Best-possible bounds of Fourier coefficients of functions of bounded variation with respect to the Haar system

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.

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Some Bounds for the Complex Čebyšev Functional of Functions of Bounded Variation

Symmetry ◽

10.3390/sym13060990 ◽

2021 ◽

Vol 13 (6) ◽

pp. 990

Author(s):

Silvestru Sever Dragomir

Keyword(s):

Bounded Variation ◽

Functions Of Bounded Variation ◽

Functional Applications ◽

Čebyšev Functional

In this paper, we provide several bounds for the modulus of the complex Čebyšev functional. Applications to the trapezoid and mid-point inequalities, that are symmetric inequalities, are also provided.

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Curvature-dependent energies: a geometric and analytical approach

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210516000202 ◽

2017 ◽

Vol 147 (3) ◽

pp. 449-503 ◽

Cited By ~ 1

Author(s):

Emilio Acerbi ◽

Domenico Mucci

Keyword(s):

Bounded Variation ◽

Analytical Approach ◽

Total Curvature ◽

Representation Formula ◽

Energy Functional ◽

Continuous Functions ◽

Explicit Representation ◽

Functions Of Bounded Variation ◽

One Dimensional ◽

Relaxed Energy

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

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