On the absolute summability of some series related to a Fourier series

Ray, B. K.

doi:10.1017/s0305004100057078

Cited by 8 publications

(2 citation statements)

References 14 publications

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“…(iii) Theorem II extends a theorem due to Ray [7,Theorem 1] on Cesàro summability of Fourier series. Ray's result corresponds to the case ß = 0, tj = 8 < 0.…”

Section: Remarks (I)supporting

confidence: 53%

Absolute Riesz summability of Fourier series. I

Dikshit¹,

Rees²

1981

Proc. Amer. Math. Soc.

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Abstract. In this paper we prove some theorems on the absolute summability of Fourier series which connect diverse \C, y\ results such as Bosanquet's classical theorem (1936), Mohanty (1952), and Ray (1970) and the recent |it, exp((log w)"+1), y\ result of Nayak (1971).It is also shown that in some sense some of the conclusions of the paper are the best possible.1.1. A series 2 Un is said to be absolutely summable by the Riesz method of 'type' exp((log u)0+x), ß > 0, and 'order' r, r > 0, and written Ja (e( for suitably chosen j = s(x), e(0, F(o>, t) = 2") = {e(u) -e(m)y~xe(m)m~x(\og rnf, where m is an integer such that m < w < m + 1.Unless otherwise specified, in what follows we use "2" to denote "2"_2" and also 2"

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“…(iii) Theorem II extends a theorem due to Ray [7,Theorem 1] on Cesàro summability of Fourier series. Ray's result corresponds to the case ß = 0, tj = 8 < 0.…”

Section: Remarks (I)supporting

confidence: 53%

Absolute Riesz summability of Fourier series. I

Dikshit¹,

Rees²

1981

Proc. Amer. Math. Soc.

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“…For Tj = 0, we see by (7), the right-hand side in (10) is zero. Since 2 n"'(log rif~r'~x, tj -5 > 0, is absolutely convergent, to prove that 2 An(x)(log nf G | F, e(co), y| (whether tj = 0 or not), it is enough to show that the integral ["(logco/co-'e-^co) 2 {e(v) -e(n)}y-xe(n)(logn)sn-x /"sin nt(log(k/t))-r' d{*(t)(log(k/t)y} du is convergent.…”

Section: Remarks (I)mentioning

confidence: 96%

Absolute Riesz Summability of Fourier Series. I

Dikshit¹,

Rees²

1981

Proceedings of the American Mathematical Society

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In this paper we prove some theorems on the absolute summability of Fourier series which connect diverse \C, y\ results such as Bosanquet's classical theorem (1936), Mohanty (1952), and Ray (1970) and the recent |it, exp((log w)"+1), y\ result of Nayak (1971). It is also shown that in some sense some of the conclusions of the paper are the best possible. 1.1. A series 2 Un is said to be absolutely summable by the Riesz method of 'type' exp((log u)0+x), ß > 0, and 'order' r, r > 0, and written Ja (e( for suitably chosen j = s(x), e(0, F(o>, t) = 2") = {e(u)-e(m)y~xe(m)m~x(\og rnf, where m is an integer such that m < w < m + 1. Unless otherwise specified, in what follows we use "2" to denote "2"_2" and also 2"

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