¯ °¿Ê

: 
1 {a n } {b n } Þ ´ (classical quasi-monotone) ¢ ¤ ∞ n=0 a n < ∞ Ö lim n→∞ nb n = 0. º f (x) = ∞ n=0 a n cos nx, g(x) = ∞ n=1 b n sin nx. ÌÞ f, g ∈ C 2π , Î ½ E n (f ) ∼ max 1 k n ka n+k + ∞ k=2n+1 a k , E n (g) ∼ max k 1 kb n+k .
2000 À, Xiao [2] ¨ ¹ 1 ℄¼È ¢³¡¡ Î¨ ¢¢ ¿ ´ ¥£ ¼ O-℄ ´ (O-regularly varying quasi-monotone), °¿ Ê É: 2006-12-22; É: 2007-07-17 
± NSERC RCD grant × AARMS, ± NSERC, ¡℄° Ò (Ã : 10471130) ÔÉ ± Ø Ø ÄÒ (Ã : PCN0613) Þ ¶ *E-mail: szhou@zjip.com 
2 f (n) ∞ n=0 f (n) + f (-n) ∞ n=0 Þ O-℄ ´ ¢ , Îº f (x) = ∞ n=-∞ f (n)e inx . ÌÞ f ∈ C 2π ¿ AEª ¥Þ ∞ n=1 | f (n) + f (-n)| < ∞ lim n→∞ n f (n) = 0. , f ∈ C 2π , ℄Ê E n (f ) ∼ max 1 k n k(| f (n + k)| + | f (-n -k)|) + max k 2n+1 k| f (k) -f (-k)| + ∞ k=2n+1 | f (k) + f (-k)|.
¿ 90 À», Fourier AE Ô, ÇAE ß£ Ì ªÙ¢ ¢¿ ¥ Ù × ¿ ¾¼ ℄, Ö£ ¥ ´°ßÄ ¢¢ Õ ¿ ¼ MS ( £¢ , monotone non-increasing sequences) ¼ CQMS( 
´ ¢ , classical quasimonotone sequences), RVQMS(O-℄ ´ ¢ , O-regularly varying quasi-monotone sequences), RBVS(Ò Ê ¢ , rest bounded variation sequences), GBVS(AE¥Ê ¢ , group bounded variation sequences), NBVS(Å´«Ê ¢ , non-onesided bounded variation sequences, ¤ °¿ Á) ¿´ 
. 
¹ ¢¢ Õ ¿ ´ ± ¡ °(Í ¿ ¯± ¤ ³ [2-8], MVBVS * Þ¦ ¢¢ ´ ¿¿Õ ): MS -RBVS -CQMS -• • • • • • -RVQMS -GBVS -NBVS -MVBVS * . Ý C := {c n } ∞ n=0 Þ Ë Ì Ë θ 0 ∈ [0, π/2) Ö n = 0, 1, . . . ¤ ¥ c n ∈ K(θ 0 ) := z : | arg z| θ 0 ¿È¢¢ . «Ê m = 0, 1, . . . ½ °¿ Ú: 2m n=m |c n -c n+1 | := 2m n=m |∆c n | C(C) |c m | + |c 2m | , ℄ C ∈ NBVS. « ( ¯ ¤ ³ [5]), CQMS RBVS, RBVS CQMS, M S RBV S ∩ CQM S RBV S ∩ RV QM S GBV S N BV S. Ì ° Ë¼¿Í℄Â¿ ¢¢ Õ ¿ Á: ¶ 1 C := {c n } ∞ n=0 Þ Ë Ì Ë θ 0 ∈ [0, π/2) Ö n = 0, 1, . . . ¤ ¥ c n ∈ K(θ 0 ) = z : | arg z| θ 0 ¿È¢¢ . Ë¢ λ 2, Ù¾ ° Ú Ì «Ê m = 0, 1, . . . ½: 2m n=m |∆c n | C(C) |c m | + |c 2m | + λm k=m |c k | k , (1) 
℄ C ∈ MVBVS * (Í Ê ¢ , strong mean value bounded variation sequences). 
