This is a short survey on osicllatory integral operators. We summarize the main development and managing techniques of the field, and give some open problems and main references in the end.
We consider the oscillatory integral operator Ta,mf（X） f（y）dy, where the function f is a Schwartz function.In this paper, the restriction theorem on Sn-1 for this operator is obtained. Moreover, we obtain a necess...We consider the oscillatory integral operator Ta,mf（X） f（y）dy, where the function f is a Schwartz function.In this paper, the restriction theorem on Sn-1 for this operator is obtained. Moreover, we obtain a necessary condition which ensures validity of the restriction theorem.展开更多
The commutators of oscillatory singular integral operators with homogeneous kernel $\frac{{\Omega (x)}}{{\left| x \right|^n }}$ are studied, where Ω is homogeneous of degree zero, has mean value zero on the unit sphe...The commutators of oscillatory singular integral operators with homogeneous kernel $\frac{{\Omega (x)}}{{\left| x \right|^n }}$ are studied, where Ω is homogeneous of degree zero, has mean value zero on the unit sphere. It is proved that Ω∈L (logL)K+1(Sn-1) is a sufficient condition under which the k-th order commutator is bounded on L2(Rn).展开更多
This paper proves a theorem on the decay rate of the oscillatory integral operator with a degenerate C^∞ phase function, thus improving a classical theorem of HSrmander. The proof invokes two new methods to resolve t...This paper proves a theorem on the decay rate of the oscillatory integral operator with a degenerate C^∞ phase function, thus improving a classical theorem of HSrmander. The proof invokes two new methods to resolve the singularity of such kind of operators： a delicate method to decompose the operator and balance the L^2 norm estimates; and a method for resolution of singularity of the convolution type. The operator is decomposed into four major pieces instead of infinite dyadic pieces, which reveals that Cotlar＇s Lemma is not essential for the L^2 estimate of the operator. In the end the conclusion is further improved from the degenerate C^∞ phase function to the degenerate C^4 phase function.展开更多
In this paper, we study a kind of oscillatory singular integral operator T with Calderon-Zygmund kernel, which had been studied by Ricci and Stein in [6], and extend their result. We get that T is bounded on L^P（R^...In this paper, we study a kind of oscillatory singular integral operator T with Calderon-Zygmund kernel, which had been studied by Ricci and Stein in [6], and extend their result. We get that T is bounded on L^P（R^n）（1〈p〈∞） when -1〈u〈 αd（1/2-｜1/p-1/2）.展开更多
In this paper, the authors establish the weighted weak and strong type norm inequalities in the set of weighted Herz-type spaces for a vector-valued analogue of the Hardy-Littlewood maximal operator; and using this, t...In this paper, the authors establish the weighted weak and strong type norm inequalities in the set of weighted Herz-type spaces for a vector-valued analogue of the Hardy-Littlewood maximal operator; and using this, the authors obtain the weightd inequalies for a wide class of sublinear singular operators defined onR n which include the Calderón-Zygmund operators as special cases. The fractional versions of these results are also given.展开更多
The concern of this paper is to study local approximation properties of the Bernstein-Durrmeyer operators Mn. We derive the complete asymptotic expansion of the operators Mn and their derivatives as n tends to infinit...The concern of this paper is to study local approximation properties of the Bernstein-Durrmeyer operators Mn. We derive the complete asymptotic expansion of the operators Mn and their derivatives as n tends to infinity. It turns out that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly in a very concise form. Our main theorem contains several earlier partial results as special cases. Finally, we obtain a Voronovskaja-type formula for simultaneous approximation by linear combinations of Mn,展开更多
In this paper, we systematically study a class of waves. We then de fine Hardy type spaces by conjugate systems for this class of waves, and study their properties. In particular, we show that they extend some class o...In this paper, we systematically study a class of waves. We then de fine Hardy type spaces by conjugate systems for this class of waves, and study their properties. In particular, we show that they extend some class of Lp estimates for the wave equation.展开更多
In this paper, the authors investigate the boundedness of the generalized fractional integrals of Pérez on the weighted Herz spaces, the weighted weak Herz spaces and the weighted Herz-type Hardy spaces for gener...In this paper, the authors investigate the boundedness of the generalized fractional integrals of Pérez on the weighted Herz spaces, the weighted weak Herz spaces and the weighted Herz-type Hardy spaces for general weights.展开更多
