Journal of Inequalities in Pure and Applied Mathematics ON THE ABSOLUTE CONVERGENCE OF SMALL GAPS FOURIER V (p) SERIES OF FUNCTIONS OF BV volume 6, issue 1, article 23, 2005. R.G. VYAS Department of Mathematics Faculty of Science The Maharaja Sayajirao University of Baroda Vadodara-390002, Gujarat, India. Received 18 October, 2004; accepted 29 November, 2004. Communicated by: L. Leindler EMail: drrgvyas@yahoo.com Abstract Contents JJ J II I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 196-04 Quit Abstract P ink x be its Fourier b Let f be a 2π periodic function in L1 [0, 2π] and ∞ k=−∞ f(nk )e series with ‘small’ gaps nk+1 − nk ≥ q ≥ 1. Here we have obtained V sufficiency conditions for the absolute convergence of such series if f is of BV (p) locally. We have also obtained a beautiful interconnection between lacunary and nonlacunary Fourier series. On the Absolute Convergence of Small Gaps Fourier V Series of Functions of BV (p) 2000 Mathematics Subject Classification: 42Axx. Key words: FourierVseries with small gaps, Absolute convergence of Fourier series and p- -bounded variation. R.G. Vyas Contents Title Page 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 3 Contents JJ J II I Go Back Close Quit Page 2 of 12 J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005 http://jipam.vu.edu.au 1. Introduction Let f be a 2π periodic function in L1 [0, 2π] and fb(n), n ∈ Z, be its Fourier coefficients. The series X (1.1) fb(nk )eink x , k∈Z wherein {nk }∞ 1 is a strictly increasing sequence of natural numbers and n−k = −nk , for all k, satisfy an inequality (1.2) (nk+1 − nk ) ≥ q ≥ 1 for all k = 0, 1, 2, . . . , On the Absolute Convergence of Small Gaps Fourier V Series of Functions of BV (p) R.G. Vyas is called the Fourier series of f with ‘small’ gaps. Obviously, if nk = k, for all k, (i.e. nk+1 − nk = q = 1, for all k), then we get non-lacunary Fourier series and if {nk } is such that (1.3) (nk+1 − nk ) → ∞ as k → ∞, then (1.1) is said to be the lacunary Fourier series. By applying the Wiener-Ingham result [1, Vol. I, p. 222] for the finite trigonometric sums with small gap (1.2) we have studied the sufficiency condiβ P tion for the convergence of the series k∈Z fb(nk ) (0 < β ≤ 2) in terms of V BV and the modulus of continuity [2, Theorem 3]. Here we have generalized this and we have also obtained a sufficiency condition if function f is of V result V BVV(p) . In 1980 Shiba [4] generalized the class BV . He introduced the class BV (p) . Title Page Contents JJ J II I Go Back Close Quit Page 3 of 12 J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005 http://jipam.vu.edu.au Definition positive real V1.1. Given an interval I, a sequence Pof non-decreasing 1 numbers = {λm } (m = 1, 2, . . .) such that m λm diverges and 1 ≤ p < ∞ V V we say that f ∈ BV (p) (that is f is a function of p − -bounded variation over (I)) if VΛp (f, I) = sup {VΛp ({Im }, f, I)} < ∞, {Im } where VΛp ({Im }, f, I) = X |f (bm ) − f (am )|p m ! p1 λm , and {Im } is a sequence of non-overlapping subintervals Im = [am , bm ] ⊂ I = [a, b]. V Note that, if p = 1, one gets the class BV (I); if λm ≡ 1 for all m, one gets the class BV (p) ; if p = 1 and λm ≡ m for all m, one gets the class Harmonic BV (I). if p = 1 and λm ≡ 1 for all m, one gets the class BV (I). Definition 1.2. For p ≥ 1, the p−integral modulus of continuity ω (p) (δ, f, I) of f over I is defined as ω (p) (δ, f, I) = sup k(Th f − f )(x)kp,I , On the Absolute Convergence of Small Gaps Fourier V Series of Functions of BV (p) R.G. Vyas Title Page Contents JJ J II I Go Back 0≤h≤δ Close where Th f (x) = f (x + h) for all x and k(·)kp,I = k(·)χI kp in which χI is the characteristic function of I and k(·)kp denotes the Lp -norm. p = ∞ gives the modulus of continuity