On absolutely convergent trigonometric series

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  • PIPPERT, R. E. Math. Zeitschr. 85, 401--406 (1964)

    On absolutely convergent trigonometric series By

    RAYMOND E. PIPPERT

    1. Introduction

    In this note, we extend Theorem A of FATOU [1] and Theorem B of SALEM [3]. Using these results, we also obtain a theorem related to Theorem C of TsucmKURA and YANO [4].

    Theorem A. Let the sequence {an} satisfy ]a,+ 1 [ =< t anl for all n. If either (i) ~ ]an cos nXo ] < oo for some x o, or (ii) ~ [a n sin nx o [ < oo for some x o 4=0 (rood n),

    then ~ [a,[ < oo. Theorem B. Let ~ An(x)-ao/2+~(a, cos nx+b, sin nx), and write p ,=

    (aZ,+b2) -~. If p,+lO) and its conjugate series ~ p. sin(nx+0n) are absolutely convergent at x=x o and x=xl , respectively. If xl -xo=(p/q)n (p/q irreducible) where p is an integer, positive, negative, or zero, and q is an odd integer, then ~ Pn< oo.

    Remarks. ZYGMUND [5; p. 232--233] gives a proof of Theorem A due to Saks, and also gives a proof of Theorem B, similar to that of Salem, utilizing Theorem A. SALEM [3] actually proves a slightly more general result, replacing the condition that pn+l0. TsucmKURA and YANO [4] show that Theorem C is no longer true if p/q is replaced by p'/q' with an even q' and p' ~0, or by an irrational number. Hence, to improve the result of Theorem C, we require an additional hypothesis (in our case a restriction on {p.}).

    2. Statement of results

    Theorem 1. Let the sequence {an} satisfy ]an+l [ < Kla,] for all n, where K is a constant independent of n. If either

    (2.1) ~ [ancosnxol

  • 402 RAYMOND E. PIPPERT;

    and write p, = (a~ + b~) ~. Suppose the sequence {p.} satisfies p,+ , =b/a, we have the desired result.

    Proof of Theorem 1. We may clearly assume 0=

  • On absolutely convergent trigonometric series 403

    Now if nk< i_ 1. Thus

    ~[an l

  • 404 RAYMOND E. ~PERT:

    ~ cp(") n

    Proof of Lemma 2. Choose a sequence of positive integers {nk} such that nk>=2nk_ l, and ~o(x)>_k+ 1 for x>nk (k=l , 2, 3, ...). Define

    T(x)=k-+ x-nk for nk~X

  • On absolutely convergent trigonometric series 405

    since the series consists of every other term of a monotone divergent series of positive terms, and our assertion is established.

    Proof of Theorem 2. We suppose first that ~ I A. (x) I < oo for x = Xo and X=Xl, and write ~A.(x) -~p. sin (n x + 0.). Now let h=xo-xl. Then

    nh=(nxo + O,)-(nxl + O,) and

    sin n h = sin (n Xo + 0,) cos (n xi + O.) - cos (n Xo + 0,) sin (n xl + 0,). Thus

    I sin n h [ < I sin (n Xo + 0,) I + [ sin (n xi + 0,) [ and

    ~. [p, sin nhl

  • 406 RAYMOND E. PIPPERT: On absolutely convergent trigonometric series

    (i) an cos nxo--*O as n~oo for some Xo, or (ii) a n sin nxo~O as n~oo for some x 0 4=0 (rood ~),

    then an-~0 as n -~m.

    The result (A) can be obta ined in a di f ferent manner [5; p. 34, Ex. 2].

    Professor S. M. SI~AH has made valuable suggestions concerning the subject matter of this note, and has, in addition, given guidance in its preparation.

    Added in Proof. For a different proof of the first part of Theorem 1, see (i) SZASZ, O.: On the absolute convergence of trigonometric series. Ann. of Math.

    47, 213--220 (1946). (ii) Stmoucm, G., and S. YANO: Notes on Fourier Analysis. XXX. On the absolute

    convergence of certain series of functions. Proc. Amer. Math. Soc. 2,380-- 389(1951).

    References [1] FATOU, P.: Sur la convergence absolue des s6ries trigonom6triques. Bull. Soc. Math.

    France 41, 47-- 53 (1913). [21 HoBsoN, E.: The Theory of Functions of a Real Variable, Vol. 2. New York: Dover

    Publications 1957. [3] SALEM, R. : The absolute convergence of trigonometrical series. Duke Math. J. 8, 317--

    334 (1941). [4] TsucmKtraA, T., and S. YANO: On the absolute convergence of trigonometrical series.

    Proc. Amer. Math. Soc. 1, 517--521 (1950). [5] ZYGMUND, A.: Trigonometric Series, Vol. 1. New York: Cambridge Univ. Press 1959.

    Department of Mathematics, University of Kansas, Lawrence, Kansas, U.S.A.

    (Received May 13, 1964)