Németh, József

English

Németh József

University of Szeged

Acta Sei. Math., 54 (1990), 291—304 Fourier series with positive coefficients and generalized Lipschitz classes J. NÉMETH*) 1 . L . LEINDLER ( [ 3 ] — [ 6 ] ) investigated the relations between function classes defined by the rate of strong approximation of functions by Fourier series and the classes determined by the modulus of continuity of the functions. Following G . G . LORENTZ [ 7 ] and R . P . BOAS [ 2 ] we shall prove theorems giving coefficient-conditions assuring that a function belong to function classes defined in terms of modulus of continuity by L. Leindler. Combining these results and those of L. Leindler mentioned above we can get coefficient-conditions for a function to belong to a function class defined by the rate of strong approximation by Fourier series. 2. Before formulating our results we give a couple of definitions, notations and theorems. Let f(x) be a continuous and 2^-periodic function and let be its Fourier series. Denote by sn=s„(x)=s„(f;x) the n-th partial sum of (1). For any positive /? and p L . LEINDLER [ 3 ] defined the following strong means and function classes *) This research was made partly while the author visited to the Ohio State University, Colum-bus, U.S.A. in academic years 1985—86 and 1986—87. Received June 27, 1988. (1) /(*) — 2 (akCoskx+bksin kx) £o 2 k=1 H(ß,p, CO) = {/: A„</; ß\P) = O(co(lln))}, 292- J. Németh where [| • || denotes the usual maximum norm and a>(S) is a modulus of continuity having the following properties co(0) = 0, « ( ^ + ¿ 2 ) ^ ( » ( ¿ j ) + c o ( 5 2 ) for any 0 S ^ S ^ s ^ 2 it. Furthermore we consider the following function classes H<° = {/: || f(x+h) -/(*)! - 0(co(h))} (H<°)*={f: \\f(x+h)+f(x-h)-2f(x)\\=0(œ(h))}. If co(ô)=ô" ( O c a ë l ) then H3" are the known Lipschitz classes. In [3] L . LEINDLER (see also in [6], p. 153) proved, among others, the following result. (2) H(P, p, èx) = H" for 0 < j < 1 I (3) H(fi,p,ô)^(Hsr for a = 1 J l f G . G . LORENTZ [7] proved in 1948 that if A N | 0 and A„ are the Fourier sine or cosine coefficients of f then f£Hs" ( 0 < a < 1) if and only if A„=0(«_ 1 _ a) . This result and some others were generalized by R. P. BOAS[2] in 1967 as follows: Let and let Xn be the Fourier sine or cosine coefficients of f . Then f£Hs" (0<a< 1) if and only if (4) 2 k = 0 ( n - % or equivalently (5) J kXk = 0(rt~*). k=l Combining this result and the result of L. Leindler mentioned above we have that if and X„ are sine or cosine coefficients of / then the following three relations are equivalent: (6) f£H(fi, p,5y, 2 h = 0(n-°); È kXk = Oin1-«), k=n *=1 if 0 < a < l and p: L. Leindler has extended results (2) and (3) from Hs" to Ha at least for certain special but more general class of moduli of continuity than a>{S)=8a. Next we give the definition of this class of moduli of continuity (see [4], [5] and in [6], p. 154). Let œa(ô) denote the modulus of continuity having the following properties for O s a s 1 : i) for any a ' > a there exists a natural number fi=/x(a') such that (7) 2" a 'o a(2_ n - ' ' ) > 2(oa(2~") holds for all 1); Fourier series with positive coefficients 293 ( 8 ) ii) for