± , {c n } ∞ n=0 ∈ NBVS, ℄ {c n } ∞ n=0 ∈ MVBVS * , ¸Á . j 0, Ò c 0 := 0, c n :=          0, 2 j n 2 j + j, 2 -j , 2 j + j < n < 2 j+1 -j, 0, 2 j+1 -j n < 2 j+1 , {c n } ∞ n=0 ∈ MVBVS * , ¸{c n } ∞ n=0 / ∈ NBVS. MS RBVS ∩ CQMS RBVS ∩ RVQMS GBVS NBVS MVBVS * .
¦ ¨ ¹ 2 Ô¿ ¢¢ ¥ RVQMS [2] , GBVS [7] , NBVS [5] 
℄¼ MVBVS * . Ý Þ 3 f (n) ∞ n=0 ∈ MVBVS * , f (n) + f (-n) ∞ n=0 ∈ MVBVS * , º f (x) = ∞ n=-∞ f (n)e inx . ÌÞ f ∈ C 2π ¿ AEª ¥Þ lim n→∞ n f (n) = 0 (2) ∞ n=1 | f (n) + f (-n)| < ∞. (3) 
, 
f ∈ C 2π , ℄Ê E n (f ) ∼ A n + max k 2n+1 k| f (k) -f (-k)| + ∞ k=2n+1 | f (k) + f (-k)|,

ÊÔ

A n := max 
1 k λn k | f (n + k)| + | f (n + k) + f (-n -k)| , λ Á 1 Ô ²¿¢. ° 1( ³ [5, ¹ 4]) f (n) ∞ n=0 ∈ NBVS, f (n) + f (-n) ∞ n=0 ∈ NBVS, º f (x) = ∞ n=-∞ f (n)e inx . ÌÞ f ∈ C 2π ¿ AEª ¥Þ (2) (3) Ú ½. , f ∈ C 2π , ℄Ê E n (f ) ∼ max 1 k n k | f (n + k)| + | f (-n -k)| + max k 2n+1 k| f (k) -f (-k)| + ∞ k=2n+1 | f (k) + f (-k)|. º f (x) = ∞ n=0 f (n) cos nx. ° ¿ Þ ¹ 3 Ì Fourier AE Ô Ë ¯¿Ê ÄAE (Ê¥ ¤ ³ [6] [9]). ° 2 { f (n)} ∈ MVBVS * ËÅÉ×¢¢ . f ∈ C 2π ¤ 2n k=n+1 f (k) = O max 1 k n k f (n + k) , ℄Ê f -S n (f ) = O(E n (f )). , £ C(x) ¯Ü Ë x Ê ¿ ¢, C ¯Ü Ë ¿ ¢¡Ê ¿ ± Ê« . 2 º¨ • 1 f (n) ∞ n=0 ∈ MVBVS * , f (n) + f (-n) ∞ n=0 ∈ MVBVS * , º f (x) = ∞ n=-∞ f (n)e inx . f ∈ C 2π , ℄Ê max k 1 k| f (±(n + k))| = O(E n (f )). ¼ t * n (x) ¯Ü f ¿ n AE ª¦Þ § µÚ, °±¤¿ Ú 1 2π π -π e ±i(n+1)x N -1 k=0 e ±ikx 2 dx = N ¾¼ N k=1 k f ∓ (n + k) + (N -k) f (∓(n + N + k)) = 1 2π π -π (f (x) -t * n (x))e ±i(n+1)x N -1 k=0 e ±ikx 2 dx N E n (f ). N E n (f ) Re N k=1 k f (∓(n + k)) + (N -k) f (∓(n + N + k)) N k=1 kRe f (n + k) + (N -k)Re f (n + N + k) N k=1 kRe f (n + k) C(θ 0 ) N k=1 k| f (n + k)|. ( 4 
