A n s)x) = a nv s v x) = c j φ j x) = v=0 c j φ j x) v=0 a nv a nv = v=j a nj c j φ j x) where v c jφ j x) is the partial sum of order v of the series.). Hence n ΔA n s)x) = a nj c j φ j x) a n,j c j φ j x) n = a nn c n φ n x) + a n,j a n,j ) c j φ j x) n = ˆa nn c n φ n x) + ˆa n,j c j φ j x) = ˆa n,j c j φ j x). Now let us clarify the reason of restriction k. From the definition of the A k summability is k. Since we shall use the Hölder s inequality with p = which must be greater than, then k < for k = we apply only k the orthogonality, until for k = we apply Schwarz s inequality). This is why we are focused only for k. The case when k > remains open.
On absolute matrix summability 497 Using the Hölder s inequality and orthogonality to the latter equality, we have that b a ΔA n s)x) k dx b a) k b b = b a) k = b a) k a a [ ) k A n s)x) A n s)x) dx ˆa n,j c j φ j x) dx ˆa n,j c j ] k. ) k Thus, the series.) b n k ΔA n s)x) k dx b a) k a n k [ ˆa n,j c j ] k converges since the last does by the assumption. From this fact and since the functions ΔA n s)x) k are non-negative, then by the Lemma.3 the series n k ΔA n s)x) k converges almost everywhere. With this the proof of Theorem. is completed. If we put.) w k) A; j) := j k n k ˆan,j n=j then the following theorem holds true. Theorem.. Let k and {Ωn)} be a positive sequence such that {Ωn)/n} is a non-increasing sequence and the series converges. If nωn) the following series c n Ω k n)w k) A; n) converges, then the orthogonal series c nφ n x) A k almost everywhere, where w k) A; n) is defined by.).
498 Xhevat Z. Krasniqi et al Proof. Applying Hölder s inequality to the inequality.) we get that b n k ΔA n s)x) k dx a K n k [ = K nωn)) k K nωn)) { K c j K { n=j ˆa n,j c j ] k [ nω k n) ) k ) Ωj) c j k j [ nω k n) nω k n) ˆa n,j } k n=j { = K c j Ω k j)w k) A; j) ˆa n,j c j ] k n k ˆan,j } k } k, ˆa n,j c j ] k which is finite by virtue of the hypothesis of the theorem. Now the the proof is an immediate result of the Lemma.3. The following corollaries follow from the main results k = ). Corollary.3. If the series { } ˆa n,j c j c n φ n x) is summable A almost everywhere. Corollary.4. Let {Ωn)} be a positive sequence such that {Ωn)/n} is a non-increasing sequence and the series converges. If the series nωn) c n Ωn)w ) A; n) c nφ n x) A almost everywhere, where w ) A; j) = j n=j n ˆa n,j.
On absolute matrix summability 499 3. Applications of the main results We can specialize the matrix A = a nv ) obtaining these means as follows. C, ) mean, when a n,v = n+ ;. Harmonic means, when a n,v = n v+) log n ; 3. C, α) means, when a n,v = n v+α+ α ) n+α α ) 4. H, p) means, when a n,v = log p n+) ; p m=0 logm v + ); 5. Nörlund means N, p n ), when a n,v = p n v P n where P n = n v=0 p v; 6. Riesz means R, p n ), when a n,v = pv P n ; 7. Generalized Nörlund means N, p, q), when a n,v = p n vq v R n where R n = n v=0 p vq n v. Let us prove now that some of known results are included in Theorem.. Namely, for a n,v = p n v P n we get ˆa n,j = a n,j a n,j = p n i n P n P n = = = i=j i=j p n i P n P n j P n P n j ) P n P n P n p n )P n j P n P n j p n j )) P n P n p n Pn P ) n j p n j. P n P n p n p n j Hence, using Theorem. for k = the following result holds. Corollary 3. []). If the series p n P n P n { p n j c n φ n x) is summable N, p almost everywhere. Pn P ) } n j c j p n p n j
500 Xhevat Z. Krasniqi et al Also, for a n,v = qv Q n one can find that ˆa n,j = a n,j a n,j = q nq j Q n Q n. Therefore, using again Theorem. for k = we obtain Corollary 3. []). If the series q n Q n Q n { c n φ n x) is summable R, q almost everywhere. Q j c j } Remark 3.3. Theorem. and Theorem. are included in Theorem. and Theorem., respectively. It is enough to consider in Theorem. and Theorem., the case k =, and the matrix A = a nv ) with its entries a n,v = p n v q v R n. Some other consequences of the main results are corollaries formulated below. Corollary 3.4. If for k the series ) k { n k p n p Pn n j P ) } k n j c j P n P n p n p n j c n φ n x) is summable N, p k almost everywhere. Remark 3.5. We note that:. If p n = for all values of n, then N, p k summability reduces to C, k summability. If k = and p n = /n + ),then N, p k is equivalent to R, log n, summability. Corollary 3.6. If for k the series ) k { n k } k q n Q Q n Q j c j n
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