共查询到20条相似文献,搜索用时 46 毫秒
1.
Wolfgang zu Castell 《Constructive Approximation》2008,27(2):217-235
Suppose that
and p > 0. In this paper we study the generalized Bessel functions for
the surface
, introduced by D.St.P. Richards. We derive a recurrence relation for these functions and utilize a series representation
to relate them to the classical symmetric functions. These generalized Bessel functions are symmetric with respect to the
action of the hyperoctahedral group Wd, which is the symmetry group of the
unit sphere. By means of this symmetry under Wd, we further express these generalized Bessel functions in terms of Bessel functions for certain finite reflection groups.
For the case in which p = 2, our representations lead to known relations for the classical Bessel functions of order (d -
2)/2. For the case in which p = 1, the generalized Bessel functions have been studied by Berens and Xu in the analysis of
summability problems for 1-radial functions, and we show how their results may be framed within our more general context. 相似文献
2.
José Luis López 《Journal of Mathematical Analysis and Applications》2011,377(1):30-42
Asymptotic expansions are given for large values of n of the generalized Bessel polynomials . The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the z-plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points z=±i/n are derived, and a new expansion in terms of modified Bessel functions is given. Earlier asymptotic expansions of the generalized Bessel polynomials by Wong and Zhang (1997) and Dunster (2001) are discussed. 相似文献
3.
Generalized Shift-Invariant Systems 总被引:1,自引:0,他引:1
A countable collection $X$ of functions in $L_2(\mbox{\footnotesize\bf
R})$ is said to be a Bessel system if the associated
analysis operator
$$
\txs{X}:L_2(\mbox{\smallbf R}^d)\to \ell_2(X) : f\mapsto (\inpro{f,x})_{x\in X}
$$
is well-defined and bounded. A Bessel system is a fundamental frame if
$\txs{X}$ is injective and its range is closed.
This paper considers the above two properties for a generalized
shift-invariant system $X$. By definition, such a system has the form
$$
X=\bigcup_{j\in J} Y_j,
$$
where each $Y_j$ is a shift-invariant system (i.e., is comprised of lattice
translates of some function(s)) and $J$ is a countable (or finite) index set.
The definition is general enough to include wavelet systems, shift-invariant
systems, Gabor systems, and many variations of wavelet systems such as
quasi-affine ones and nonstationary ones.
The main theme of this paper is the fiberization of $\txs{X}$,
which allows one to study the frame and Bessel properties of $X$ via the spectral
properties of a collection of finite-order Hermitian nonnegative matrices. 相似文献
4.
This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations.
Let
be the Euclidean Dirac operator in the n-dimensional flat space
the radial symmetric Euler operator and α and λ be arbitrary non-zero complex parameters. The goal of this paper is to describe
explicitly the structure of the solutions to the PDE system
in terms of arbitrary complex order Bessel functions and homogeneous monogenic polynomials.
Received: 27 October 2005 相似文献
5.
Konstantinos Chrysafinos 《BIT Numerical Mathematics》2007,47(3):487-505
Error estimates for Galerkin discretizations of parabolic integro-differential equations are presented under minimal regularity
assumptions. The analysis is applicable in case that the full Galerkin matrix A associated to the integral operator is replaced by a compressed “sparse” matrix using wavelet basis techniques. In particular, a semi-discrete (in space) scheme and a fully-discrete scheme which is discontinuous
in time but conforming in space are analyzed.
AMS subject classification (2000) 65R20, 65M60 相似文献
6.
Sebastian Bogner Bernd Fritzsche Bernd Kirstein 《Complex Analysis and Operator Theory》2007,1(1):55-95
The main theme of this paper is to characterize distinguished subclasses of the matricial Schur class
in terms of Taylor coefficients. Starting point of our investigations is the observation that the Taylor coefficient sequences
of functions from
are exactly the infinite p × q Schur sequences. We draw our attention mainly to the subclass
of
which consists of all p × q Schur functions for which the corresponding Taylor coefficient sequences are nondegenerate p × q Schur sequences. Using an appropriate adaptation of the Schur–Potapov algorithm for functions belonging to
to infinite sequences of complex p × q matrices we obtain an one-to-one correspondence between infinite nondegenerate p × q Schur sequences and the set of all infinite sequences (Ej)j=0∞ of strictly contractive complex p × q matrices. Taking into account the construction of
this gives us an one-to-one correspondence between
and the set of all infinite sequences (Ej)j=0∞ of strictly contractive complex p × q matrices. Hereby, (Ej)j =0∞ is called the sequence of Schur–Potapov parameters (shortly SP-parameters) of f.