) ÈÌ f (n) ∞ n=0 ∈ MVBVS * , Ì [N/2] + 1 j N -1 ( N 4) Ê f (n + N ) N -1 k=j ∆ f (n + k) + f (n + j) n+N -1 k=n+j ∆ f (k) + f (n + j) 2(n+j) k=n+j ∆ f (k) + f (n + j) = O λ(n+j) k=n+j f (k) k + f (n + j) + f (2n + 2j) = O λ(n+N -1) k=n+[N/2]+1 f (k) k + f (n + j) + f (2n + 2j) . Â , (4) ½ N f (n + N ) = O N λ(n+N -1) k=n+[N/2]+1 f (k) k + N -1 j=[N/2]+1 f (n + j) + f (2n + 2j) = O N λ(n+N -1)-n k=[N/2]+1 k f (n + k) (n + k) k + O 1 N N -1 j=[N/2]+1 j f (n + j) + 2j f (2n + 2j) = O N ((λ -1) n + λ (N -1)) (n + [N/2] + 1) ([N/2] + 1) E (λ-1)n+λ(N -1) (f ) + O (E n (f ) + E 2n (f )) = O (E n (f )) . ÌÞÊ (¹ N < 4 Õ± ½) N f (n + N ) = O (E n (f )) , N 1, Ø max k 1 k| f (n + k)| = O(E n (f )). (5) 
•¦Ì (4) Ú, ± ¾¼ 
2N E n (f ) N k=1 k Re f (n + k) + Re f (-n -k) C(θ 0 ) N k=1 k f (n + k) + f (-n -k) . ßÀ¼ f (n) + f (-n) ∞ n=0 ∈ MVBVS * , •¦Ì (5) Ú¿ , Ð¾ max k 1 k| f (n + k) + f (-n -k)| = O(E n (f )), max k 1 |k f (-n -k)| max k 1 k| f (n + k) + f (-n -k)| + max k 1 k| f (n + k)| = O(E n (f )), Ã¹ ©. • 2 ( ³ [2, ¹ 4]) f ∈ C 2π Ì Ë 0 θ 1 < π/2 ¤ f (n) + f (-n) ∈ K(θ 1 ), n = 1, 2, . . ., ℄ ½ ∞ k=2n+1 f (k) + f (-k) = O(E n (f )). • 3 f (n) ∞ n=0 ∈ MVBVS * , f (n) + f (-n) ∞ n=0 ∈ MVBVS * , ℄ Ì À x ∈ [0, π]¡ n k=1 f (±(n + k)) sin kx = O (A n ) Ñ ½. ¼ x = 0 Ñ x = π ¿Ï½ÞÄÀ¿, Â Ð x ∈ (0, π) . ¹ 0 < x π/n Õ, Ú | sin x| x n k=1 f (n + k) sin kx π n n k=1 k f (n + k) π max 1 k n k f (n + k) . (6) 
¹ π/n < x < π Õ, Ë ¢ m < n, Ù¾ m π/x < m + 1, ℄Ê 1) n k=1 f (n + k) sin kx π m m k=1 k f (n + k) + n k=m+1 f (n + k) sin kx π m m k=1 k f (n + k) + [n/λ] k=m+1 f (n + k) sin kx + n k=[n/λ]+1 f (n + k) =: T 1 + T 2 + T 3 .
±, 
|T 1 | π max 1 k n k| f (n + k)|, (7) 
T 3 C(λ) max 1 k n k| f (n + k)|. (8) 
ÄAE Abel , ¾¼ 
T 2 = [n/λ]-1 k=m+1 ∆ f (n + k) k j=1 sin jx + f (n + [n/λ]) [n/λ] j=1 sin jx -f (n + m + 1) m j=1 sin jx. Â Ì m π/x < m + 1, k j=1 sin jx = O(x -1 ) = O(m + 1), ℄ |T 2 | = O (m + 1) [n/λ]-1 k=m+1 |∆ f (n + k)| + max 1 k n k| f (n + k)| . ( 9 
)
¢Ý log n := ln n/ ln 2. È { f (n)} ∞ n=0 ∈ MVBVS * ¾¼ ( Ó £¾± [n/λ] Ë¢) 1) m + 1 [n/λ], ¸ §Í T 3 ÃÛ Ü ¬ ¬ ¬ È n k=m+1 f (n + k) sin kx ¬ ¬ ¬.
