Convergence to Zero of Exponential Sums with Positive Integer Coefficients and Approximation by Sums of Shifts of a Single Function on the Line

Бородин, Петр Анатольевич; Konyagin, Sergeĭ

doi:10.1007/s10476-018-0204-2

Cited by 18 publications

(5 citation statements)

References 9 publications

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“…Theorem 4.6 (Borodin [17]). There is a function f : R → R such that the sums n k=1 f (x − a k ) of its shifts are dense in all real spaces L p (R) for 2 ⩽ p < ∞ and also in the real space C 0 (R).…”

Section: Shifts On the Linementioning

confidence: 99%

“…For p = 1 it is obviously not valid, since shifts of one function do not form an all-round set in L 1 (R): the functional f → R f (x) dx takes the values of the same sign at all these shifts. In L ∞ (R), where an analogue of Theorem 4.6 is not valid either, the role of the forbidding functional is played by the Banach limit at +∞ [17].…”

Section: Shifts On the Linementioning

confidence: 99%

“…Theorem 4.6 fails in the complex spaces L p (R) [17]; however, an analogue of it is valid for Hardy spaces in the half-plane.…”

Section: Shifts On the Linementioning

confidence: 99%

“…Theorem 5.6 (Konyagin [17]). There exists a sequence of trigonometric polynomials This very difficult theorem raises the natural question on the existence of a sequence of trigonometric polynomials with positive integer coefficients that would converge to zero in the L 2 -norm or even uniformly on each interval [δ, 2π − δ], δ ∈ (0, π).…”

Section: Density Of Polynomials With Integer Coefficientsmentioning

confidence: 99%

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Density of quantized approximations

Borodin,

Shklyaev

2023

Russian Math. Surveys

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The paper contains a review of known results and proofs of new results on conditions on a set $M$ in a Banach space $X$ that are necessary or sufficient for the additive semigroup $R(M)=\{x_1+…+x_n\colon x_k\in M, n\in {\mathbb N}\}$ to be dense in $X$. We prove, in particular, that if $M$ is a rectifiable curve in a uniformly smooth real space $X$, and $M$ does not lie entirely in any closed half-space, then $R(M)$ is dense in $X$. We present known and new results on the approximation by simple partial fractions (logarithmic derivatives of polynomials) in various spaces of functions of a complex variable. Meanwhile, some well-known theorems, in particular, Korevaar's theorem, are derived from new general results on the density of a semigroup. We also study approximation by sums of shifts of one function, which are a natural generalization of simple partial fractions. Bibliography: 79 titles.

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Section: Shifts On the Linementioning

confidence: 99%

Section: Shifts On the Linementioning

confidence: 99%

“…Theorem 4.6 fails in the complex spaces L p (R) [17]; however, an analogue of it is valid for Hardy spaces in the half-plane.…”

Section: Shifts On the Linementioning

confidence: 99%

Section: Density Of Polynomials With Integer Coefficientsmentioning

confidence: 99%

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Density of quantized approximations

Borodin,

Shklyaev

2023

Russian Math. Surveys

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“…Èìååòñÿ îïðåäåëåííûé èíòåðåñ ê ðàçëîaeåíèÿì â ðÿä ñ öåëûìè êîýôôèöèåíòàìè. Íàïðèìåð, â [6] ïðèâîäèòñÿ ðåçóëüòàò î ñóùåñòâîâàíèè ïîñëåäîâàòåëüíîñòè òðèãîíîìåòðè÷åñêèõ ìíîãî÷ëåíîâ ñ öåëûìè ïîëîaeèòåëüíûìè êîýôôèöèåíòàìè, êîòîðàÿ ñõîäèòñÿ ê íóëþ ïî÷òè âñþäó.…”

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Series of Fourier type with integer coefficients by systems of dilates and translates of one function in Lp, p ≥ 1

Filippov¹

2019

Iz. VUZ. Matematika

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Series of Fourier type with integer coecients by systems of dilates and translates of one function in L p , p ≥ 1 Abstract. We consider systems of functions obtained from dilates and translations of one function in the spaces L p (0, 1), 1 ≤ p < ∞. We obtain results on the representation with respect to these systems with integer coefficients of the Fourier type series. The approximation of the elements of the spaces L p (0, 1), 1 ≤ p < ∞, according to the proposed methods has the property of image compression, that is, many coefficients are zero in this case. These studies may be of interest to specialists in the transfer and processing of digital information.

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Approximation by sums of shifts and dilations of a single function and neural networks

Shklyaev

2023

Journal of Approximation Theory

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Convergence to Zero of Exponential Sums with Positive Integer Coefficients and Approximation by Sums of Shifts of a Single Function on the Line

Cited by 18 publications

References 9 publications

Density of quantized approximations

Density of quantized approximations

Series of Fourier type with integer coefficients by systems of dilates and translates of one function in Lp, p ≥ 1

Approximation by sums of shifts and dilations of a single function and neural networks

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