Commentationes Mathematicae Universitatis Carolinae

Transcription

1 Commentationes Mathematicae Universitatis Carolinae Bogdan Szal Trigonometric approximation by Nörlund type means in L p -norm Commentationes Mathematicae Universitatis Carolinae, Vol. 5 (29), No. 4, Persistent URL: Terms of use: Charles University in Prague, Faculty of Mathematics and Physics, 29 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 Comment.Math.Univ.Carolin. 5,4(29) Trigonometric approximation by Nörlund type means in L p -norm Bogdan Szal Abstract. We show that the same degree of approximation as in the theorems proved by L. Leindler [Trigonometric approximation in L p -norm, J. Math. Anal. Appl. 32 (25), 29 36] and P. Chandra [Trigonometric approximation of functions in L p -norm, J. Math. Anal. Appl. 275 (22), 3 26] is valid for a more general class of lower triangular matrices. We also prove that these theorems are true under weakened assumptions. Keywords: class Lip(α, p), trigonometric approximation, L p -norm Classification: 42A, 4A25. Introduction Let f be 2π-periodic and f L p [, 2π] = L p for p. Denote by S n (f) = S n (f; x) = a n 2 + (a k coskx + b k sinkx) k= n U k (f; x) the partial sum of the first (n + ) terms of the Fourier series of f L p (p ) at a point x, and by ω p (f; δ) = { sup < h δ 2π 2π } f(x + h) f(x) p p dx the integral modulus of continuity of f L p. If, for α >, ω p (f; δ) = O(δ α ), then we write f Lip(α, p) (p ). Throughout the paper L p will denote L p -norm, defined by f L p = { 2π 2π } f(x) p p dx (f L p (p )). We shall consider the approximation of f L p by trigonometrical polynomials T n (f; x), where T n (f; x) = T n (f, A; x) := n a n,n k S k (f; x) (n =,, 2,...)

3 576 B. Szal and A := (a n,k ) is a lower triangular infinite matrix of real numbers such that: (.) a n,k for k n, a n,k = for k > n (k, n =,, 2...) and (.2) n a n,k = (n =,, 2...). If a n,n k = p n k P n, where P n = p + p + + p n (n ), then we denote the corresponding trigonometrical polynomials by N n (f; x) = P n n p n k S k (f; x) (n =,, 2...). The case a n,k = n+ for k n and a n,k = for k > n of T n (f; x) yields σ n (f; x) = n + We shall also use the notations n S k (f; x) (n =,, 2...). a k = a k a k+, k a n,k = a n,k a n,k+ and we shall write I I 2 if there exists a positive constant K such that I KI 2. A nonnegative sequence c := (c n ) is called almost monotone decreasing (increasing) if there exists a constant K := K(c), depending on the sequence c only, such that for all n m c n Kc m (Kc n c m ). For such sequences we shall write c AMDS and c AMIS, respectively. When we write that a sequence (a n,k ) belongs to one of the above classes, it means that it satisfies some of the above conditions with respect to k =,, 2,..., n for all n. Let A n,k = n k+ i=n k a n,i. If (A n,k ) AMDS ((A n,k ) AMIS), then we shall say that (a n,k ) is an almost monotone decreasing (increasing) upper mean sequence, briefly (a n,k ) AMDUMS ((a n,k ) AMIUMS). A sequence c := (c n ) of nonnegative numbers will be called the head bounded variation sequence, or briefly c HBVS, if it has the property (.3) m c k K(c)c m for all natural numbers m, or only for all m N if the sequence c has only finite nonzero terms and the last nonzero term is c N.