[n/λ]-1 
k=m+1 |∆ f (n + k)| [log([n/λ]-1)] k=[log(m+1)]+1 2 k j=2 k-1 |∆ f (n + j)| + [n/λ]-1 k=([n/λ]-1)/2 |∆ f (n + k)| = O [log([n/λ]-1)] k=[log(m+1)]+1 λ(n+2 k-1 ) j=n+2 k-1 | f (j)| j + f (n + 2 k-1 ) + f (2n + 2 k ) + O λ(n+([n/λ]-1)/2) k=n+([n/λ]-1)/2 | f (k)| k + O f (n + ([n/λ] -1)/2) + f (2n + ([n/λ] -1)) = O max 1 k λn k| f (n + k)| [log([n/λ]-1)] k=[log(m+1)]+1 λ(n+2 k-1 )-n j=2 k-1 1 (n + j) j + 2 -k + O max 1 k λn k| f (n + k)| λ(n+([n/λ]-1)/2)-n k=([n/λ]-1)/2 1 (n + k) k + 1 n = O (m + 1) -1 max 1 k λn k| f (n + k)| . (10) 
£ , ( 6)-(10) Ú, ±¾ 
n k=1 f (n + k) sin kx = O max 1 k λn k| f (n + k)| . ( 11 
) º n k=1 f (-n -k) sin kx = n k=1 f (n + k) + f (-n -k) sin kx - n k=1 f (n + k) sin kx. ÄAE ¡ÏÛÚ, ßÀ¼ f (n) ∞ n=0 ∈ MVBVS * Ö f (n) + f (-n) ∞ n=0 ∈ MVBVS * , Ð¾ n k=1 f (-n -k) sin kx = O max 1 k λn k| f (n + k) + f (-n -k)| + max 1 k λn k| f (n + k)| . ( 12 
) , Ã¹ 3 ¿ È (11) (12) Ø± . • 4 f ∈ C 2π , f (n) ∞ n=0 ∈ MVBVS * , ℄ ∞ k=m f (k) sin kx = O max k m k| f (k)| Ì À x ∈ [0, π] Ö m 1 Ñ ½. ¼ Ò J(x) = ∞ k=m f (k) sin kx. ÈÌ¹ x = 0 Ö x = π Õ, J(x) = 0, ± ¨x ´Ó (0, π) Ó £¾. N = [1/x] Îº 2) J(x) = N -1 k=m f (k) sin kx + ∞ k=N f (k) sin kx =: K 1 (x) + K 2 (x). 2) º N m Ö, Ü K 2 AE ²ÇÍ Ü J(x) Ý ε m = max k m k| f (k)|. N = [1/x] Ø±Ð¾ K 1 (x) x N -1 k=m k| f (k)| < x(N -1)ε m ε m . ÄAE Abel , Ã¹ 3 ¿ •¦ ± |K 2 (x)| ∞ k=N ∆ f (k) k ν=1 sin νx + f (N ) N ν=1 sin νx = O x -1 ∞ j=0 2 j+1 N k=2 j N ∆ f (k) + f (N ) = O ε m + x -1 ∞ j=0 λ2 j N k=2 j N f (k) k + f (2 j N ) + f (2 j+1 N ) = O ε m + ε m ∞ j=0 2 -j = O (ε m ) ½, Ã¹ 4. C:= {c n }Þ Ë (1) ÚÔ C (C) m λm k=[m/λ] |a k | k ³ ÊË ¿Ú ¤«Ê Á 1 ¥¿È¢¢ , ℄ C Ê Ê (mean value bounded variation), Ý C ∈ MVBVS. • 5( ³ [8, ¹ 4]) { f (n)} ∞ n=0 ∈ MVBVS Þ Ë θ 0 ∈ [0, π/2) ¤ f (n) + f (-n) ∈ K(θ 0 ) (n = 1, 