Communicated by Daniel Alpay.
Submitted: August 17, 2006; Accepted: September 13, 2006 相似文献
7.
Clemens Markett 《Constructive Approximation》1989,5(1):383-404
One of the most far-reaching qualities of an orthogonal system is the presence of an explicit product formula. It can be utilized to establish a convolution structure and hence is essential for the harmonic analysis of the corresponding orthogonal expansion. As yet a convolution structure for Fourier-Bessel series is unknown, maybe in view of the unpractical nature of the corresponding expanding functions called Fourier-Bessel functions. It is shown in this paper that for the half-integral values of the parameter
,n=0, 1, 2,, the Fourier-Bessel functions possess a product formula, the kernel of which splits up into two different parts. While the first part is still the well-known kernel of Sonine's product formula of Bessel functions, the second part is new and reflects the boundary constraints of the Fourier-Bessel differential equation. It is given, essentially, as a finite sum over triple products of Bessel polynomials. The representation is explicit up to coefficients which are calculated here for the first two nontrivial cases
and
. As a consequence, a positive convolution structure is established for
. The method of proof is based on solving a hyperbolic initial boundary value problem.Communicated by Tom H. Koornwinder. 相似文献
8.
We study the null solutions of iterated applications of the spherical (Atiyah-Singer) Dirac operator on locally defined polynomial forms on the unit sphere of ; functions valued in the universal Clifford algebra , here called spherical k-regular functions. We construct the kernel functions, get the integral representation formula and Cauchy integral formula
of spherical k-regular functions, and as applications, the weak solutions of higher order inhomogeneous spherical (Atiyah-Singer) Dirac
equations . We obtain, in particular, the weak solution of an inhomogeneous spherical Poisson equation Δ
s
g = f.
This work was partially supported by NNSF of China (No.10471107) and RFDP of Higher Education (No.20060486001). 相似文献
9.
Andrew Raich 《Mathematische Zeitschrift》2007,256(1):193-220
Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional
kernels. We prove that the kernels are C
∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.
相似文献
10.
11.
Amol Sasane 《Acta Appl Math》2008,103(2):161-168
In this article, we prove that the Krull dimension of several commonly used classes of transfer functions of infinite dimensional
linear control systems is infinite. On the other hand, we also show that the weak Krull dimension of the Hardy algebra
, the disk algebra
and the Wiener algebra
is equal to 1.
A. Sasane is supported by the Nuffield Grant NAL/32420. 相似文献
12.
Pawel Kröger 《Potential Analysis》1996,5(1):103-108
We study the first eigenfunction
1 of the Dirichlet Laplacian on a convex domain in Euclidean space. Elementary properties of Bessel functions yield that
if D is a sector in Euclidean plane with area 1 and the angle tends to 0. We aim to characterize those domains D such that
is large in terms of the ratio of the first eigenvalue of D and the infimum of the first eigenvalues of all subdomains D of D with given volume.Research supported by the Deutsche Forschungsgemeinschaft 相似文献
13.
Jean-André Marti 《Acta Appl Math》2009,105(3):267-302
We introduce a general context involving a presheaf
and a subpresheaf ℬ of
. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic
techniques) can be interpretated as the ℬ-local analysis of sections of
.
But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a “frequential
microlocal analysis” and into a “microlocal asymptotic analysis”. The frequential microlocal analysis based on the Fourier
transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in
the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques.
The microlocal asymptotic analysis is a new spectral study of singularities. It can inherit from the algebraic structure of
ℬ some good properties with respect to nonlinear operations.
相似文献
14.
Ahmed Ainouche 《Graphs and Combinatorics》2009,25(2):129-137
Let G = (V, E) be a any simple, undirected graph on n ≥ 3 vertices with the degree sequence . We consider the class of graphs satisfying the condition where , is a positive integer. It is known that is hamiltonian if θ ≤ δ. In this paper,
相似文献
(i) | we give a necessary and sufficient condition, easy to check, ensuring that is nonhamiltonian and we characterize all the exceptional sub-classes. |
(ii) | we prove that is either bipartite or contains cycles of all lengths from 3 to c(G), the length of a longest cycle in G. |
15.
