Integral Representation for Bessel’s Functions of the First Kind and Neumann SeriesMicheli, Enrico
2018 Results in Mathematics
doi: 10.1007/s00025-018-0826-5
A Fourier-type integral representation for Bessel’s functions of the first kind and complex order is obtained by using the Gegenbauer extension of Poisson’s integral representation for the Bessel function along with a suitable trigonometric integral representation of Gegenbauer’s polynomials. By using this representation, expansions in series of Bessel’s functions of various functions which are related to the incomplete gamma function can be obtained in a unified way. Neumann series are then considered and a new closed-form integral representation for this class of series is given. The density function of this representation is the generating function of the sequence of coefficients of the Neumann series on the unit circle. Examples of new closed-form integral representations of special functions are thus presented.
$$C^\infty $$ C ∞ -Convergence of Conformal Mappings for Conformally Equivalent Triangular LatticesBücking, Ulrike
2018 Results in Mathematics
doi: 10.1007/s00025-018-0845-2
Two triangle meshes are conformally equivalent if for any pair of incident triangles the absolute values of the corresponding cross-ratios of the four vertices agree. Such a pair can be considered as preimage and image of a discrete conformal map. In this article we study discrete conformal maps which are defined on parts of a triangular lattice T with strictly acute angles. That is, T is an infinite triangulation of the plane with congruent strictly acute triangles. A smooth conformal map f can be approximated on a compact subset by such discrete conformal maps
$$f^\varepsilon $$
f
ε
, defined on a part of
$$\varepsilon T$$
ε
T
, see Bücking (in: Bobenko (ed) Advances in discrete differential geometry. Springer, Berlin, pp 133–149, 2016). We improve this result and show that the convergence is in fact in
$$C^\infty $$
C
∞
. Furthermore, we describe how the cross-ratios of the four vertices for pairs of incident triangles are related to the Schwarzian derivative of f.
On Functional Equations Characterizing Derivations: Methods and ExamplesGselmann, Eszter; Kiss, Gergely; Vincze, Csaba
2018 Results in Mathematics
doi: 10.1007/s00025-018-0833-6
Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive functions satisfying some further functional equations as well. It is an important question that how these morphisms can be characterized among additive mappings in general. The paper contains some multivariate characterizations of higher order derivations. The univariate characterizations are given as consequences by the diagonalization of the multivariate formulas. This method allows us to refine the process of computing the solutions of univariate functional equations of the form
$$\begin{aligned} \sum _{k=1}^{n}x^{p_{k}}f_{k}(x^{q_{k}})=0, \end{aligned}$$
∑
k
=
1
n
x
p
k
f
k
(
x
q
k
)
=
0
,
where
$$p_k$$
p
k
and
$$q_k$$
q
k
(
$$k=1, \ldots , n$$
k
=
1
,
…
,
n
) are given nonnegative integers and the unknown functions
$$f_{1}, \ldots , f_{n}:R\rightarrow R$$
f
1
,
…
,
f
n
:
R
→
R
are supposed to be additive on the ring R. It is illustrated by some explicit examples too. As another application of the multivariate setting we use spectral analysis and spectral synthesis in the space of the additive solutions to prove that it is spanned by differential operators. The results are uniformly based on the investigation of the multivariate version of the functional equations.
A New Method for Refining the Shafer’s Equality and Bounding the Definite IntegralsChen, Xiao-Diao; Jin, Song; Li-Geng, Chen; Wang, Yigang
2018 Results in Mathematics
doi: 10.1007/s00025-018-0836-3
This paper presents an interpolation-based method for bounding some smooth functions, including the arctangent functions related to Shafer’s inequality. Given the form of new bounding functions, the interpolation technique is also utilized for determining the corresponding unknown coefficients, and the resulting functions bound the given function very well under some preset condition. Two applications are shown, one is to refine Shafer’s inequality, and the other is to approximate the definite integrals of some special functions; both of them have wide applications in computer science, mathematics, physical sciences and engineering. Experimental results show that the new bounds achieve much better bounds than those of prevailing methods.
Some Results on Fusion Frames and g-FramesNga, Nguyen
2018 Results in Mathematics
doi: 10.1007/s00025-018-0839-0
In this paper, we discuss some aspects where fusion frames and g-frames behave differently from frames. Several counterexamples to make clear their different behaviour are given. We also improve some results on g-frames. Moreover, we extend the notion of redundancy to g-frames and show that most of the desirable properties of lower and upper redundancies on frames and fusion frames can carry over g-frames. We also study the relationship between redundancy of g-frames and their dual g-frames, redundancy for infinite g-frames and the excess of g-frames.