2, . . .) ¿È¢¢ , ℄ f ∈ C 2π Î lim n→∞ f -S n (f ) = 0 ¿ AEª ¥Þ (2) (3) Ú ½. • 6 { f (n)} ∞ n=0 ∈ MVBVS * Þ Ë θ 0 ∈ [0, π/2) ¤ f (n) + f (-n) ∈ K(θ 0 ) (n = 1, 2, . . .) ¿È¢¢ , ℄ f ∈ C 2π Î lim n→∞ f -S n (f ) = 0 ¿ AEª ¥Þ (2) (3) Ú ½. ¼ Ã¹ 5 , Ð À¢ { f (n)} ∞ n=0 ∈ MVBVS * Þ Ê ¢ ({ f (n)} ∈ MVBVS). n Â¢, Ò n * = n, Ç℄Ò n * = n -1. { f (n)} ∞ n=0 ∈ MVBVS * , Ì À m j 2m, Ê f (m) j-1 k=m ∆ f (k) + f (j) = O j * k=j * /2 ∆ f (k) + f (j) = O λj * k=j * /2 f (k) k + f (j * ) + f (j * /2) + f (j) , (13) 
•¦ , f (2m) 
2m k=j ∆ f (k) + f (j) = O 2j k=j ∆ f (k) + f (j) = O λj k=j f (k) k + f (j) + f (2j) . (14) (13) 
(14) ÚØ¾¼ f (m) + f (2m) = O 2λm k=m * /2 f (k) k . (15) 
¨( 15) Ú³ (1) ÚÊ 
2m n=m |∆ f (n)| = O 2λm k=[m/(2λ)] f (k) k , ØÀ { f (n)} ∈ MVBVS. • 7 f (n) ∞ n=0 ∈ MVBVS * , f (n) + f (-n) ∞ n=0 ∈ MVBVS * , ℄ ∞ k=2n+1 f (k) -f (-k) sin kx = O max k 2n+1 k f (k) -f (-k) + ∞ k=2n+1 f (k) + f (-k) . ¼ ßÀ¼ f (k) -f (-k) = 2 f (k) -( f (k) + f (-k)), Ö f (n) ∞ n=0 f (n) + f (-n) ∞ n=0 Ì MVBVS * ¿ ¥, Ã¹ 4 ¾ ∞ k=2n+1 f (k) -f (-k) sin kx 4n+1 k=2n+1 f (k) -f (-k) sin kx + ∞ k=4n+2 f (k) -f (-k) sin kx = O 4n+1 k=2n+1 f (k) -f (-k) + O max k 4n+2 k| f (k)| + O max k 4n+2 k| f (k) + f (-k)| = O max k 2n+1 k f (k) -f (-k) + O max k 4n+2 k| f (k)| + O max k 4n+2 k| f (k) + f (-k)| . max k 4n+2 k| f (k)| = k 0 | f (k 0 )|. Â k 0 4n + 2, k 0 | f (k 0 )| 1 2 k 0 f (k 0 ) -f (-k 0 ) + 1 2 k 0 f (k 0 ) + f (-k 0 ) max k 2n+1 k f (k) -f (-k) + 1 2 k 0 f (k 0 ) + f (-k 0 ) . È (15) ÚÐ¾ f (k 0 ) + f (-k 0 ) = O 2λk0 k=k * 0 /2 f (k) + f (-k) k , ÌÞ k 0 f (k 0 ) + f (-k 0 ) = O 2λk0 k=k * 0 /2 f (k) + f (-k) = O ∞ k=2n+1 f (k) + f (-k) . ¢ Ìµ, ¤¼ max k 4n+2 k| f (k)| = O max k 2n+1 k f (k) -f (-k) + ∞ k=2n+1 f (k) + f (-k) ½. { f (k) + f (-k)} ∈ MVBVS * ¿ ¥, È•¦¿ , Ê max k 4n+2 k f (k) + f (-k) = O ∞ k=2n+1 f (k) + f (-k) . ¢ «Ê ¡ÏÛÚ Ã¹¿ . 3 AE½ 3 ¬¼ ª ¾. f ∈ C 2π , ÃAEÃ¹ 1, (2) Ú± ½, ÈÃ¹ 2 ±¾ ∞ k=2n+1 | f (k) + f (-k)| = O(E n (f )), ( 3 
