scholarly journals Approximation of Conjugate Functions by General Linear Operators of Their Fourier Series at the Lebesgue Points

2014 ◽  
Vol 47 (4) ◽  
Author(s):  
Włodzimierz Łenski ◽  
Bogdan Szal
Keyword(s):  
Fourier Series ◽  
Linear Operators ◽  
General Linear ◽  
Conjugate Functions ◽  
Moduli Of Continuity ◽  
Pointwise Estimates ◽  
Lebesgue Points ◽  
Norm Approximation

AbstractThe pointwise estimates of the deviations r T͂n,A,Bf (·) - f͂͂ (·) and T͂n,A,Bf (·) - f͂͂ (·,ε) in terms of moduli of continuity ω̃f and r ω̃f are proved. Analogical results on norm approximation with remarks and corollary are also given. These results generalized a theorem of Mittal [3, Theorem 1, p. 437].

scholarly journals On Pointwise Approximation of Conjugate Functions by Some Hump Matrix Means of Conjugate Fourier Series

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
W. Łenski ◽  
B. Szal
Keyword(s):  
Fourier Series ◽  
Conjugate Functions ◽  
Pointwise Approximation ◽  
Conjugate Fourier Series ◽  
Summability Methods ◽  
Matrix Means ◽  
Special Cases ◽  
Norm Approximation

The results generalizing some theorems onN, pnE, γsummability are shown. The same degrees of pointwise approximation as in earlier papers by weaker assumptions on considered functions and examined summability methods are obtained. From presented pointwise results, the estimation on norm approximation is derived. Some special cases as corollaries are also formulated.

scholarly journals On almost-Hermite-Fejér-interpolation: pointwise estimates

1982 ◽  
Vol 25 (3) ◽  
pp. 405-423 ◽  
Author(s):  
Heinz H. Gonska
Keyword(s):  
Continuous Function ◽  
Positive Operators ◽  
Linear Operators ◽  
Moduli Of Continuity ◽  
Continuous Linear ◽  
Pointwise Estimates ◽  
General Technique ◽  
Different Types ◽  
Continuous Linear Operators

We give a brief survey of the results obtained by numerous authors in so-called almost-Hermite-Fejér-interpolation and deal mainly with new quantitative assertions.These are based upon more general theorems for certain continuous linear operators which yield estimates involving different types of moduli of continuity.Our paper shows that in the case of almost-Hermite-Fejér-interpolation the underlying general technique can be used to treat three essentially different cases: sequences of positive operators, which converge uniformly for every continuous function on [−1, 1], sequences of non-positive operators doing the same, and sequences of operators which converge on proper subspaces of C[−1, 1] only.

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

scholarly journals Lebesgue points of $$\ell _1$$-Cesàro summability of d-dimensional Fourier series

2021 ◽  
Vol 6 (3) ◽  
Author(s):  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Lebesgue Point ◽  
Cesàro Means ◽  
Cesàro Summability ◽  
Cesaro Means ◽  
Cesaro Summability ◽  
Lebesgue Points ◽  
Cesáro Means

AbstractWe generalize the classical Lebesgue’s theorem and prove that the $$\ell _1$$ ℓ 1 -Cesàro means of the Fourier series of the multi-dimensional function $$f\in L_1({{\mathbb {T}}}^d)$$ f ∈ L 1 ( T d ) converge to f at each strong $$\omega $$ ω -Lebesgue point.

scholarly journals Pointwise estimates for higher order convexity preserving polynomial approximation

1994 ◽  
Vol 36 (2) ◽  
pp. 213-233 ◽  
Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska
Keyword(s):  
Polynomial Approximation ◽  
Convex Functions ◽  
Higher Order ◽  
Second Order ◽  
Convolution Type ◽  
Shape Preservation ◽  
Moduli Of Continuity ◽  
Pointwise Estimates ◽  
Higher Order Convexity ◽  
Boolean Sum

AbstractDeVore-Gopengauz-type operators have attracted some interest over the recent years. Here we investigate their relationship to shape preservation. We construct certain positive convolution-type operators Hn, s, j which leave the cones of j-convex functions invariant and give Timan-type inequalities for these. We also consider Boolean sum modifications of the operators Hn, s, j show that they basically have the same shape preservation behavior while interpolating at the endpoints of [−1, 1], and also satisfy Telyakovskiῐ- and DeVore-Gopengauz-type inequalities involving the first and second order moduli of continuity, respectively. Our results thus generalize related results by Lorentz and Zeller, Shvedov, Beatson, DeVore, Yu and Leviatan.

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