Narrow Results

In this paper, we apply Markov-modulated models to value continuous-installment options of the European style with a partial differential equation approach. Under regime-switching models and the opportunity for continuing or stopping to pay installments, the valuation problem can be formulated as coupled partial differential equations (CPDE) with free boundary features, which in many ways is similar to the free boundary problem for vanilla American options due to the possibility of early exercise. In this paper, to value the continuous-installment options under the proposed model with a numerical approach, we first express the truncated CPDE as a linear complementarity problem (LCP), and then a finite element method is applied to solve the resulting variational inequality. We studied the existence and uniqueness of the solution and analyzed the stability of the proposed method under some appropriate assumptions, then we illustrated the error estimates on the appropriate spaces. We presented some numerical results to examine the rate of convergence and accuracy of the proposed method for the pricing problem under the regime-switching model.

In the present paper, we introduce the commutators of local bilinear maximal operator, local bilinear maximal commutators and their fractional variants. Under the conditions that the symbol function belongs to the first-order Sobolev spaces, the bounds for the above commutators on the Sobolev spaces are established. These results are based on several new pointwise estimates for the weak gradients of the above commutators.

In this paper, utilizing the theory of Watson transform and Watson convolution, we explore the Watson wavelet convolution product and its related properties. The relation between the Watson Wavelet convolution product and Watson convolution is also computed. Watson wavelet transform and its inversion formula are analyzed heuristically. Watson two-wavelet multipliers and its trace class are derived from Watson wavelet convolution product

This paper addresses the existence of solutions u ∈ H¹(ℝ₊;ℝ^N) of ODE systems , with boundary condition u₁(0) = ξ, where u₁ is a (vector) component of u. Under general conditions, the problem corresponds to a functional equation involving a Fredholm operator with calculable index, which is proper on the closed bounded subsets of H¹(ℝ₊;ℝ^N). When the index is 0 and the solutions are bounded a priori, the existence follows from an available degree theory for such operators. Specific conditions are given that guarantee the existence of a priori bounds and second order equations with Dirichlet, Neumann or initial value conditions are discussed as applications.

In this paper, we study the local regularity of very weak solution of the elliptic equation D_j(a_ij(x)D_iu) = f - D_ig_i. Using the bootstrap argument and the difference quotient method, we obtain that if , and with 1 < p < ∞, then . Furthermore, we consider the higher regularity of u.

The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Since the difference quotient is based on shifting the function, it cannot be generalized to the variable exponent case. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the variable exponent Sobolev space.

In this paper, the authors characterize the Sobolev spaces $W^{\alpha ,p}({ℝ}^n)$ with $\alpha \in (0,2]$ and $p\in (max\{1,\frac{2n}{2\alpha +n}\},\infty )$ via a generalized Lusin area function and its corresponding Littlewood–Paley $g_{\lambda }^{\ast }$-function. The range $p\in (max\{1,\frac{2n}{2\alpha +n}\},\infty )$ is also proved to be nearly sharp in the sense that these new characterizations are not true when $\frac{2n}{2\alpha +n}>1$ and $p\in (1,\frac{2n}{2\alpha +n})$. Moreover, in the endpoint case $p=\frac{2n}{2\alpha +n}$, the authors also obtain some weak type estimates. Since these generalized Littlewood–Paley functions are of wide generality, these results provide some new choices for introducing the notions of fractional Sobolev spaces on metric measure spaces.

The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz–Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.

In this paper, we construct a class of compactly supported wavelets by taking trigonometric B-splines as the scaling function. The duals of these wavelets are also constructed. With the help of these duals, we show that the collection of dilations and translations of such a wavelet forms a Riesz basis of 𝕃²(ℝ). Moreover, when a particular differential operator is applied to the wavelet, it also generates a Riesz basis for a particular generalized Sobolev space. Most of the proofs are based on three assumptions which are mild generalizations of three important lemmas of Jia et al. [Compactly supported wavelet bases for Sobolev spaces, Appl. Comput. Harmon. Anal. 15 (2003) 224–241].

The standard Sobolev space $W_2^s({ℝ}^d)$, with arbitrary positive integers $s$ and $d$ for which $s>d/2$, has the reproducing kernel

This paper aims to explore the optimal nonlinear approximation on the class of functions in Sobolev space in the probabilistic and average case settings. Specifically, we investigate the optimal approximation through the utilization of a set with finite pseudo-dimension, measured by the Kolmogorov probabilistic nonlinear $(n,\delta )$-width (or say, the Kolmogorov probabilistic pseudo-$(n,\delta )$-width). Furthermore, we provide an estimation of the exact asymptotic order of the Kolmogorov probabilistic nonlinear $(n,\delta )$-width and $p$-average nonlinear $n$-width on the class of functions in Sobolev space.

