H. Rezaei, Z. Zafarasa
In the present paper by the Fourier transform we show that every linear differential equations of $n$-th order has a solution in $L^1(\Bbb{R})$ which is infinitely differentiable in $\Bbb{R} \setminus \{0\}$. Moreover the Hyers-Ulam stability of such equations on $L^1(\Bbb{R})$ is investigated.
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Let $μ$ be a compactly supported absolutely continuous probability measure on ${\Bbb R}^n$, we show that $μ$ admits Fourier frames if and only if its Radon-Nikodym derivative is upper and lower bounded almost everywhere on its support. As a consequence, we prove that if an equal weight absolutely continuous self-similar measure on ${\Bbb R}^1$ admits Fourier frame, then the measure must be a characteristic function of self-similar tile. In particular, this shows for almost everywhere $1/2<λ<1$, the $λ$-Bernoulli convolutions cannot admit Fourier frames.
Wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulation operators, respectively. They have been extensively studied. However, dilation-and-modulation systems have not, and they cannot be derived from wavelet or Gabor systems. In this paper, we investigate a class of dilation-and-modulation systems in the causal signal space $L^{2}(\Bbb R_{+})$. $L^{2}(\Bbb R_{+})$ can be identified a subspace of $L^{2}(\Bbb R)$ consisting of all $L^{2}(\Bbb R)$-functions supported on $\Bbb R_{+}$, and is unclosed under the Fourier transform. So the Fourier transform metho…
The Fourier transform of order statistics with applications to Lorentz spaces
本文给出了$\Bbb R^n$中顺序统计量的Fourier变换公式，证明了在$\Bbb R^n$中的坐标平面外，所有这些Fourier变换都等于一个常数倍。对于$a_1\geq ... \geq a_n\ge0$和$q>0,$，用$\ell_{w,q}^n$表示具有$\|(x_1,...,x_n)\| = (a_1 (x_1^{*})^q +...+ a_n (x_n^{*})^q)^{1/q}$范数的$n$维洛伦兹空间，其中，$(x_1^{*},...,x_n^{*})$为非利用上述公式和Banach空间等距可嵌入性的傅里叶变换准则，空间转化为$L_q$ \cite{10}，证明了对于$n\geq 3$和$q\leq 1,$，空间$\ell_{w,q}^n$与$L_q$的子空间等距当且仅当$a_1,...,a_n$形成等差数列。对于$q>1,$所有的数字$a_i$必须是相等的，使$\ell_{w,q}^n = \ell_q^n.$因此，洛伦兹函数空间$L_{w,q}(0,1)$是等距的一个子空间$L_q$当且仅当{ \it either} $0<q<\infty$和权重$w$是一个常数函数（使$L_{w,q}= L_q$），{ \it either} $q\le 1$和$w(t)$是一个递减线性函数。最后，我们把我们的结果与正定函数理论联系起来。
Weighted Gelfand pairs, weighted spherical Fourier transform and multipliers
本文通过加权Lebesgue空间中的双不变函数Banach代数引入广义Gelfand对，构造球面Fourier变换，并据此刻画相关代数的乘子。
A new generalized function space in which all Gelfand-Shilov classes $S^{\prime 0}_α$ ($α>1$) of analytic functionals are embedded is introduced. This space of {\it ultrafunctionals} does not possess a natural nontrivial topology and cannot be obtained via duality from any test function space. A canonical isomorphism between the spaces of hyperfunctions and ultrafunctionals on $R^k$ is constructed that extends the Fourier transformation of Roumieu-type ultradistributions and is naturally interpreted as the Fourier transformation of hyperfunctions. The notion of carrier cone that replaces the n…
Dynamical Sampling in Shift-Invariant Spaces Associated with multi-dimensional Special Affine Fourier Transform
该研究针对多维特殊仿射傅里叶变换（SAFT）相关平移不变空间的多变量动态采样问题，推导了函数可通过对应动态采样值稳定恢复的充要条件，并辅以示例验证，拓展了SAFT在采样理论中的应用。
A space of entire functions of several complex variables rapidly decreasing on ${\mathbb R}^n$ and such that their growth along $i{\mathbb R}^n$ is majorized with a help of a family of weight functions is considered in the paper. For this space an equivalent description in terms of estimates on all partial derivatives of functions on ${\mathbb R}^n$ and Paley-Wiener type theorem are obtained.
We discuss different representations of the White noise spaces $(E)_β$, $0 \leq β< 1$ by introducing generalized Wick tensors. As an application we state a generalization of the Mehler formula for the Ornstein-Uhlenbeck semigroup.
A space of entire functions of several complex variables rapidly decreasing on ${\mathbb R}^n$ and such that their growth along $i{\mathbb R}^n$ is majorized with a help of a family of weight functions (not radial in general) is considered in the paper. For this space an equivalent description in terms of estimates on all partial derivatives of functions on ${\mathbb R}^n$ and Paley-Wiener type theorem are given.
For any polynomial $p$ on $\mathbf{C}^{n}$, a variety $ V_{p} = \{z \in \mathbf{C}^{n} ; p(z)=0 \} $ will be considered. Let $\text{Exp}(V_{p})$ be the space of holomorphic functions of expotential growth on $V_{p}$. We shall prove that the Fourier-Borel transformation yields an isomorphism of the dual space $\text{Exp}'(V_{p})$ with the space of holomorphic solutions $\mathcal{O}_{\partial p}(\mathbf{C}^{n})$ with respect to the differential operator $\partial p$ which is obtained by replacing each variable $z_{j}$ with $\partial / \partial z_{j}$ in $p$ when $p$ is a reduced polynomial. The …
Trigonometric weighted generalized convolution operator associated with Fourier cosine-sine and Kontorovich-Lebedev transformations
本研究引入带三角权重γ=sin y、关联傅里叶余弦-正弦与Kontorovich-Lebedev变换的广义卷积算子，证明其在双参数Lebesgue空间的有界性，给出加权L_p范数估计并得到相关卷积积分微分方程的L_1可解条件。
In this work, a class of semiclassical Fourier Integral Operators (FIOs) with complex phase associated to some canonical transformation of the phase space $T^*\R^d$ is constructed. Upon some general boundedness assumptions on the symbol and the canonical transformation, their continuity (as operators) from the Schwartz class into itself and from $L^2$ into itself are proven.
We study the Beurling and Fourier transforms on subspaces of $L^2({\mathbb C})$ defined by an invariance property with respect to the root-of-unity group. This leads to generalizations of these transformations acting unitarily on weighted $L^2$-spaces over $\mathbb C$.
There is a canonical unitary transformation from $L^2(\R)$ onto the Fock space $F^2$, called the Bargmann transform. We study the action of the Bargmann transform on several classical integral operators on $L^2(\R)$, including the fractional Fourier transform, the fractional Hilbert transform, and the wavelet transform.
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