every natural number v there exists a natural number N(v) such that 2vsttt>a(2-n-v) ^ 2(ox(2—) if n > N(v). L . LEINDLER in [4] , [5] (see also [6], p. 1 5 4 ) proved the following relation gen-eralizing results (2) and (3). It is clear thât in order to get coefficient conditions of type (6) for f£H(ß,p, a>J instead of Hiß, p, <5a) it is sufficient to generalize the mentioned Boas results to class Hm". These results are formulated in the next paragraph. Throughout the rest of this paper g(x), fix), <p(x) will denote continuous 2% periodic functions; furthermore g(x) and fix) always denote the sum of sine series and cosine series, respectively. And (p(x) denotes the sum of either sine or cosine series while /„ will denote the Fourier coefficients of g(x),f(x) or cp(x). ' T h e o r e m 1. Let A„ëO. Then cpÇ.Hœ" (0«x<l) if and only if (9) Hiß, p, œx) = for (10) Hiß, p, mj = (№»•)* for <0, 3. Theorems 2 = 0(ncoa(i/n)). k=l ' Theorem 2. Let A„s0. Then (13a) if and only if g€J5r»i (13b) 2 kXk = 0(^(1/70). Theorem 3. Let A„sO. Then (14a) if and only if (14b) <p<iiH°»y i* 2K = 0(a>1{\ln)), 5 294- J. Németh Theorem 4. Let A„sO. Then (15a) f i H ^ if and only if (15b) 2K = 0(^(1 /«) ) and ¿fcA4sinfoe - 0 ( ^ ( 1 / « ) ) . *=n ft = l Theorem 5. Let A„sO. Then (16a) f t H 0 * if and only if (16b) ¿ f j * = 0 M l / n ) ) . k=n Furthermore ( 1 7 ) g£H<°<> implies (18) 2 kXk = O(nco0(l/n)), and from k=1 (19) 2 ^ = 0(0*0/»)) k=n (20) g tH"» follows. Remark 1. Combining relation (9) of L. Leindler and Theorem 1 we get generalizations of the relations under (6). Namely that if then for 0 < a - = l with P>ctp if and only if i or equivalently 2 h = 0{(oa(Un)), 2 kXk = 0(no)a(l/ri)). k=l Remark 2. Using Theorem 1 we can prove the following result. If hZH°" ( 0 « x < l ) and a 00 + 2 ak 0 0 5 kx+bk sin kx *=1 Fourier series with positive coefficients 295 is the Fourier series of h(x) and ak^0,bk^0 then (21) Us„(*)-A(*)i = o K ( i / 4 It should be noted that for arbitrary ak, bk according to the well-known Lebesgue result only QS .(*)-A(x)l=0(ffla(l/»i)).logii can be obtained. The proof of this remark is very simple. Really, consider ||sn(x)-A(x)|| =5 ||sB(x)-an(x)|| + ||<Tn(x)-/i(x)|| = I+II. Using (12) and the fact that 1 " s„ (x) - o„ (x) = ——— 2 (kak cos kx+kbk sin kx) w+l *=1 I can be estimated as follows I s —J-r- J k(ak + bk) = 0(<b.(1/b)). Here we used that if hZH10", then both its sine-part and cosine-part also belong to Ha". And finally from (9) lI = 0{coXm)-Thus we have (21). 4. Lemmas Lemma 1. (Lemma 2.6 of [6], p. 39.) For any nonnegative sequence {a„} the inequality (22) Z a n s K a m (m = 1, 2, ...; K > 0) n—i holds if and only if there exist a positive number c and a natural number fi such that for any n (23) a„+1 a ca„ and (24) an+ll ^ 2a„ are valid. r* 296- J. Németh L e m m a 2. If fik=0 and <5>/?