Quantitative Behavior of Transforms with Norm-Bounded Partial Sums

1980 ◽  
Vol 102 (2) ◽  
pp. 321 ◽  
Author(s):  
Louis Pigno ◽  
Brent Smith
Keyword(s):  
Partial Sums ◽  
Bounded Partial Sums

Bounded Partial Sums?

PRIMUS ◽  
2008 ◽  
Vol 18 (3) ◽  
pp. 276-282
Author(s):  
Michael A. Brilleslyper ◽  
Robert H. Wolverton
Keyword(s):  
Partial Sums ◽  
Bounded Partial Sums

scholarly journals Correlations of multiplicative functions and applications

2017 ◽  
Vol 153 (8) ◽  
pp. 1622-1657 ◽  
Author(s):  
Oleksiy Klurman
Keyword(s):  
Asymptotic Formula ◽  
Multiplicative Function ◽  
Divided Difference ◽  
Circle Method ◽  
Partial Sums ◽  
Multiplicative Functions ◽  
Image Position ◽  
Bounded Partial Sums

We give an asymptotic formula for correlations $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$ where $f,\ldots ,f_{m}$ are bounded ‘pretentious’ multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative functions $f:\mathbb{N}\rightarrow \{-1,+1\}$ with bounded partial sums. This answers a question of Erdős from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of the multiplicative function is zero, then either $f(n)=n^{s}$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of Kátai. Third, we apply our theorem to count the number of representations of $n=a+b$, where $a,b$ belong to some multiplicative subsets of $\mathbb{N}$. This gives a new ‘circle method-free’ proof of a result of Brüdern.

scholarly journals On Multiplicative Functions with Bounded Partial Sums

Integers ◽  
2012 ◽  
Vol 12 (4) ◽  
Author(s):  
Joseph Vandehey
Keyword(s):  
Multiplicative Function ◽  
Partial Sums ◽  
Multiplicative Functions ◽  
Bounded Partial Sums

Abstract.Consider a multiplicative function

scholarly journals Strongly Bounded Partial Sums

2014 ◽  
Vol 33 (2) ◽  
pp. 205-213
Author(s):  
Charles Swartz
Keyword(s):  
Partial Sums ◽  
Bounded Partial Sums

scholarly journals Abel and Riesz Transforms of Series Having Bounded Partial Sums

1954 ◽  
Vol 3 (1) ◽  
pp. 47-72
Author(s):  
Ralph Agnew
Keyword(s):  
Riesz Transforms ◽  
Partial Sums ◽  
Bounded Partial Sums

scholarly journals On a new proof and sharpening of a result of Fejer on bounded partial sums

1988 ◽  
Vol 107 ◽  
pp. 237-251
Author(s):  
P. Oliver ◽  
Q.I. Rahman ◽  
R.S. Varga
Keyword(s):  
Partial Sums ◽  
Bounded Partial Sums

scholarly journals On the coefficients in Fourier-Stieltjes series with norm-bounded partial sums

1985 ◽  
Vol 49 (2) ◽  
pp. 257-265 ◽  
Author(s):  
John J. F. Fournier
Keyword(s):  
Partial Sums ◽  
Stieltjes Series ◽  
Bounded Partial Sums

Abel Transformations Into I1

1982 ◽  
Vol 25 (4) ◽  
pp. 421-427 ◽  
Author(s):  
J. A. Fridy
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Partial Sums ◽  
Sequence Variant ◽  
Abel Summability ◽  
Summability Methods ◽  
The Matrix ◽  
Bounded Partial Sums

AbstractLet t be a sequence in (0,1) that converges to 0, and define the Abel matrix At by ank = tn(1-tn)k. The matrix At determines a sequence-to-sequence variant of the classical Abel summability method. The purpose of this paper is to study these transformations as l-l summability methods: e.g., At maps l1 into l1 if and only if t is in l1. The Abel matrices are shown to be stronger l-l methods than the Euler-Knopp means and the Nӧrlund means. Indeed, if t is in l1 and Σ xk has bounded partial sums, then Atx is in l1. Also, the Abel matrix is shown to be translative in an l-l sense, and an l-l Tauberian theorem is proved for At.

On Rational Approximation of Markov Functions by Partial Sums of Fourier Series on a Chebyshev–Markov System

2020 ◽  
Vol 108 (3-4) ◽  
pp. 566-578
Author(s):  
E. A. Rovba ◽  
P. G. Potseiko
Keyword(s):  
Fourier Series ◽  
Rational Approximation ◽  
Partial Sums ◽  
Markov System
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