In this paper the author presents a method for the numerical solution of a 2-D Cauchy principal value of the formwhere S is a domain with a continuous boundary. By usmg polar coordinates, the integral is reduced to th...In this paper the author presents a method for the numerical solution of a 2-D Cauchy principal value of the formwhere S is a domain with a continuous boundary. By usmg polar coordinates, the integral is reduced to the formwhere denotes the finite-part of the integral. We construct the relative product rule based onquasi-interpolating splines.Convergence results are proved and numerical examples are given.展开更多
A local bivariate C1 quasi-interpolating spline operator with a four directional mesh is considered and studied. Based on the above operator we present cubature formulas for 2-D singular integrals, defined in the Hada...A local bivariate C1 quasi-interpolating spline operator with a four directional mesh is considered and studied. Based on the above operator we present cubature formulas for 2-D singular integrals, defined in the Hadamard finite part sense. Convergence results are obtained for a wide class of functions. Moreover numerical tests are given.展开更多
In this paper we study the sequence of Bernstein operators in the case when the binomial coefficients are substituted by general ones satisfying a similar recursive rule. Besides the characterization of the convergenc...In this paper we study the sequence of Bernstein operators in the case when the binomial coefficients are substituted by general ones satisfying a similar recursive rule. Besides the characterization of the convergence and the approximation properties of the sequence of operators obtained in this manner, the main application regards the approximation of the solutions of suitable second-order parabolic problems.展开更多
The two-dimensional classical Hardy space Hp(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space Hp(T×T) to Lp(T2) ...The two-dimensional classical Hardy space Hp(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space Hp(T×T) to Lp(T2) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H 1 # (T×T), L1(T2)), where the Hardy space H 1 # (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H 1 # (T×T)?LlogL(T 2) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces Hp(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈Hp(T×T), the (C, α, β) means convergs to f in Hp(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.展开更多
In this study, we use inexact newton methods to find solutions of nonlinear, nondifferenti-able operator equations on Banach spaces with a convergence structure. This technique involves the introduction of a generaliz...In this study, we use inexact newton methods to find solutions of nonlinear, nondifferenti-able operator equations on Banach spaces with a convergence structure. This technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way the metric properties of the examined problem can be analyzed more precisely. Moreover , this approach allmvs us to derive from the same theorem, on the one hand, semi-local results of Kantorovich-type, and on the other hand, global results based on mono-tonicity considerations. Furthermore, ive show that special cases of our results reduce to the corresponding ones already in the literature. Finally > our results are used to solve integral equations that cannot be solved with existing methods.展开更多
It has been argued that Chebyshev polynomials are ideal to use as approximating functions to obtain solutions of integral equations and convolution integrals on account of their fast convergence. Using the standard de...It has been argued that Chebyshev polynomials are ideal to use as approximating functions to obtain solutions of integral equations and convolution integrals on account of their fast convergence. Using the standard deviation as a measure of the accuracy of the approximation and the CPU time as a measure of the speed, we find that for reasonable accuracy Legendre polynomials are more efficient. '展开更多
文摘This is a short survey on osicllatory integral operators. We summarize the main development and managing techniques of the field, and give some open problems and main references in the end.
文摘We consider the oscillatory integral operator Ta,mf（X） f（y）dy, where the function f is a Schwartz function.In this paper, the restriction theorem on Sn-1 for this operator is obtained. Moreover, we obtain a necessary condition which ensures validity of the restriction theorem.
文摘The commutators of oscillatory singular integral operators with homogeneous kernel $\frac{{\Omega (x)}}{{\left| x \right|^n }}$ are studied, where Ω is homogeneous of degree zero, has mean value zero on the unit sphere. It is proved that Ω∈L (logL)K+1(Sn-1) is a sufficient condition under which the k-th order commutator is bounded on L2(Rn).