>0 then (25a) 2 = 0{ita,-f(Mn)) is equivalent to (25b) ¿ A = O(®a-,0/»))-k=n P r o o f . First we suppose that (25b) is true. By using Abel-rearrangement ([1], p. 71) we have (26) i f t k ' s i (k°-(k-1)0 ¿ ^ + 2 = / + / / . *=1 fc = l V=fc V = 1 It is obvious by (25b) and the definition of <as-f{t) that II does not exceed K.n0<ot^{\ln)*\ And / can be estimated as follows n [logn] (27) / S Ai 2 k'^co^l/k) ^K, 2 (2m)i_1 • 2mco i_ i( l /2m) = *=1 m=l [logn] = 2 2 © i_p(l/2m) K3n cos^p(l/n), m = l where the last estimation can be obtained by using property (7) of cox(ő) and Lemma 1. So taking into account (26) and (27), (25a) really follows from (25b). Now we sup-pose (25a). Again Abel-rearrangement gives that (28) i ft = 1 ftok-' = 2 (k->-(k + i)-°) 21'ni + rr* 2 k=m k—m k=m ft=l Making n tend to infinity from (25a) and (28) we get (29) 2 fk = 2 ( k " i - ( f e + i ) - a ) 2 Isto = h. k=m k=m 1=1 can be estimated by (25a) as follows (30) AsAi 2 ^^-.„(l/A:)^ 2 co^il/l") ^ Kzco^pil/m). k=m n=[Iogm] The last step can be derived from property (8) of coa(ö). From (29) and (30) we have (25b). Thus Lemma 2 is completely proved. *' K,KltKa,... will denote positive absolute constants. Fourier series with positive coefficients 297 w Lemma 3. If fik^0, 2 anà 0< /Ts l , then (31a) 2 Hktt - c o s kx) = 0(cop(xj) k=1 is equivalent to (3ib) | f t = o W V 4 k=n Proof . Supposing first (31a), we have that (32) l f k ^ k l k 2 - J O S k X = 0 ( x ~ ^ p ( x ) ) holds for any positive x. Since A : ^ / - 2 ( l - c o s on (0, 1), from (32) it follows that [i/*] „ ; , (33) Z = 0(x-2<o„(*)). k=l Putting x = 1 \n we get (34) 2 k2(ik = 0(n2a)p(l/n)) k—1 which, by using Lemma 2, is equivalent to (35) j ? f t = O M l / i » ) ) fc=n which proves (31a)=>(31b). Now we suppose that (31b) is valid. But (31b), by using Lemma 2, is equiv-alent to (36) ¿ f c » f t = 0(n«ffl,(l/n)). *=i Using (31b) and (36) we have that CO [1/*] 00 2 tik(l-cos kx) ^ 2 fail—cos kx)+2 2 Vk = k—l k=l fc=[V*l = *2 1 1 * * i ¿ m * ) ) k=1 K'X* *=[1/*] which gives that (31b) really implies (31a). Thus Lemma 3 is completed. * 298- J. Németh 5. Proofs of the theorems P r o o f of T h e o r e m 1. We detail the proof just for cosine, the line of the proof for sine is very similar. Suppose that ( 0 < a < 1). Then since by the Paley's theorem from the oo continuity of f(x) it follows that 2 ^ * < 0 0 > we have k = 1 (37) | / (*)- / (0)1 = 2 - c o s kx) = 0(cox(x)). k=l By Lemma 3 the right-hand side equality of (37) is equivalent to (11). In virtue of Lemma 2 (11) is equivalent to (12). Thus the necessary part of Theorem 1 is proved. Now we suppose that (11) holds and put (38) ¡f(x+2h)—f(x)l - I 2K [cos k(x+2h)-cos kx]I = k=l , Wh] = 2 \ 2 lkSmk(x+h)-smkh Us 2 2 2 2 ^ s i n k / z + 2 2 K-*=1 t=l k=l k=lllh] The second term in the last row of (38) is 0(coa(h)) (see (11)). The first one can be handled as follows: [i/M wo cjn uu [i/i>] 2 ¿k-sinkh = h 2 k?.k-nr^K.h(2 kkk) = 0{(Da{h))-fc=l k=1 Krt Jt=l In the last step we used again (12) and Lemma 2. So from (38) we have /€//""«. The proof of Theorem 1 for cosine series is completed. P r o o f of T h e o r e m 2. Let Using \g(x)\ == KMx), term by term integration gives (39) | / g(t)dt\ = 2 1 —cos kx) = ¿»(XCOJOC)). o *= 1 From (39) it follows that [iM (40) 2 k-1Ak(l-coskx) = 0(x<o1(x)). k=1 (40) can be written as follows (41) x ^ f kXk l ~ ™ k X = Oixco^x)). % Fourier series with positive coefficients 299 By using the same argument as before from (41) we get W>] (1 1 (42) Z ^ = 0 - 0 ) 1 « • k=i \x / Putting