4 Trigonometric approximation by Nörlund type means in L p -norm 577 Therefore we assume that the sequence (K(α n )) n= there exists a constant K such that is bounded, that is, that K (α n ) K holds for all n, where K(α n ) denote the sequence of constants appearing in the inequalities (.3) for the sequence α n := (a nk ). Now we can consider the conditions to be used later on. We assume that for all n and m n m holds if α n := (a nk ) belongs to HBVS. It is clear that and k a nk Ka nm NDS HBVS AMIS NIS AMDS, where NDS (NIS) is the class of nonnegative and nondecreasing (nonincreasing) sequences. In the present paper we show some embedding relations between the classes AMDS and AMDUMS and the classes AMIS and AMIUMS. In 937 E. Quade [8] proved that, if f Lip(α, p) for < α, then σ n (f) f L p = O(n a ) for either p > and < α or p = and < α <. He also showed that, if p = α =, then σ n (f) f L = O(n log(n + )). There are several generalizations of the above result for p > (see, for example [], [2], [3], [6] and [7]). In [4], P. Chandra extended the work of E. Quade and proved the following theorems: Theorem. Let f Lip(α, p) and let (p n ) be positive such that (.4) (n + )p n = O (P n ). If either (i) p >, < α, and (ii) (p n ) is monotonic, or (i) p =, < α <, and (ii) (p n ) is nondecreasing, then N n (f) f L p = O ( n α). Theorem 2. Let f Lip(, ) and let (p n ) with (.4) be positive, and ((n + ) η p n ) NDS for some η >.

5 578 B. Szal Then R n (f) f L = O ( n ), where R n (f; x) = P n n p ks k (f; x) and P n = p + p + + p n (n ). In [5] L. Leindler obtained the same degree of approximation as in Theorem, where the conditions of monotonicity were replaced by weaker assumptions. Namely, he proved the following theorem: Theorem 3. Let f Lip(α, p) and let (p n ) be positive. If one of the conditions (i) p >, < α < and (p n ) AMDS, (ii) p >, < α <, (p n ) AMIS and (.4) holds, (iii) p >, α = and k= k p k = O(P n ), (iv) p >, α =, p k = O(P n /n) and (.4) holds, (v) p =, < α < and k= p k = O(P n /n) holds, then N n (f) f L p = O ( n α). In this paper we shall show that the same degree of approximation as in the above theorems is valid for a more general class of lower triangular matrices. We shall also prove that the cited theorems are true with weakened assumptions. 2. Statement of the results Our main results are the following. Theorem 4. The following properties hold: (i) if (a n,k ) AMDS, then (a n,k ) AMIUMS, (ii) if (a n,k ) AMIS, then (a n,k ) AMDUMS, (iii) if a n,k = O(n ), then A n,k = O(n ), (iv) if k= k a n,k = O(), then A n,k = O(n ). Theorem 5. Let f Lip(α, p) and (.), (.2) hold. If one of the conditions (i) p >, < α < and (a n,k ) AMDUMS, (ii) p >, < α <, (a n,k ) AMIUMS and (n + )a n,n = O(), (iii) p >, α = and ka n,k = O(n ), (iv) p =, < α < and k= a n,k = O(n ), (v) p = α =, ((k + ) β a n,n k ) HBVS for some β > and (n + )a n, = O() holds, then (2.) T n (f) f L p = O ( n α). In the particular case a n,k = p k P n, where P n = p + p + + p n and P n,k = n k+ i=n k p i, we can derive from Theorem 5 the following corollary:

6 Trigonometric approximation by Nörlund type means in L p -norm 579 Corollary. Let f Lip(α, p) and let (p k ) be positive. If one of the conditions (i) p >, < α < and (p k ) AMDUMS, (ii) p >, < α <, (p k ) AMIUMS and (n + )p n = O(P n ), (iii) p >, α = and kp n,k = O(P n /n), (iv) p =, < α < and k= p k = O(P n /n), (v) p = α =, ((k+) β p n k ) HBVS for some β > and (n+)p = O(P n ) holds, then N n (f) f L p = O ( n α). Remark. By Theorem 4 we can observe that Theorem 3 and consequently Theorem follow from Corollary (i) (iv). Moreover, since NDC HBVS we can derive from Corollary (v) an analogous estimate as in Theorem 2 for the deviation N n (f) f in L p -norm. 3. Auxiliary results We shall use the following lemmas in proofs of Theorems 4 and 5. Lemma ([8, Theorem 4]). If f Lip(α, p), p, < α, then, for any positive integer n, f may be approximated in L p -space by a trigonometrical polynomial t n of order n such that f t n L p = O ( n α). Lemma 2 ([8, Theorem 5(i)]). If f Lip(α, ), < α <, then σ n (f) f L = O ( n α). Lemma 3 ([8, p. 54, last line]). If f Lip(, p) (p > ), then σ n (f) S n (f) L p = O ( n ). Lemma 4 ([8, Theorem 6(i), p. 54]). If f Lip(α, p), < α, p >, then S n (f) f L p = O ( n α). Lemma 5. Let (.) and (.2) hold and let one of the following assumptions (i) (a n,k ) AMDUMS, (ii) (a n,k ) AMIUMS and (n + )a n,n = O() be satisfied. Then n (k + ) α a n,n k = O ( (n + ) α) holds for all < α <.

7 58 B. Szal Proof: Let r = [ n 2 ]. Then, if (.) and (.2) hold, n (k + ) α a n,n k r n (k + ) α a n,n k + (r + ) α r (k + ) α a n,n k + (r + ) α. By Abel s transformation and (.2), we get n r (k + ) α { a n,n k (k + ) α (k + 2) α} k r + (r + ) α a n,n k + (r + ) α r { (k + ) α (k + 2) α} n r i=n k a n,n i k=r+ (k + 2) α (k + ) α (k + ) α (k + 2) α A n,k + 2(r + ) α. a n,n k a n,i + 2 (r + ) α Using Lagrange s mean value theorem for the function f(x) = x α ( < α < ) on the interval (k +, k + 2) we obtain n (k + ) α a n,n k If (a n,k ) AMDUMS, then n r α (k + 2) α A n,k + 2 (r + ) α. r (k + ) α a n,n k A n,r + (r + ) α (k + 2) α r + n k=n r r a n,k (k + 2) α + (r + ) α (r + ) α (n + ) α. When (a n,k ) AMIUMS and (n + )a n,n = O() we get n r (k + ) α a n,n k A n, (k + 2) α + (r + ) α r a n,n (k + 2) α + (r + ) α

8 Trigonometric approximation by Nörlund type means in L p -norm 58 a n,n (r + ) α + (r + ) α (n + ) α. This ends the proof. 4. Proofs of the results 4. Proof of Theorem 4. (i) If (a n,k ) AMDS, then Ka n,m a n,l for m l and all n. We prove that A n,m max{, K 2 }A n,l for m l and all n. For m = l this is true. Let m < l. Then Thus (l + ) n m + and (a n,k ) AMIUMS. a n,i = (m + ) n (m + ) (m + ) n { n { n max {, K 2} (m + ) a n,i + (l m) n a n,i a n,i + K (l m)a n,n m } n m a n,i + K 2 n i=n l a n,i max {, K 2} l + i=n l a n,i. n i=n l a n,i a n,i } (ii) Let (a n,k ) AMIS. We get that a n,m Ka n,l for m l and all n. We show that min{, }A n,m A n,l for m l and all n. If m = l, then it is true. K Suppose m < l, then 2 we have (l + ) n a n,i = (m + ) (m + ) (m + ) { min, n { n { n K 2 a n,i + (l m) n a n,i a n,i + K (l m)a n,n m } a n,i + n m K 2 a n,i } (m + ) i=n l n a n,i. i=n l }