We consider some classes of 2π-periodic functions defined by a class of operators having certain oscillation properties, which include the classical Sobolev class and a class of analytic functions which can not be represented as a convolution class as its special cases. Let be the largest integer not bigger than x. We prove that on these classes of functions the rectangular formula
is optimal among all quadrature formulae of the form
where the nodes 0 ≤ t
1 < ... < t
n
< 2π and the coefficients (weights) are arbitrary, i = 1,...,n, j = 0,1,..., ν
i
− 1, and (ν1,...,ν
n
) is a system of positive integers satisfying the condition . In particular, the rectangular formula is optimal for these classes of functions among all quadrature formulae of the form
with free nodes 0 ≤ t
1 < ... < t
N
< 2π and arbitrary weights . Moreover, we exactly determine the error estimates of the optimal quadrature formulae on these classes of functions.Project supported by the National Natural Science Foundation of China (Grant No. 10671019) and Research Fund for the Doctoral Program Higher Education (Grant No. 20050027007). 相似文献
16.
Wavelet–type transform associated with singular Laplace–Bessel differential operator
is introduced and the relevant Calderón–type reproducing formula is established. Representations of the generalized Bessel potentials
0)$
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and their inverses via the wavelet–type transform are obtained. 相似文献
17.
Daniel Girela José Ángel Peláez Fernando Pérez-González Jouni Rättyä 《Integral Equations and Operator Theory》2008,61(4):511-547
In this paper we study the positive Borel measures μ on the unit disc in for which the Bloch space is continuously included in , 0 < p < ∞. We call such measures p-Bloch-Carleson measures. We give two conditions on a measure μ in terms of certain logarithmic integrals one of which is a necessary condition and the other a sufficient condition for μ being a p-Bloch-Carleson measure. We also give a complete characterization of the p-Bloch-Carleson measures within certain special classes of measures. It is also shown that, for p > 1, the p-Bloch-Carleson measures are exactly those for which the Toeplitz operator , defined by , maps continuously into the Bergman space A
1, . Furthermore, we prove that if p > 1, α >-1 and ω is a weight which satisfies the Bekollé-Bonami -condition, then the measure defined by is a p-Bloch-Carleson-measure.
We also consider the Banach space of those functions f which are analytic in and satisfy , as . The Bloch space is contained in . We describe the p-Carleson measures for and study weighted composition operators and a class of integration operators acting in this space. We determine which of
these operators map continuously to the weighted Bergman space and show that they are automatically compact.
This research is partially supported by several grants from “the Ministerio de Educación y Ciencia, Spain” (MTM2005-07347,
MTM2007-60854, MTM2006-26627-E, MTM2007-30904-E and Ingenio Mathematica (i-MATH) No. CSD2006-00032); from “La Junta de Andalucía”
(FQM210 and P06-FQM01504); from “the Academy of Finland” (210245) and from the European Networking Programme “HCAA” of the
European Science Foundation. 相似文献
18.
The aim of this note is to prove the following theorem. Let
where P(ix) is a nonnegative homogeneous elliptic polynomial on R
d
and V is a nonnegative polynomial potential. Then for every 1 < p < ∞ and every α > 0 there exist constants C
1, C
2 > 0 such that
and
for f in the Schwartz class . We take advantage of the Christ inversion theorem for singular integral operators with a small amount of smoothness on
nilpotent Lie groups, the maximal subelliptic L
2-estimates for the generators of stable semi-groups of measures, and the principle of transference of Coifman–Weiss.
In memory of Tadek Pytlik, our teacher and friend.
Research supported by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis,
Nonlinear Analysis and Probability” MTKD-CT-2004-013389 and by Polish funds for science in years 2005–2008 (research project
1P03A03029). 相似文献
19.
We study functions f(z) holomorphic in having the property f(z) ≠ 0 for 0 < Im z < 1 and we obtain lower bounds for |f(z)| for 0 < Im z < 1. In our analysis we deal with scalar functions f(z) as well as with operator valued holomorphic functions I + A(z) assuming that A(z) is a trace class operator for and I + A(z) is invertible for 0 < Im z < 1 and is unitary for .
A. Borichev was partially supported by the ANR project DYNOP. 相似文献
20.
Josef Dick 《Annali di Matematica Pura ed Applicata》2008,187(3):385-403
The Koksma–Hlawka inequality states that the error of numerical integration by a quasi-Monte Carlo rule is bounded above by
the variation of the function times the star-discrepancy. In practical applications though functions often do not have bounded
variation. Hence here we relax the smoothness assumptions required in the Koksma–Hlawka inequality. We introduce Banach spaces
of functions whose fractional derivative of order is in . We show that if α is an integer and p = 2 then one obtains the usual Sobolev space. Using these fractional Banach spaces we generalize the Koksma–Hlawka inequality
to functions whose partial fractional derivatives are in . Hence we can also obtain an upper bound on the integration error even for certain functions which do not have bounded variation
but satisfy weaker smoothness conditions.
相似文献