基金the State Key Laboratory of Software Development Environmentthe Grant No.SKLSDE-07-004 under the National Basic Research Program of China (the 973 Program Grant No.2005CB321901)
文摘This paper proves a theorem on the decay rate of the oscillatory integral operator with a degenerate C^∞ phase function, thus improving a classical theorem of HSrmander. The proof invokes two new methods to resolve the singularity of such kind of operators： a delicate method to decompose the operator and balance the L^2 norm estimates; and a method for resolution of singularity of the convolution type. The operator is decomposed into four major pieces instead of infinite dyadic pieces, which reveals that Cotlar＇s Lemma is not essential for the L^2 estimate of the operator. In the end the conclusion is further improved from the degenerate C^∞ phase function to the degenerate C^4 phase function.
文摘In this paper, we study a kind of oscillatory singular integral operator T with Calderon-Zygmund kernel, which had been studied by Ricci and Stein in [6], and extend their result. We get that T is bounded on L^P（R^n）（1〈p〈∞） when -1〈u〈 αd（1/2-｜1/p-1/2）.
文摘In this paper, the authors establish the weighted weak and strong type norm inequalities in the set of weighted Herz-type spaces for a vector-valued analogue of the Hardy-Littlewood maximal operator; and using this, the authors obtain the weightd inequalies for a wide class of sublinear singular operators defined onR n which include the Calderón-Zygmund operators as special cases. The fractional versions of these results are also given.
文摘The concern of this paper is to study local approximation properties of the Bernstein-Durrmeyer operators Mn. We derive the complete asymptotic expansion of the operators Mn and their derivatives as n tends to infinity. It turns out that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly in a very concise form. Our main theorem contains several earlier partial results as special cases. Finally, we obtain a Voronovskaja-type formula for simultaneous approximation by linear combinations of Mn,
文摘In this paper, we systematically study a class of waves. We then de fine Hardy type spaces by conjugate systems for this class of waves, and study their properties. In particular, we show that they extend some class of Lp estimates for the wave equation.
文摘In this paper, the authors investigate the boundedness of the generalized fractional integrals of Pérez on the weighted Herz spaces, the weighted weak Herz spaces and the weighted Herz-type Hardy spaces for general weights.
文摘In this paper the author presents a method for the numerical solution of a 2-D Cauchy principal value of the formwhere S is a domain with a continuous boundary. By usmg polar coordinates, the integral is reduced to the formwhere denotes the finite-part of the integral. We construct the relative product rule based onquasi-interpolating splines.Convergence results are proved and numerical examples are given.
文摘A local bivariate C1 quasi-interpolating spline operator with a four directional mesh is considered and studied. Based on the above operator we present cubature formulas for 2-D singular integrals, defined in the Hadamard finite part sense. Convergence results are obtained for a wide class of functions. Moreover numerical tests are given.
文摘In this paper we study the sequence of Bernstein operators in the case when the binomial coefficients are substituted by general ones satisfying a similar recursive rule. Besides the characterization of the convergence and the approximation properties of the sequence of operators obtained in this manner, the main application regards the approximation of the solutions of suitable second-order parabolic problems.
文摘The two-dimensional classical Hardy space Hp(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space Hp(T×T) to Lp(T2) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H 1 # (T×T), L1(T2)), where the Hardy space H 1 # (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H 1 # (T×T)?LlogL(T 2) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces Hp(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈Hp(T×T), the (C, α, β) means convergs to f in Hp(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.
文摘In this study, we use inexact newton methods to find solutions of nonlinear, nondifferenti-able operator equations on Banach spaces with a convergence structure. This technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way the metric properties of the examined problem can be analyzed more precisely. Moreover , this approach allmvs us to derive from the same theorem, on the one hand, semi-local results of Kantorovich-type, and on the other hand, global results based on mono-tonicity considerations. Furthermore, ive show that special cases of our results reduce to the corresponding ones already in the literature. Finally > our results are used to solve integral equations that cannot be solved with existing methods.
文摘It has been argued that Chebyshev polynomials are ideal to use as approximating functions to obtain solutions of integral equations and convolution integrals on account of their fast convergence. Using the standard deviation as a measure of the accuracy of the approximation and the CPU time as a measure of the speed, we find that for reasonable accuracy Legendre polynomials are more efficient. '