Cascade algorithms play an important role in wavelet analysis and computer graphics. The paper considers the convergence of cascade algorithms in Sobolev spaces. With the help of the factorization of matrix masks, we give a sufficient condition for the convergence. The condition is expressed in the time domain. More importantly, an algorithm for the construction of convergent cascade algorithms in Sobolev space starting from any matrix mask satisfying a mild condition is presented. Examples are given to illustrate our theorems.

In this paper, we study two problems: one is of the worst case cubature error over the Sobolev classes on the sphere and another is of the approximation by hyperinterpolation operators on the sphere in the Sobolev space setting. We obtain lower estimates for the worst case cubature error over the Sobolev classes, which are optimal in the sense of order. We also obtain optimal estimates for the approximation by hyperinterpolation operators on the sphere in the Sobolev space setting.

It is well known that the domain of Fourier transform (FT) can be extended to the Schwartz space $\mathcal{𝒮}(R)$ for convenience. As a generation of FT, it is necessary to detect the linear canonical transform (LCT) on a new space for obtaining the similar properties like FT on $\mathcal{𝒮}(R)$. Therefore, a space ${\mathcal{𝒮}}_{A_1}(R)$ generalized from $\mathcal{𝒮}(R)$ is introduced firstly, and further we prove that LCT is a homeomorphism from ${\mathcal{𝒮}}_{A_1}(R)$ onto itself. The linear canonical wavelet transform (LCWT) is a newly proposed transform based on the convolution theorem in LCT domain. Moreover, we propose an equivalent definition of LCWT associated with LCT and further study some properties of LCWT on ${\mathcal{𝒮}}_{A_1}(R)$. Based on these properties, we finally prove that LCWT is a linear continuous operator on the spaces of $L^{p,A_1}$ and $H_{A_1}^{s,p}$.

The Sobolev space over local fields $H^s(𝕂)$ is defined. A multiresolution analysis for the Sobolev space is developed. Orthonormal wavelets with respect to these space are constructed and an example is also presented.

We are concerned with the construction and study of composition of wavelet transform (CWT) associated with generalized fractional Hankel-type transform. Parseval’s identity is obtained for the CWT. Plancherel’s formula is also proposed. This paper goes further to describe the boundedness property of the CWT on Zemanian type and Sobolev type spaces. Some particular cases are derived.

In this paper, the boundedness and compactness of the Hankel wavelet multiplier associated with the unitary representation on $L^p$ space for $1\le p\le \infty$, are obtained by using the Hankel transform technique. The application of Hankel wavelet multipliers in form of the Landau–Pollak–Slepian operator is given. With the help of the Hankel wavelet multiplier, the Sobolev space associated with the Hankel transform is constructed and its various properties are studied.

The well-posedness theory for hyperbolic systems of first-order quasilinear PDE's with ODE's boundary conditions (on a bounded interval) is discussed. Such systems occur in multi-scale blood flow models, as well as valveless pumping and fluid mechanics. The theory is presented in the setting of Sobolev spaces H^m (m ≥ 3 being an integer), which is an appropriate set-up when it comes to proving existence of smooth solutions using energy estimates. A blow-up criterion is also derived, stating that if the maximal time of existence is finite, then the state leaves every compact subset of the hyperbolicity region, or its first-order derivatives blow-up. Finally, we discuss physical examples which fit in the general framework presented.

Inequalities of Korn's type on a surface with boundary have been proved in many papers using different techniques (see e.g. [4, 5, 11]). The author proves here an inequality of Korn's type on a compact surface without boundary. The idea is to use a finite number of maps for defining the surface and the inequality of Korn's type without boundary conditions for every map and to recast these in a general functional analysis setting about quotient spaces.

The main goal of this paper is to study properties of the linear canonical transform (LCT) on Schwartz-type space ${{𝒮}}_A(ℝ)$. The symbol class ${{𝒮}}_{\rho ,\sigma }^{m,A}$ is introduced. The pseudo-differential operator (p.d.o.) involving LCT is defined and also some of its properties including boundedness are investigated in Sobolev-type space. Kernel and integral representation of p.d.o. are obtained. Some applications of LCT to generalized partial differential equations and canonical convolution integral equation have been solved.