9 582 B. Szal Hence min {, K 2 } m + n and (a n,k ) AMDUMS. (iii) An elementary calculation yields that (4.) (4.2) Clearly, A n,k = n i=n k = (k + )(k + 2) n (k + 2) i=n k (k + )(k + 2) a n,i (k + )a n,n k = holds for any k n. Using (4.2) and (4.) we have A n,k = (k + )(k + 2) (k + )(k + 2) a n,i l + a n,i (k + ) n i=n k n i=n l n a n,i i=n k a n,i a n,i (k + )a n,n k. k (i + )(a n,n i a n,n i ) k (i + )(a n,n i a n,n i ) k (i + ) a n,n i a n,n i (k + ) a n,n k a n,n k (i + )(i + 2) i=k a n,n k a n,n k = a n,k a n,k+. If a n,k = O(n ), then we obtain that A n,k = O(n ), too. (iv) Let r = [ n 2 ]. Using (4.2) we get A n,k = r (k + )(k + 2) (k + )(k + 2) k (i + ) a n,n i a n,n i k (i + ) a n,n i a n,n i

10 Trigonometric approximation by Nörlund type means in L p -norm (k + )(k + 2) k=r An elementary calculation yields that If k (i + ) a n,n i a n,n i := I + I 2. r I (k + ) a n,n k a n,n k (i + )(i + 2) r a n,n k a n,n k = n k=n r k a n,k. i=k k=n r a n,k a n,k+ (4.3) k a n,k = O(), k= then I = O(n ). On the other hand, I 2 = k=r + (k + )(k + 2) { r (i + ) a n,n i a n,n i } k (i + ) a n,n i a n,n i := I 2 + I 22. i=r Furthermore, using again (4.3), we get and I 22 I 2 k=r n k=r r (k + ) k=n r (k + ) a n,n i a n,n i n r r + k a n,k = O ( n ) k i=r a n,n i a n,n i r + n r k a n,k a n,k+ = ( n ). n k= k=n r k=r i=r a n,k a n,k+ k a n,n i a n,n i

11 584 B. Szal Summing up our partial result, we obtain that A n,k = O(n ). The proof is complete. 4.2 Proof of Theorem 5. We prove the cases (i) and (ii) together using Lemmas 4 and 5. Since we have T n (f) f L p T n (f; x) f(x) = n a n,n k (S k (f; x) f(x)), n a n,n k S k (f) f L p n (k + ) α a n,n k = O ( n α) and this is (2.). Next we consider the case (iii). Using two times Abel s transformation and (.2) we get T n (f; x) f(x) = n a n,n k (S k (f; x) f(x)) = (S k (f; x) S k+ (f; x)) = (S k (f; x) S k+ (f; x)) k a n,n i + S n (f; x) f(x) n i=n k a n,i + S n (f; x) f(x) = S n (f; x) f (x) (k + )U k+ (f; x)a n,k = S n (f; x) f (x) (A n,k A n,k+ ) A n, (k + )U k+ (f; x) = S n (f; x) f(x) (A n,k A n,k+ ) n n i= (k + )U k+ (f; x). a n,i k (i + )U i+ (f; x) k (i + )U i+ (f; x)

12 Trigonometric approximation by Nörlund type means in L p -norm 585 Hence (4.4) Since T n (f) f L p S n (f) f L p k+ + A n,k A n,k+ iu i (f) i= + n ku k (f; x) n by Lemma 3 we get n (4.5) ku k (f) k= k= σ n (f; x) S n (f; x) = n + L p By (4.4), (4.5) and Lemma 4 we have that If ka n,k = O(n ), then and (2.) holds. Now, we prove the cases (iv). By Abel s transformation T n (f; x) f(x) = L p. n ku k (f; x), k= L p = (n + ) σ n (f) S n (f) L p = O (). T n (f) f L p n + A n,k A n,k+. T n (f) f L p = O ( n ) n a n,n k (S k (f; x) f(x)) = (a n,n k a n,n k ) k (S i (f; x) f(x)) n + a n, (S k (f; x) f(x)) = (a n,n k a n,n k )(k + )(σ k (f; x) f(x)) + a n, (n + )(σ n (f; x) f(x)).