x=l/tt we have from (42) 2 kkk = 0 ( ^ ( 1 / « ) ) 4 = 1 which proves that from (13a) follows (13b). Now we suppose that (13b) is ful-filled. Put (43) \g(x+2h) - g(x)| = 1 2 4 [sin fc(*+2A)-sin kx]I = *=1 ~ [1/h] » = 2\2 Ak cos k(x+h) sin kh\ S 2 2 4sinfcA+ 2 kk = I+II. jt=i »=1 *=[i/fc] By using (13b) we have that IV*] sin kh ( 1 \ (44) I = 2A 2 ^ = h 0 { T = For II to be estimated by Kco^h) we can use the same argument as in the second part of the proof Lemma 2. Namely taking ¿ = 1 and 5—p=l we get that (13b) implies 2 ^ = 0(0x0/»)) k=n which gives that (45) / / = C > M / 0 ) . Thus (43), (44) and (45) give that geHm>. Theorem 2 is completed. P roof of Theorem 3. First we prove the theorem for cosine series. Suppose that (14a) holds, that is, \f(x+h)+f(x-h)-2f(x)\ ^ Kcoxih) from which we get (46) \f(h) - m \ s KcoAh) in other words (47) 2 4 ( 1 - c o s kh) = O K (/i)) *=i holds. 300- J. Németh From (47) by using Lemma 3; (14b) follows, that was to be proved. Now we assume (14b) and estimate the following difference by using Lemma 3 at the last step: (48) \f(x+2h) +fix—2h) — 2f(x)\ = 41 £ Xk sin2 kh cos kxI fc=i sin8 kh = 2 2 V 1 - c o s 2kh) = OfaQi)). k=1 fc=l Thus the proof of Theorem 3 is completed for cosine series. The proof for sine series in direction from (14b) to (14a) can be done in the same way as for cosine series, since (49) |g(je+2/j)+g(*-2/z)-2g(;c)| = 4 1 J sin kx sin2 kh\. *=i So we detail only the other direction. Suppose that (50) g e i H ^ r , that is, (51) \gix+h)+gix-h)-2gix)\ = 0(o}1ih)). Write (51) in the following form (52) 2 1 2 K sin kx(l - c o s kh)| = Ofaih)). k—1 By integrating term by term in (52) we get (53) 2 K l~°°skXil - c o s kh) = Oixco.ih)). From (53) we have . » i . 1 cos lex (54) 2^kXk (1 - c o s kh) = Oixco^h)). Using (54) it follows that Vlx] (55) 2 xkXk(l - c o s kh) = 0(co^h)). k=1 Putting h=x in (55) [i/*]' (56) 2 hkXk(l-coskh) = 0(aj1(h)) Fourier series with positive coefficients 301 can be obtained which gives (57) = 0(ajL(h)). From (57) taking h — l/n (58) 2 fc34 = 0(^(0,01»)) k =1 follows. By using Lemma 2 (58) implies (14b), which was to be proved. Thus Theorem 3 is completely proved. P r o o f of T h e o r e m 4. First we prove the necessity of the conditions, namely we suppose that (15a) holds. From Theorem 3 using the relation #<0ic(/y<Bl)* it follows that (59) ¿ J * = 0(ffli(l/»)). k~n So it remains just to prove (60) || 2 fc4 sin fcx|| = O(n£o1(l/n)). k=1 Set [l//.] \f(x+h)-f(x)\ = 2 2 Aksin k(x+h) sin kh+0( 2 4)-k=l fc=[l//i] From (15a) and (59) we get [i/fc] (61) || 2 4 sin k(x+h) sin kh\\ = .OUa^h)). «=1 Since sin kh=kh+0(k3hs) we have from (61) that Ei/fc] [i/fc] (62) ||A 2 4 ^ s i n k ( x + h ) + h 3 2 4^3sin/C(X+/J)|| = Oiw^h)) k = l k = 1 and having in view Lemma 2 we get [l/h] (63) \\h3 2 4fc3sin k(x+h)\\ = O^Qi)). *=i Using (63) we have from (62) [i Ml (64) ||A 2 k h sin fc(*+A)|| = O f a i h ) ) . k=1 302- J. Németh Since sin k(x+h)=sin kx-OiWh*) sin kx+0(kh) cos kx, (64) has the following form ,, [l/A] [1//.J (65) \\h 2 kK sin kx-h 2 kXkO(k2h2) sin kx+ k=l ft=l" I1'*' +h 2 kXkO(kh)cos kx\\ = OfaQi)). fc^i Taking into account that from (59) by using Lemma 2 (66) 2 = 0(^0^(1/«)) k=1 