On Discontinuous Riesz Means of Type <i>n</i>

Kuttner, B.

doi:10.1112/jlms/s1-37.1.354

Cited by 21 publications

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“…Much attention has been given in the past to the inclusion C 5 R , K > -2 , that is, to the Orders k of mutual inclusion, and, in case of strict inclusion, to how C^ can be specified within R* . B.Kuttner's proof in [13] for C = R* , -2 < k < 2, and C ^ r* ^ 2 ^ k , wi1i be adapted to I C I C K T K yield H^ , 0 < k < 3 , and = H^ , 3 ^ k . (For the common ränge of strict inclusion, see below.)…”

Section: Basio Notation and Objeotivesmentioning

confidence: 99%

On the Nörlund means generated by real powers of n

Körle¹

1987

Analysis

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Nörlund methods H^;-fW, (n+1)'^ generalize harmonia limitation H^. They ahow atvong analogy to thc methoda R* (N^ ) of diaarete cacithmetia R i e s z meana, as regards their relation to the C e 3 ä r o methods C^. Also^ though not oomparable for k ^ 3, R* and H^ ahare the Kroneaker Tauberian oondition. By way of «a « '= "o " S " ' üc < 0 < ß < 3 , the Position of harmonia limitability betueen oonvergenoe and proper Cesäro one ia represented by monotony with reapeot to an inoreaaing parameter, ThuSf AHS Classification: AO G 05

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Section: Basio Notation and Objeotivesmentioning

confidence: 99%

On the Nörlund means generated by real powers of n

Körle¹

1987

Analysis

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“…3. According to Kakeya a power series 2, p n z n , Po>Pi>P2> ••-, n=0 p n > 0 (the series is Abel summable for \z\ < 1, z =£ 1), has no roots for |z| ^ 1 [3,4]. If we assume that {p n } is totally monotone, then our theorem shows, with the notation no roots for | z \ ^ 1 if | e | < \.…”

Section: N=0 N=lmentioning

confidence: 99%

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confidence: 99%

On the Modulus of Power Series of a Certain Type

Peyerimhoff

1965

Journal of the London Mathematical Society

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In a recent publication B. Kuttner [4] has proven that the functions QO f K (z) which are defined for | z | < l by £ (n-\-l) K z n have the property n=0 f K (z) ^ 0 for | z | < 1, 1 < K < 2. It is the purpose of this paper to give a new proof of this result. We obtain it as a special case of a theorem on the modulus of power series where the coefficients are totally monotone. This theorem also yields a generahzation of Kakeya's theorem [3] under the same assumption about the coefficients.1. We remark that the power series which are considered in what follows (with one exception in §3) have unique analytic extensions into the whole z-plane with the possible exception of z > 1 [and for/(z) = 2 a n z n , a n^Q , we will write /(l) = oo if 2 a n diverges]. A sequence {q n } is totally monotone if A k q n^Q for k, n = 0, 1, ... (Ag n = g n -q n+1 , A & q n = A*" 1 q n -A*" 1 q n+1 , k = 2,3,...), and by a theorem of Hausdorff [2] this is equivalent with a representation Jo g(t) monotonic increasing for 0 <£ < 1 (g(l) < oo) . THEOREM. J / / ( z ) = tt o + £ q n -i zn > a o rea^ an d {Qn} totally monotone, n = l then \f(re i0 )\^f(-r)pf(-l) for 0

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“…of discontinuous (R, X n , -1) summability is already available. Now it is known [5], that (C, k) and (R, n, k) are equivalent for -1 < k < 2, and Dr Kuttner has shown me a proof, similar to that of [5], that (R, n, -1) implies (C, -1 ) but that the converse implication is false. Thus (C, X n , -1 ) is not equivalent to (R, X n , -1 ) even when X n = n.…”

Section: = -mentioning

confidence: 99%

Generalized Cesaro means of order −1

Maddox¹

1966

Proceedings of the Glasgow Mathematical Association

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A series £ a n is said to be summable (C, -1) to s if it converges to 5 and na n = o(l) [8]. It is well known that this definition is equivalent to t n^s (n->oo), where t n -s n +na n , s n = <*o + ••• + a n-The series is summable | C, -1 | to s if the sequence t = {t n } is of bounded variation (t e B.V.), i.e. £ | At n | = £ | ?"-/"_! | < oo, and £Af n = lim t n = j . t An equivalent condition is £ | a n | < oo, £a,, = s and £ | A(wa n ) | < oo. For, suppose that Yfin -s | C, -11. Since {^n} is the sequence of (C, l)-means of {;"} and since | C, 0 | c | C, 1 |, we have £ | a n | < oo and Yfln = s > whence £ I A(«a n ) | < oo. Conversely, £ | a n | < oo, Yfln = s a n d S I ^( wfl n) I < °o imply t e B.V. and £Af n = j+lim «a n . But lim na n = 0, since Z I On I < 00-Now let Yfln be a given series, with s n = a o + ... +a n , and define the sequence {t n } so that s. is the discontinuous Riesz mean of order 1 of t.:where 0 ^ X o < X y < ... < X n -* oo. Then we have = -n -.We shall say that £)*" = i(C, X n , -1) if and only if t n -+ 5 (n -»oo). By the regularity of (£, A n , 1) summability it is easily seen that an equivalent definition is that £a n converges to s and Hn a n = °(1)-If ^n = «»(C, ^«> ~ 1) reduces to (C, -1 ) , so that the new method generalizes the Cesaro method of order -1. We have used the notation (C, k n , -1) rather than (R, A n , -1) since a definition? of discontinuous (R, X n , -1) summability is already available. Now it is known [5], that (C, k) and (R, n, k) are equivalent for -1 < k < 2, and Dr Kuttner has shown me a proof, similar to that of [5], that (R, n, -1) implies (C, -1 ) but that the converse implication is false. Thus (C, X n , -1 ) is not equivalent to (R, X n , -1 ) even when X n = n.Using (1) we define £a n = 5 | C, X n , -1 | if and only if / e B.V. and t n -> s. Thus we have the inclusion | C, X n , -11 c (C, X n , -1 ) . We now give an equivalent condition for | C, X n , -1 | summability. THEOREM 1. Y. a n = s\C,X n ,-l\ if and only ifZ \ a n \ < oo, £a B = J a«^Z | AOyO | < oo.t All summations run from 0 to oo, and we take /_ i =0.

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On Discontinuous Riesz Means of Type n

Cited by 21 publications

References 0 publications

On the Nörlund means generated by real powers of n

On the Nörlund means generated by real powers of n

On the Modulus of Power Series of a Certain Type

Generalized Cesaro means of order −1

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