13 586 B. Szal Using Lemma 2 we get T n (f) f L a n,n k a n,n k (k + ) σ k (f) f L + a n, (n + ) σ n (f) f L a n,n k a n,n k (k + ) α + a n, (n + ) α ( ) (n + ) α a n,n k a n,n k + a n, = (n + ) α k= a n,k, where a n, =. When the assumptions (iv) hold we get T n (f) f L = O ( n α). This ends the proof of the case (iv). Finally, we prove the case (v). Let t n be a trigonometrical polynomial of Lemma of the present paper. Then for m n, S m (t n ; x) = t m and S m (f; x) t m = S m (f t n ; x). Thus n n T n (f; x) a n,n k t k (x) = a n,n k S k (f t n ; x), where S k (f t n ; x) = π 2π {f(x + u) t n (x + u)} sin ( ) k + 2 u 2 sin u du. 2 By the general form of Minkowski inequality we get

14 Trigonometric approximation by Nörlund type means in L p -norm 587 (4.6) T n(f) 2π 2 = 2π 2 2π 2π n = π f t n L a n,n k t k L K n (u) du K n (u) 2π π 2π 2π f(x + u) t n (x + u) dx f(x) t n (x) dx K n (u) du = 2 π f t n L K n (u) du ( π/n = 2 π f t n L π K n (u) du + K n (u) du π/n ) where = 2 π f t n L (I + I 2 ), K n (u) = n sin ( ) k + 2 u a n,n k 2 sin u. 2 Now, we estimate the quantities I and I 2. By (.2) (4.7) I π/n n (k + )a n,n k du = O(). If ((k + ) β a n,n k ) HBVS, then ((k + ) β a n,n k ) AMIS. Hence, for l m n, ( ) β m + Ka n,n m a n,n l a n,n l. l + Thus (a n,n k ) AMIS. Using this and the assumption (n + )a n, = O() we obtain that (4.8) I 2 a n, π π/n u 2 du = O(). Combining (4.6) (4.8) we have n (4.9) T n (f) a n,n k t k f t n L. L

15 588 B. Szal Further, by using (4.9) and Lemma for p = α =, we get T n (f) f L By Abel s transformation n T n n(f) a n,n k t k L + a n,n k t k f L n n + a n,n k t k f n n + a n,n k t k f L L n n + a n,n k (k + ). T n (f) f L n + a n,n k (k + ) β a n,n k (k + 2) β + a n, (n + ) β n (k + ) β k (i + ) β n + (n + )β a n,n k (k + ) β a n,n k (k + 2) β + a n,. Since ((k + ) β a n,n k ) HBVS and (n + )a n, = O(), we obtain T n (f) f L = O ( n ) and (2.) holds. This completes the proof of Theorem 5. References [] Chandra P., Approximation by Nörlund operators, Mat. Vestnik 38 (986), [2] Chandra P., Functions of classes L p and Lip(α, p) and their Riesz means, Riv. Mat. Univ. Parma (4) 2 (986), [3] Chandra P., A note on degree of approximation by Nörlund and Riesz operators, Mat. Vestnik 42 (99), 9. [4] Chandra P., Trigonometric approximation of functions in L p -norm, J. Math. Anal. Appl. 275 (22), [5] Leindler L., Trigonometric approximation in L p -norm, J. Math. Anal. Appl. 32 (25), [6] Mohapatra R.N., Russell D.C., Some direct and inverse theorem in approximation of functions, J. Austral. Math. Soc. (Ser. A) 34 (983), [7] Sahney B.N., Rao V.V., Error bounds in the approximation of functions, Bull. Austral. Math. Soc. 6 (972), 8.

16 Trigonometric approximation by Nörlund type means in L p -norm 589 [8] Quade E.S., Trigonometric approximation in the mean, Duke Math. J. 3 (937), University of Zielona Góra, Faculty of Mathematics, Computer Science and Econometrics, ul. Szafrana 4a, Zielona Góra, Poland B.Szal@wmie.uz.zgora.pl (Received November 2, 28, revised October 2, 29)