follows, the norm of the second term in the left-hand side of (65) can be estimated as follows „ [llhl I. [1"'1 / , V (67) \\h 2 klkO(k2h2) sin kx\\ s Kh3 2 k3Xk = OfaQij). k=1 *=1 Similarly by using 2 = 0(^(1/«)) k=l instead of (66) we can get that the magnitude of the third term of (65) in norm is 0(a)1(/0). Using this last estimation and (65), (67) we have (60). The sufficiency of conditions (15b) can be proved in very similar way as the necessity, so we omit it. Thus Theorem 4 is completed. oo Proof of Theorem 5. Let f(x)— 2 h cos kx and suppose that f^H<°<>. k=l Then we have | / ( / i )- /(0) | ^ Kco0(h), that is, 2 At(l—cos kh)sK(o0(h). k=1 Integrating both sides on (0, x) we have °° A (68) 2 -r-ikx- sin kx) Kx(o0 (x). k = l k Since kx-sin kx^O so we have from (68) (69) 2 ~T~ (kx~sin kjc) - Kxco0(x). k=2n k Fourier series with positive coefficients 303 Putting 1/w For x and taking into account that k . k 1 k , . , s in—S —— for k S 2n n n 2 n we get (70) Z h ^ K ^ i U n ) k = 2n which gives (16b). Now we suppose that (16b) holds and we prove f£H<0<>. First we note that we can notice that the first part of the proof of Lemma 2 remains valid if we take S - p=0 and <5 = 1. So from (16b) we have (71) ¿ W , = O(no)0(l/«)). And now estimate the following difference using (16b) and (71) . \f(x+2h)-f(x)\ = 12 4[cos k(x+2h)-cos kx]\ = k =1 oo oo = 2\2 h sin k(x + h) sin kh\^ 2\2 h sin kh\ = k=1 *=i ti/fc] S 2 2 4 sin/eft + 2 4 = O(co0(h)) k=l k=[llh] which proves that (16b) implies (16a). Now we prove (18) from (17). Suppose that H™0. Using the estimation (72) |g(x)| Kco0(x), term by term integration gives from (72) that (73) ¿ " 4 ^ 0 - c o s kx) S Kxco0(x). fc=x k From (73) [1M mJx\ k=1 * follows as at the proof of (33) which taking x=l/n gives (18). The proof of (20) from (19) can be done in the very same way as (16a) from (16b), so we omit it. Theo-rem 5 is completed. 304 J. Németh: Fourier series with positive coefficients References [1] G. ALEXITS, Convergence problems of orthogonal series, Akadémiai Kiadó (Budapest, 1961). [2] R. P. BOAS, Fourier series with positive coefficients, J. Math. Anal., 17 (1967), 463—483. [3] L. LEINDLER, Strong approximation and classes of functions, Mitteilungen Math. Seminar dessen, 1 3 2 (1978) , 2 9 — 3 8 . [4] L. LEINDLER, Strong approximation and generalized Lipschitz classes, Functional analysis and approximation (Oberwohlfach, 1980), Birkhäuser (Basel—Boston, 1981), pp. 343—350. [5] L. LEINDLER, Strong approximation and generalized Zygmund class, Acta. Sei. Math., 43 (198.1), 3 0 1 — 3 0 9 . [6] L. LEINDLER; Strong approximation by Fourier series. Akadémiai Kiadó (Budapest, 1985). [7] G. G. LORENTZ, Fourier-Koeffizienten und Funktionen Klassen, Math. Z„ 51 (1948), 135— 149. [8] A. ZYGMUND, Trigonometrie Series. I—II, (Cambridge, 1968). BOLYAI INSTITUTE ARADI VÉRTANÚK TERE 1 6720 SZEGED, HUNGARY 

Fourier series with positive coefficients and generalized Lipschitz classes

http://acta.bibl.u-szeged.hu/15281/1/math_054_fasc_003_004_291-304.pdf

Fourier series with positive coefficients and generalized Lipschitz classes

Abstract

Similar works

Full text

Available Versions

University of Szeged