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A P Solodov - One of the best experts on this subject based on the ideXlab platform.
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sharp two sided estimate for the sum of a Sine Series with convex slowly varying sequence of coefficients
Analysis Mathematica, 2020Co-Authors: A P SolodovAbstract:The sum of a Sine Series $$g\left({b,x} \right) = \sum\nolimits_{k = 1}^\infty {}$$ bk sin kx with coefficients forming a convex sequence b is known to be positive on the interval (0,π). To estimate its values near zero Telyakovskiĭ used the piecewise-continuous function $$\sigma \left({{\bf{b}},x} \right) = \left({1/m\left(x \right)} \right)\sum\nolimits_{k = 1}^{m\left(x \right) - 1} {{k^2}\left({{b_k} - {b_{k + 1}}} \right),\,\,m\left(x \right) = \left[{\pi /x} \right]}$$ . He showed that in some neighborhood of zero the difference g(b,x) − (bm(x)/2) cot(x/2) can be estimated from both sides two-sided in terms of the function σ(b,x) with absolute constants. In the present paper, sharp values of these constants on the class of convex slowly varying sequences b are found. A sharp two-sided estimate for the sum of a Sine Series on this class is obtained. Examples that demonstrate good accuracy of the obtained two-sided estimates are given.
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Exact Constants in Telyakovskii’s Two-Sided Estimate of the Sum of a Sine Series with Convex Sequence of Coefficients
Mathematical Notes, 2020Co-Authors: A P SolodovAbstract:It is known that the sum of the Sine Series $$g\left( {{\rm{b,}}\,x} \right) = \sum\nolimits_{k = 1}^\infty {{b_k}} $$bk sin kx whose coefficients constitute a convex sequence b is positive on the interval (0, π). To estimate its values in a neighborhood of zero, Telyakovskii used the piecewise continuous function $$\sigma \left( {{\mathop{\rm b}\nolimits} ,\,x} \right) = {1 \over {m\left( x \right)}}\sum\limits_{k = 1}^{m\left( x \right) - 1} {{k^2}\left( {{b_k} - {b_{k + 1}}} \right),\,\,\,\,\,\,\,\,\,\,\,m\left( x \right) = \left[ {{{\rm{\pi }} \over x}} \right].} $$ He showed that the difference g(b, x) − (bm(x)/2)cot(x/2) in a neighborhood of zero admits a two-sided estimate in terms of the function a(b,x) with absolute constants. The exact values of these constants for the class of convex sequences b are obtained in this paper.
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exact constants in telyakovskii s two sided estimate of the sum of a Sine Series with convex sequence of coefficients
Mathematical Notes, 2020Co-Authors: A P SolodovAbstract:It is known that the sum of the Sine Series $$g\left( {{\rm{b,}}\,x} \right) = \sum\nolimits_{k = 1}^\infty {{b_k}} $$bk sin kx whose coefficients constitute a convex sequence b is positive on the interval (0, π). To estimate its values in a neighborhood of zero, Telyakovskii used the piecewise continuous function $$\sigma \left( {{\mathop{\rm b}\nolimits} ,\,x} \right) = {1 \over {m\left( x \right)}}\sum\limits_{k = 1}^{m\left( x \right) - 1} {{k^2}\left( {{b_k} - {b_{k + 1}}} \right),\,\,\,\,\,\,\,\,\,\,\,m\left( x \right) = \left[ {{{\rm{\pi }} \over x}} \right].} $$ He showed that the difference g(b, x) − (bm(x)/2)cot(x/2) in a neighborhood of zero admits a two-sided estimate in terms of the function a(b,x) with absolute constants. The exact values of these constants for the class of convex sequences b are obtained in this paper.
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Estimates with Sharp Constants of the Sums of Sine Series with Monotone Coefficients of Certain Classes in Terms of the Salem Majorant
Mathematical Notes, 2018Co-Authors: A. Yu. Popov, A P SolodovAbstract:New estimates of the sums of Sine Series with monotone coefficients of special classes in terms of the Salem majorant are obtained. The asymptotic sharpness of the obtained estimates for sequences of coefficients from the classes under consideration is proved.
