In this paper we prove some improved Caffarelli–Kohn–Nirenberg inequalities and uncertainty principle for complex- and vector-valued functions on
R
n
, which is a further study of the results in Dang ...et al. (J Funct Anal 265:2239-2266, 2013). In particular, we introduce an analogue of “phase derivative" for vector-valued functions. Moreover, using the introduced “phase derivative", we extend the extra-strong uncertainty principle to cases for complex- and vector-valued functions defined on
S
n
,
n
≥
2
.
In the Clifford algebra setting the present study develops three reproducing kernel Hilbert spaces of the Paley–Wiener type, namely the Paley–Wiener spaces, the Hardy spaces on strips, and the ...Bergman spaces on strips. In particular, we give spectrum characterizations and representation formulas of the functions in those spaces and estimation of their respective reproducing kernels.
In the Clifford algebra setting, the present study develops efficient approximations by linear combinations of the parameterized kernel functions in monogenic reproducing kernel Hilbert spaces of ...Paley–Wiener type, which include the Paley–Wiener space, the Hardy space on strips, and the Bergman space on strips.
In this paper we establish the three balls theorem for functions u satisfying Du=λu in Clifford analysis, where D is the Dirac operator. As an application, we generalize Hadamard's three circles ...theorem to monogenic function in Rn+1.
In this paper, we study the Bergman kernel Bφ(x,y) of generalized Bargmann-Fock spaces in the setting of Clifford algebra. The versions of L2-estimate method and weighted subharmonic inequality for ...Clifford algebra are established. Consequently we show the existence of Bφ(x,y) and then give some estimates on- and off- the diagonal. As a by-product, we also obtain an upper estimate of the weighted harmonic Bergman kernel.
The Fourier type expansions on tubes Mai, Weixiong; Qian, Tao
Complex variables and elliptic equations,
02/2022, Volume:
67, Issue:
2
Journal Article
Peer reviewed
In view of recent developments of the study of reproducing kernel Hilbert spaces, in particular with the context the Hardy spaces on tubes, aspects of rational approximation for functions of finite ...energy in several complex and several real variables are developed.
One important problem in the theory of Hardy space is to find the best rational approximation of a given order to a function in the Hardy space H2 on the unit disk. It is equivalent to finding the ...best Blaschke form with free poles. The cyclic adaptive Fourier decomposition method is based on the grid search technique. Its approximative precision is limited by the grid spacing. This paper proposes two enhanced methods of the cyclic adaptive Fourier decomposition. The proposed algorithms utilize the gradient descent optimization to tune the best pole-tuple on the mesh grids, reaching higher precision. Their performances are confirmed by several examples.
In this paper, we revisit some fundamental properties of linear canonical transform (abbreviated as LCT). In particular, we prove the additive property rigorously for LCT in the higher dimensional ...case (abbreviated as MLCT). We also consider the
‐theory of MLCT with
. Specifically, the inversion theorem of MLCT by the related Gauss and Abel means is studied, and the pointwise convergence of approximate identities with respect to convolution for MLCT is also obtained. As applications, we study the
‐type Heisenberg‐Pauli‐Weyl uncertainty principles and the
‐type Donoho‐Stark uncertainty principles for MLCT.
In this paper, we revisit some fundamental properties of linear canonical transform (abbreviated as LCT). In particular, we prove the additive property rigorously for LCT in the higher dimensional ...case (abbreviated as MLCT). We also consider the
Lp$$ {L}^p $$‐theory of MLCT with
1≤p≤2$$ 1\le p\le 2 $$. Specifically, the inversion theorem of MLCT by the related Gauss and Abel means is studied, and the pointwise convergence of approximate identities with respect to convolution for MLCT is also obtained. As applications, we study the
Lp$$ {L}^p $$‐type Heisenberg‐Pauli‐Weyl uncertainty principles and the
Lp$$ {L}^p $$‐type Donoho‐Stark uncertainty principles for MLCT.
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