ω(δ, f, I). We prove the following theorems. Quit Page 4 of 12 J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005 http://jipam.vu.edu.au Theorem 1.1. Let f ∈ L[−π, π] possess a Fourier V series with ‘small’ gaps 2π (1.2) and I be a subinterval of length δ1 > q . If f ∈ BV (I) and β2 ω( n1k , f, I) < ∞, Pnk 1 k k=1 j=1 λj ∞ X then X (1.4) β <∞ fb(nk ) (0 < β ≤ 2). k∈Z P k 1 Since {λj } is non-decreasing, one gets nj=1 ≥ λnnk and hence our earlier λj k theorem [2, Theorem 3] follows from Theorem 1.1. Theorem 1.1 with β = 1 and λn ≡ 1 shows that the Fourier series of f with ‘small’ gaps condition (1.2) (respectively (1.3)) converges absolutely if the hypothesis of the Stechkin theorem [5, Vol. II, p. 196] is satisfied only in a subinterval of [0, 2π] of length > 2π (respectively of arbitrary positive length). q V Theorem 1.2. Let f and I be as in Theorem 1.1. If f ∈ BV (p) (I), 1 ≤ p < 2r, 1 < r < ∞ and ∞ X k=1 where 1 r + 1 s ω ((2−p)s+p) k P 1 , f, I nk nk j=1 = 1, then (1.4) holds. 2−p/r β2 r1 1 λj < ∞, On the Absolute Convergence of Small Gaps Fourier V Series of Functions of BV (p) R.G. Vyas Title Page Contents JJ J II I Go Back Close Quit Page 5 of 12 J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005 http://jipam.vu.edu.au Theorem 1.2 with β = 1 is a ‘small’ gaps analogue of the Schramm and Waterman result [3, Theorem 1]. We need the following lemmas to prove the theorems. Lemma 1.3 ([2, Lemma 4]). Let f and I be as in Theorem 1.1. If f ∈ L2 (I) then X 2 fb(nk ) ≤ Aδ |I|−1 kf k22,I , (1.5) k∈Z On the Absolute Convergence of Small Gaps Fourier V Series of Functions of BV (p) where Aδ depends only on δ. Lemma 1.4. If |nk | > p then for t ∈ N one has π p Z sin2t |nk | h dh ≥ 0 R.G. Vyas π 2t+1 p . Title Page Contents Proof. Obvious. Lemma 1.5 (Stechkin, refer to [6]). If un ≥ 0 for n ∈ N, un 6= 0 and a function F (u) is concave, increasing, and F (0) = 0, then ∞ X F (un ) ≤ 2 1 Lemma 1.6. If f ∈ ∞ X 1 V BV (p) JJ J II I Go Back F un + un+1 + · · · n . (I) implies f is bounded over I. Close Quit Page 6 of 12 J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005 http://jipam.vu.edu.au Proof. Observe that p p |f (x)| ≤ 2 ≤ 2p |f (b) − f (x)|p |f (x) − f (a)|p + λ2 |f (a)| + λ1 λ1 λ2 p |f (a)| + λ2 V∧p (f, I) p Hence the lemma follows. Proof of Theorem 1.1. Let I = x0 − δ21 , x0 + δ21 for some x0 and δ2 be such that 0 < 2π < δ2 < δ1 . Put δ3 = δ1 − δ2 and J = x0 − δ22 , x0 + δ22 . Suppose q integers T and j satisfy (1.6) 4π |nT | > δ3 On the Absolute Convergence of Small Gaps Fourier V Series of Functions of BV (p) R.G. Vyas δ3 |nT | and 0 ≤ j ≤ . 4π Title Page V Since f ∈ BV (I) implies f is bounded over I by Lemma 1.6 (for p = 1), we have f ∈ L2 (I), so that (1.5) holds and f ∈ L2 [−π, π]. If we put fj = (T2jh f − T(2j−1)h f ) then fj ∈ L2 (I) and the Fourier series of fj also possesses gaps (1.2). Hence by Lemma 1.3 we get X 2 nk h 2 2 ˆ (1.7) f (nk ) sin = O kfj k2,J 2 k∈Z Contents JJ J II I Go Back Close Quit because ink (2j− 12 h) fˆj (nk ) = 2ifˆ(nk )e sin nk h 2 . Page 7 of 12 J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005 http://jipam.vu.edu.au Integrating both the sides of (1.7) over (0, nπT ) with respect to h and using Lemma 1.4, we get Z π ∞ X 2 nT fˆ(nk ) = O(nT ) (1.8) k fj k22,J dh. 0 |nk |≥nT Multiplying both the sides of the equation by λ1j and then taking summation over j, we get ! ∞ Z π X |fj |2 X 1 X 2 nT dh. (1.9) fˆ(nk ) = O(nT ) λ λ j j 0 j j |nk |≥nT 1,J Now, since x ∈ J and h ∈ (0, nπT ) we have |fj (x)| = O(ω( n1T , f, I)), for each V P |f (x)| j of the summation; since x ∈ J and f ∈ BV (I) we have j jλj = O(1) because for each j the points x + 2jh and x + (2j − 1)h lie in I for h ∈ (0, nπT ) and x ∈ J ⊂ I. Therefore ! ! X X |fj (x)|2 1 |fj (x)| =O ω , f, I λ n