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a sharp lower bound for the sum of a Sine Series with convex coefficients
Sbornik Mathematics, 2016Co-Authors: A P SolodovAbstract:The sum of a Sine Series with coefficients forming a convex sequence is known to be positive on the interval . Its values near zero are conventionally evaluated using the Salem function , . In this paper it is proved that is not a minorant for . The modified Salem function is shown to satisfy the lower bound in some right neighbourhood of zero. This estimate is shown to be sharp on the class of convex sequences . Moreover, the upper bound for is refined on the class of monotone sequences . Bibliography: 11 titles.
A. Yu. Popov - One of the best experts on this subject based on the ideXlab platform.
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Two-Sided Estimates of the $$L^\infty$$ -Norm of the Sum of a Sine Series with Monotone Coefficients $$\{b_k\}$$ via the $$\ell^\infty$$ -Norm of the Sequence $$\{kb_k\}$$
Mathematical Notes, 2020Co-Authors: E D Alferova, A. Yu. PopovAbstract:We refine the classical boundedness criterion for sums of Sine Series with monotone coefficients $$b_k$$ : the sum of a Series is bounded on $$\mathbb R$$ if and only if the sequence $${\{kb_k\}}$$ is bounded. We derive a two-sided estimate of the Chebyshev norm of the sum of a Series via a special norm of the sequence $$\{kb_k\}$$ . The resulting upper bound is sharp, and the constant in the lower bound differs from the exact value by at most $$0.2$$ .
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Estimates with Sharp Constants of the Sums of Sine Series with Monotone Coefficients of Certain Classes in Terms of the Salem Majorant
Mathematical Notes, 2018Co-Authors: A. Yu. Popov, A P SolodovAbstract:New estimates of the sums of Sine Series with monotone coefficients of special classes in terms of the Salem majorant are obtained. The asymptotic sharpness of the obtained estimates for sequences of coefficients from the classes under consideration is proved.
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Estimate for the integral of the absolute value of a Sine Series with monotone coefficients
Proceedings of the Steklov Institute of Mathematics, 2013Co-Authors: A. Yu. Popov, S. A. TelyakovskiiAbstract:Two-sided estimates are established for the integral of the absolute value of a Sine Series with monotonically decreasing coefficients. The integral of the absolute value of the conjugate Dirichlet kernel is represented as a finite sum.
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Estimates of the Sums of Sine Series with Monotone Coefficients of Certain Classes
Mathematical Notes, 2003Co-Authors: A. Yu. PopovAbstract:In this paper, we obtain two-sided estimates with sharp constants of the sums of Sine Series with a convex sequence of coefficients tending to zero.
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Estimates of the Sums of Sine Series with Monotone Coefficients of Certain Classes
Mathematical Notes, 2003Co-Authors: A. Yu. PopovAbstract:summary:In this paper we obtain estimates of the sum of double Sine Series near the origin, with monotone coefficients tending to zero. In particular (if the coefficients $a_{k,l}$ satisfy certain conditions) the following order equality is proved $$ g(x,y)\sim mna_{m,n}+\frac mn\sum _{l=1}^{n-1}la_{m,l}+\frac nm\sum _{k=1}^{m-1}ka_{k,n}+\frac 1{mn}\sum _{l=1}^{n-1}\sum _{k=1}^{m-1}kla_{k,l}, $$ where $x\in (\frac {\pi }{m+1}, \frac {\pi }m]$, $ y\in (\frac {\pi }{n+1}, \frac {\pi }n]$, $ m, n=1,2,\dots $
Bogdan Szal - One of the best experts on this subject based on the ideXlab platform.
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On weighted integrability of double Sine Series
Journal of Mathematical Analysis and Applications, 2019Co-Authors: Krzysztof Duzinkiewicz, Bogdan SzalAbstract:Abstract We introduce a new class of double sequences D G M ( α , β , γ , r ) called Double General Monotone and some new class of weight functions to study the weighted integrability of double Sine Series. Some results of D. Yu, P. Zhou and S. Zhou are also generalized.
