GIBBS' PHENOMENON FOR A FAMILY OF SUMMABILITY METHODS. (Received December 14, 1965)

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1 Tohoku Math. Journ. Vol. 18, No. 1, 1966 GIBBS' PHENOMENON FOR A FAMILY OF SUMMABILITY METHODS KYUHEI IKENO (Received December 14, 1965) (1.1) We can prove the above formula (1.1) by the same calculation as the one which we use in order to obtain Lebesgue constant for a family of summability methods (see [3]). Gibbs' phenomenon for this family is independent of the parameter a of F(a, q(p)). Since sn(x) is odd function of x, a similar phenomenon occurs in the left-hand neighourhood of x=0.

2 104 K. IKENO (2.1) (2.2) (2.4) 3. Two lemmas. In order to prove the formula (1.1), we require the following two lemmas.

3 GIBBS' PHENOMENON FOR A FAMILY OF SUMMABILITY METHODS 105 (3.1) The proof follows from (2.1) and (2.2) by a simple calculation. LEMMA 3.2. If {xp} satisfies the condition which is mentioned in section 1 and p tends to infinity, then we have (3.2) and (3.3) PROOF. From the condition on {xp}, for sufficiently large p, we get Similary, we get for sufficiently large p, 4. Gibbs' phenomenon. In this section we consider the Gibbs' phenomenon for a family of summability methods whose matrix [cpk] belongs to F(a,q(p)).

4 106 K. IKENO means of a family of summability methods whose matrix belongs to F(a,q(p)). (1.1) PROOF. From lemma 3.1, (2.2) and (2.4), we have we put I1(xp) and I2(xp) as follows: Applying large p, lemma 3.2 and (2.2) to I1(xp) and I2(xp), we get for sufficiently and

5 GIBBS' PHENOMENON FOR A FAMILY OF SUMMABILITY METHODS 107 Then we obtain (4.1) When we put n=k-q(p), we have Using the property of Theta function [8], we get (4.2) From (4.1) and (4.2), we get (4.3) We put f(u,p) and Dp(xp) as follows:

6 108 K. IKENO (4.4) Applying integration by parts to Dp(xp), we get (4.5) (4.6) From (4.5) and (4.6), we get (4.7) Consequently we get from (4.3), (4.4) and (4.7) for sufficiently large p (4.8) Thus the theorem has been proved when q=q(p) is integer. Next we shall consider the other case. Let [q] denote the integral part of q=q(p) and q0=[q]+1. We put D1(xp), D2(xp), D3(xp) and D4(xp) as follows: (4.9)

7 GIBBS' PHENOMENON FOR A FAMILY OF SUMMABILITY METHODS 109 Hence the following estimation results: Then we get (4.10) Hence the following estimation results just as in the case of i):

8 110 K. IKENO Then we get (4.11) Next we shall estimate D3(xp), D4(xp) and we get (4.12)

9 GIBBS' PHENOMENON FOR A FAMILY OF SUMMABILITY METHODS 111 Thus we have obtained Gibbs' phenomenon for a family of summability methods whose matrix [cpk] belongs to F(a,q(p)). From the theorem, we get the following results: i) Borel-transformation (see L. Lorch [4]). The summability matrix of Borel-transformation is defined by where p>0, a=1/2 and q(p)=p (see A. Meir [5]). If we define Bp(x) by the linear transformation of sn(x) by means of Borel-transformation, we have from (1.1) ii) Valiron-transformation. The summability matrix of Valiron-transformation is defined by If we define Vp(x) by the linear transformation of sn(x) by means of Valiron-transformation, we have

10 112 K. IKENO iii) Euler-transformation (see O. Szasz [7]). The summability matrix of Euler-transformation is defined by iv) Taylor-transformation (see K.Ishiguro [1]). The summability matrix of Taylor-transformation is defined by

11 GIBBS' PHENOMENON FOR A FAMILY OF SUMMABILITY METHODS 113 REFERENCES [1] K. ISHIGURO, Zur Gibbsschen Erscheinung fur das Kreisverfahren, Math. Zeitschr., 76 (1961), [3] K. IKENO, Lebesgue constants for a family of summability methods, Tohoku Math. Journ., 17 (1965), [4] L. LORCH, The Gibbs phenomenon for Borel means, Proc. Amer. Math. Soc., 8 (1957), [5] A. MEIR, Tanberian constants for a family of transformations, Ann. of Math., 78 (1963), [6] C.L. MIRACLE, The Gibbs phenomenon for Taylor means and for [F, dn] means, Canad. Journ. Math., 12 (1960), [7] O. SZASZ, On the Gibbs' phenomenon for Euler means, Acta Univ. Szeged, 12 (1950), [8] E.T. WHITTAKER AND G.N. WATSON, A Course of Modern Analysis, Cambridge University Press, AKITA UNIVERSITY.