λj j T j j 1 =O ω , f, I . nT It follows now from (1.9) that RnT = X |nk ≥nT On the Absolute Convergence of Small Gaps Fourier V Series of Functions of BV (p) R.G. Vyas Title Page Contents JJ J II I Go Back Close Quit fˆ(nk ) 2 1 ω nT , f, I = O PnT 1 . j=1 λj Page 8 of 12 J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005 http://jipam.vu.edu.au 2 Finally, Lemma 1.5 with uk = fˆ(nk ) (k ∈ Z) and F (u) = uβ/2 gives ∞ X fˆ(nk ) β =2 |k|=1 ∞ X fˆ(nk ) Rnk k β/2 F 2 k=1 ≤4 ≤4 ∞ X F k=1 ∞ X k=1 = O(1) Rnk k On the Absolute Convergence of Small Gaps Fourier V Series of Functions of BV (p) ∞ X ω( n1k , f, I) k=1 1 k k(Σnj=1 ) λj !(β/2) R.G. Vyas . Title Page This proves the theorem. Contents Proof of Theorem 1.2. Since f ∈ BV (p) (I), Lemma 1.6 implies f is bounded over I. Therefore f ∈ L2 (I), and hence (1.5) holds so that f ∈ L2 [−π, π]. Using the notations and procedure of Theorem 1.1 we get (1.9). Since 2 = (2−p)s+p + pr , by using Hölder’s inequality, we get from (1.9) s V Z 2 Z |fj (x)| dx ≤ J (2−p)s+p |fj (x)| 1s Z |fj (x)| dx dx J 1/r J Z ≤ Ωh,J J |fj (x)|p dx p r1 r1 JJ J II I Go Back Close Quit Page 9 of 12 , J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005 http://jipam.vu.edu.au where Ωh,J = (ω (2−p)s+p (h, f, J))2r−p . This together with (1.9) implies, putting B= X 2 fb(nk ) sin2 k∈Z nk h 2 that B≤ 1/r Ωh,J Z , r1 p |fj (x)| dx . On the Absolute Convergence of Small Gaps Fourier V Series of Functions of BV (p) J Thus Z r B ≤ Ωh,J p |fj (x)| dx . R.G. Vyas J Now multiplying both the sides of the equation by λ1j and then taking the summation over j = 1 to nT (T ∈ N) we get R P |fj (x)|p Ωh,J J dx j λj P 1 Br ≤ . j λj Title Page Contents JJ J II I Go Back Therefore B≤ Ω Ph,J1 j λj ! r1 Z J p X |fj (x)| j λj ! r1 ! dx . Substituting back the value of B and then integrating both the sides of the equa- Close Quit Page 10 of 12 J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005 http://jipam.vu.edu.au tion with respect to h over (0, nπT ), we get (1.10) X k∈Z 2 Z π/nT fb(nk ) sin 2 0 |nk | h 2 r1 Ω1/n ,J = O P T 1 dh π/nT Z 0 j λj X |fj (x)|p Z J λj j ! r1 ! dx dh. Observe that for x in J, h in (0, nπT ) and for each j of the summation the points V x + 2jh and x + (2j − 1)h lie in I; moreover,f ∈ BV (p) (I) implies R.G. Vyas X |fj (x)|p j = O(1). λj Title Page Therefore, it follows from (1.10) and Lemma 1.4 that RnT ≡ X 2 fb(nk ) =O Ω1/n ,I PnT T 1 ω RnT = O Contents ! r1 . j=1 λj |nk |≥nT Thus On the Absolute Convergence of Small Gaps Fourier V Series of Functions of BV (p) (2−p)s+p P 1 nT , f, I r1 nT 1 j=1 λj 2−p/r JJ J II I Go Back Close . Now proceeding as in the proof of Theorem 1.1, the theorem is proved using Lemma 1.5. Quit Page 11 of 12 J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005 http://jipam.vu.edu.au References [1] A. ZYMUND, Trigonometric Series, 2nd ed., Cambridge Univ. Press, Cambridge, 1979 (reprint). [2] J.R. PATADIA AND R.G. VYAS, Fourier series with small gaps and functions of generalized variations, J. Math. Anal. and Appl., 182(1) (1994), 113–126. [3] M. SCHRAM AND V D. WATERMAN, V Absolute convergence of Fourier se(p) ries of functions of BV and Φ BV , Acta. Math. Hungar, 40 (1982), 273–276. On the Absolute Convergence of Small Gaps Fourier V Series of Functions of BV (p) R.G. Vyas [4] M. SHIBA, On the absolute convergence of Fourier series of functions of V (p) class BV , Sci. Rep. Fukushima Univ., 30 (1980), 7–10. [5] N.K. BARRY, A Treatise on Trigonometric Series, York,1964. Pergamon, New [6] N.V. PATEL AND V.M. SHAH, A note on the absolute convergence of lacunary Fourier series, Proc. Amer. Math. Soc., 93 (1985), 433–439. Title Page Contents JJ J II I Go Back Close Quit Page 12 of 12 J. Ineq. Pure and Appl. Math. 6(1) Art. 23, 2005 http://jipam.vu.edu.au