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1
Content available remote

Geodesics in the Heisenberg Group

100%
Hajłasz P., Zimmerman S.
Analysis and Geometry in Metric Spaces
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2015
|
tom 3
|
nr 1
EN
We provide a new and elementary proof for the structure of geodesics in the Heisenberg group Hn. The proof is based on a new isoperimetric inequality for closed curves in R2n.We also prove that the Carnot- Carathéodory metric is real analytic away from the center of the group.
2
Content available remote

On least squares estimation of Fourier coefficients and of the regression function

100%
Popiński W.
Applicationes Mathematicae
|
1993-1995
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tom 22
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nr 1
91-102
EN
The problem of nonparametric function fitting with the observation model $y_i = f(x_i) + η_i$, i=1,...,n, is considered, where $η_i$ are independent random variables with zero mean value and finite variance, and $x_i \in [a,b] \subset \R^1$, i=1,...,n, form a random sample from a distribution with density $ϱ \in L^1[a,b]$ and are independent of the errors $η_i$, i=1,...,n. The asymptotic properties of the estimator $\widehat{f}_{N(n)}(x) = \sum_{k=1}^{N(n)} \widehat{c}_ke_k(x)$ for $f \in L^2[a,b]$ and $\widehat{c}^{N(n)}=( \widehat{c}_1,..., \widehat{c}_{N(n)})^T$ obtained by the least squares method as well as the limits in probability of the estimators $\widehat{c}_k$, k=1,...,N, for fixed N, are studied in the case when the functions $e_k$, k=1,2,..., forming a complete orthonormal system in $L^2\[a,b\]$ are analytic.
3
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Solving dual integral equations on Lebesgue spaces

100%
Ciaurri Ó., Guadalupe J. J., Pérez M., Varona J. L.
Studia Mathematica
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2000
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tom 142
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nr 3
253-267
EN
We study dual integral equations associated with Hankel transforms, that is, dual integral equations of Titchmarsh's type. We reformulate these equations giving a better description in terms of continuous operators on $L^p$ spaces, and we solve them in these spaces. The solution is given both as an operator described in terms of integrals and as a series $∑_{n=0}^{∞} c_n J_{μ+2n+1}$ which converges in the $L^p$-norm and almost everywhere, where $J_ν$ denotes the Bessel function of order ν. Finally, we study the uniqueness of the solution.
4
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On the $L_1$-convergence of Fourier series

100%
Fridli S.
Studia Mathematica
|
1997
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tom 125
|
nr 2
161-174
EN
Since the trigonometric Fourier series of an integrable function does not necessarily converge to the function in the mean, several additional conditions have been devised to guarantee the convergence. For instance, sufficient conditions can be constructed by using the Fourier coefficients or the integral modulus of the corresponding function. In this paper we give a Hardy-Karamata type Tauberian condition on the Fourier coefficients and prove that it implies the convergence of the Fourier series in integral norm, almost everywhere, and if the function itself is in the real Hardy space, then also in the Hardy norm. We also compare it to the previously known conditions.
5
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On the rate of summability of Fourier series by means of the method of the Voronoi-Nörlund

100%
Stoiński S. l.
Commentationes Mathematicae
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2009
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tom 49
|
nr 1
EN
In this note we present the theorem on the rate of pointwise summability of Fourier series by means of generalized Voronoi-Nörlund sums.
6
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A note on the MRBSVS class and its application to the uniform convergence and boundedness of sine series

100%
Szal B.
Commentationes Mathematicae
|
2009
|
tom 49
|
nr 1
EN
A new class of \(\gamma\) rest bounded second variation sequences is defined. Some relationships between classes of considered sequences are proved. The results of Leindler [3] and author [8] are extended to our new class.
7
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Fourier series of functions involving higher-order ordered Bell polynomials

100%
Kim T., Kim D. S., Jang G.-W., Jang L. C.
Open Mathematics
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2017
|
tom 15
|
nr 1
1606-1617
EN
In 1859, Cayley introduced the ordered Bell numbers which have been used in many problems in number theory and enumerative combinatorics. The ordered Bell polynomials were defined as a natural companion to the ordered Bell numbers (also known as the preferred arrangement numbers). In this paper, we study Fourier series of functions related to higher-order ordered Bell polynomials and derive their Fourier series expansions. In addition, we express each of them in terms of Bernoulli functions.
8
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A novel recursive method to reconstruct multivariate functions on the unit cube

100%
Zhang Z.
Open Mathematics
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2017
|
tom 15
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nr 1
1568-1577
EN
Due to discontinuity on the boundary, traditional Fourier approximation does not work efficiently for d−variate functions on [0, 1]d. In this paper, we will give a recursive method to reconstruct/approximate functions on [0, 1]d well. The main process is as follows: We reconstruct a d−variate function by using all of its (d−1)–variate boundary functions and few d–variate Fourier coefficients. We reconstruct each (d−1)–variate boundary function given in the preceding reconstruction by using all of its (d−2)–variate boundary functions and few (d−1)–variate Fourier coefficients. Continuing this procedure, we finally reconstruct each univariate boundary function in the preceding reconstruction by using values of the function at two ends and few univariate Fourier coefficients. Our recursive method can reconstruct multivariate functions on the unit cube with much smaller error than traditional Fourier methods.
9
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A note on the uniform convergence and boundedness of a generalized class of sine series

100%
Szal B.
Commentationes Mathematicae
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2008
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tom 48
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nr 1
EN
In this paper we essentially extend the Leindler’s results concerning the uniform convergence and boundedness of a certain class of sine series.
10
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Content available

On the approximation of conjugate functions from \(L^{p}\) by some special matrix means of conjugate Fourier series

100%
Krantz R., Rzepka A.
Commentationes Mathematicae
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2014
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tom 54
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nr 2
EN
The results corresponding to some theorems of W. Łenski and B. Szal are shown. From the presented pointwise results the estimates on norm approximation are derived. Some special cases as corollaries are also formulated.
11
Content available remote

On the Bézout equation in the ring of periodic distributions

88%
Rupp R., Sasane A.
Topological Algebra and its Applications
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2016
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tom 4
|
nr 1
1-8
EN
A corona type theorem is given for the ring D'A(Rd) of periodic distributions in Rd in terms of the sequence of Fourier coefficients of these distributions,which have at most polynomial growth. It is also shown that the Bass stable rank and the topological stable rank of D'A(Rd) are both equal to 1.
12
Content available remote

Determining the weights of a Fourier series neural network on the basis of the multidimensional discrete Fourier transform

88%
Halawa K.
International Journal of Applied Mathematics and Computer Science
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2008
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tom 18
|
nr 3
369-375
EN
This paper presents a method for training a Fourier series neural network on the basis of the multidimensional discrete Fourier transform. The proposed method is characterized by low computational complexity. The article shows how the method can be used for modelling dynamic systems.
13
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A survey of certain results on strong approximation by orthogonal series

51%
Leindler L.
Open Mathematics
|
2004
|
tom 2
|
nr 3
448-477
EN
This is a survey of results in a particular direction of the theory of strong approximation by orthogonal series, related mostly with author's contributions to the subject.
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