) Ú ½. AE¾. ± Ã¹ 6 ¬ . ² f ∈ C 2π , Ã¹ 1 2 A n + max k 2n+1 k| f (k) -f (-k)| + ∞ k=2n+1 | f (k) + f (-k)| C(λ) max k 1 k f ± (n + k) + ∞ k=2n+1 f (k) + f (-k) = O(E n (f )). ° §½Á•¿ Ú. ×º f (x) °: f (x) = 2n k=-2n f (k)e ikx + i ∞ k=2n+1 f (k) -f (-k) sin kx + 1 2 ∞ k=2n+1 f (k) + f (-k) e ikx + e -ikx , ℄Ê E n (f ) n k=1 f (n + k)e i(n+k)x + f (-n -k)e -i(n+k)x - n k=1 f (n + k)e i(n-k)x + f (-n -k)e -i(n-k)x + ∞ k=2n+1 f (k) -f (-k) sin kx + ∞ k=2n+1 | f (k) + f (-k)| 2 n k=1 f (n + k) sin kx + 2 n k=1 f (-n -k) sin(-kx) + ∞ k=2n+1 f (k) -f (-k) sin kx + ∞ k=2n+1 | f (k) + f (-k)|. ÄAEÃ¹ 3 Ð¾ n k=1 f (n + k) sin kx + n k=1 f (-n -k) sin(-kx) = O (A n ) . Â , ÈÃ¹ 7 ¾¼ E n (f ) = O A n + max k 2n+1 k f (k) -f (-k) + ∞ k=2n+1 f (k) + f (-k) , ¹¿ . ° 1 ¬¼ ¹ 1 ¿ °¿Ý× A n = O max 1 k n k f (n + k) + f (-n -k) + O max k 2n+1 k f (k) + max k 2n+1 k f (k) + f (-k) , Ð max k 2n+1 k f (k) = O max 1 k n k f (n + k) + max k 2n+1 k f (k) -f (-k) + ∞ k=2n+1 f (k) + f (-k) , (16) 



  k f (n + k) + f (-n -k) + ∞ k=2n+1 f (k) + f (-k) . (17) max k 2n+1 k f (k) = k 0 | f (k 0 )|. 2n + 1 k 0 4n, k) + f (4n) = O f (2n) + f (4n) , Â k 0 | f (k 0 )| = O max 1 k n k f (n + k) + 4n f (4n) . (18) ÈÌ 4n f (4n) = 2n f (4n) -f (-4n) + 2n f (4n) + f (-4n) k) + f (-k) + f (j) + f (-j) 2j k=j ∆ f (k) + f (-k) + f (j) + f (-j) = O f (j) + f (-j) + f (2j) + f (-2j) , Ì À 2n + 1 j 4n -1, 2n f (4n) + f (-4n) = O + 1, È (20) Ú•¦¿ »±Ð¾ k 0 f (k 0 ) + f (k 0 ) = O k0 j=[k0/2]+1 f (j) + f (-j) + f (2j) + f (-2j) = O ∞ k=2n+1 f (k) + f (-k) .(21)¢ (18)-(21) ÚØ¾¼ (16) Ú. È ¥ f (n) + f (-n) ∞ n=0 ∈ NBVS, •¦¿ (17) Ú. ª ± ³
© ¹ Fourier, Û Ae, Û©ae