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On the uniform convergence of double Sine Series
arXiv: Classical Analysis and ODEs, 2015Co-Authors: Krzysztof Duzinkiewicz, Bogdan SzalAbstract:The fundamental theorem in the theory of the uniform convergence of Sine Series is due to Chaundy and Jolliffe from 1916 (see [1]). Several authors gave conditions for this problem supposing that coefficients are monotone, non-negative or more recently, general monotone (see [8], [6] and [3], for example). There are also results for the regular convergence of double Sine Series to by uniform in case the coefficients are monotone or general monotone double sequences. In this article we give new sufficient conditions for the uniformity of the regular convergence of double Sine Series, which are necessary as well in case the coefficients are non-negative. We shall generalize those results defining a new class of double sequences for the coefficients.
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Uniform convergence of double Sine Series
2013Co-Authors: Krzysztof Duzinkiewicz, Bogdan SzalAbstract:is nbn → 0. Several authors gave conditions for this problem supposing that coefficients are monotone, non-negative or more recently, general monotone (see [2] and [3], for example). There are also result for the regular convergence of double Sine Series to by uniform in case the coefficients are monotone or general monotone double sequences. In this presentations we give new sufficient conditions for the uniformity of the regular convergence of double Sine Series, which are necessary as well in case the coefficients are non-negative. The new results also bring necessary and sufficient conditions for the uniform regular convergence of double trigonometric Series in complex form. We shall generalize those results defining a new class of double sequences for the coefficients.
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A new class of numerical sequences and its applications to uniform convergence of Sine Series
Mathematische Nachrichten, 2011Co-Authors: Bogdan SzalAbstract:In the present paper we introduce a new class of sequences called GM (�,r), which is the generalization of a class considered by Tikhonov in [13]. Moreover, we obtained in this note sufficient and necessary conditions f or uniform convergence of Sine Series with (�,r) −general monotone coefficients.
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a note on the uniform convergence and boundedness of a generalized class of Sine Series
Commentationes Mathematicae, 2008Co-Authors: Bogdan SzalAbstract:In this paper we essentially extend the Leindler’s results concerning the uniform convergence and boundedness of a certain class of Sine Series. 2000 Mathematics Subject Classification: 40A30, 42A10.
Sergiusz Kęska - One of the best experts on this subject based on the ideXlab platform.
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On the Uniform Convergence of Sine Series with Square Root
Journal of Function Spaces, 2019Co-Authors: Sergiusz KęskaAbstract:Chaundy and Jolliffe proved that if {ck}k=1∞ is a nonincreasing real sequence with limk→∞ck=0, then the Series ∑k=1∞cksinkx converges uniformly if and only if kck→0. The purpose of this paper is to show that kck→0 is a necessary and sufficient condition for the uniform convergence of Series ∑k=1∞cksinkθ in θ∈[0,π]. However for ∑k=1∞cksink2θ it is not true in θ∈[0,π].
Yu A Popov - One of the best experts on this subject based on the ideXlab platform.
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two sided estimates of the l infty norm of the sum of a Sine Series with monotone coefficients b_k via the ell infty norm of the sequence kb_k
Mathematical Notes, 2020Co-Authors: E D Alferova, Yu A PopovAbstract:We refine the classical boundedness criterion for sums of Sine Series with monotone coefficients $$b_k$$ : the sum of a Series is bounded on $$\mathbb R$$ if and only if the sequence $${\{kb_k\}}$$ is bounded. We derive a two-sided estimate of the Chebyshev norm of the sum of a Series via a special norm of the sequence $$\{kb_k\}$$ . The resulting upper bound is sharp, and the constant in the lower bound differs from the exact value by at most $$0.2$$ .
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estimates of the least positive root of the sum of a Sine Series with monotone coefficients
Mathematical Notes, 2014Co-Authors: Yu A PopovAbstract:In this paper, the extremal problem of finding the infimum of the positive roots of the sum of a Sine Series with monotone coefficients for special subclasses of such Series is solved.
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exact lower estimate of the upper limit of the ratio of the sum of Sine Series with monotone coefficients to its majorant
Moscow University Mathematics Bulletin, 2014Co-Authors: Yu A Popov, A P SolodovAbstract:An unimprovable lower estimate of the upper limit of the ratio of the sum of Sine Series with monotone coefficients to its majorant is obtained.