Rong-qing Jia - Academia.edu (original) (raw)
Papers by Rong-qing Jia
Refinement equations play an important role in computer graphics and wavelet analysis.In this pap... more Refinement equations play an important role in computer graphics and wavelet analysis.In this paper we investigate multivariate refinement equations associated with a dilationmatrix and a finitely supported refinement mask. We characterize the L p -convergenceof a subdivision scheme in terms of the p-norm joint spectral radius of a collection of matricesassociated with the refinement mask. In particular, the 2-norm joint
Advances in Computational Mathematics, 2006
In this paper a pair of wavelets are constructed on the basis of Hermite cubic splines. These wav... more In this paper a pair of wavelets are constructed on the basis of Hermite cubic splines. These wavelets are in C1 and supported on (−1, 1). Moreover, one wavelet is symmetric, and the other is antisymmetric. These spline wavelets are then adapted to the interval (0, 1) .T he construction of boundary wavelets is remarkably simple. Furthermore, global stability of
Mathematics of Computation, 1998
We consider solutions of a system of refinement equations written in the form
This paper studies the spectrum of continuous renement opera- tors and relates their spectral pro... more This paper studies the spectrum of continuous renement opera- tors and relates their spectral properties with the solutions of the corresponding continuous renement equations.
SIAM Journal on Matrix Analysis and Applications, 1999
We consider the smoothness of solutions of a system of re nement equations written in the form as... more We consider the smoothness of solutions of a system of re nement equations written in the form as = X 2Z Z a( ) (2 ? )
Proceedings of the American Mathematical Society, 1994
ABSTRACT Let G be a semigroup of commuting linear operators on a linear space S with the group op... more ABSTRACT Let G be a semigroup of commuting linear operators on a linear space S with the group operation of composition. The solvability of the system of equations l i f=φ i , i=1,⋯,r, where l i ∈G and φ i ∈S, was considered by Dahmen and Micchelli in their studies of the dimension of the kernel space of certain linear operators. The compatibility conditions l j φ i =l i φ j , i≠j, are necessary for the system to have a solution in S. However, in general, they do not provide sufficient conditions. We discuss what kinds of conditions on operators will make the compatibility sufficient for such systems to be solvable in S.
Proceedings of the American Mathematical Society, 1985
The local approximation order from a scale (S h ) of approximating functions on IR m is character... more The local approximation order from a scale (S h ) of approximating functions on IR m is characterized in terms of the linear spac (and its Fourier transform) of the finitely many compactly supported functions ϕ whose integer translates ϕ(· − j), j ∈ Z Z m , span the space S = S 1 from which the scale is derived. This provides a correction of similar results stated and proved, in part, by Strang and Fix.
Mathematics of Computation, 1998
Wavelets are generated from refinable functions by using multiresolution analysis. In this paper ... more Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in W k−1 1 (R s ) provides approximation order k.
Linear Algebra and its Applications, 1982
nonfinite totally positive matrix A is shown to have one and only one "main diagonal." This means... more nonfinite totally positive matrix A is shown to have one and only one "main diagonal." This means that exactly one diagonal of A has the property that all finite sections of A principal with respect to this diagonal are invertible and their inverses converge boundedly and entrywise to A I. This is shown to imply restrictions on the possible shapes of such a matrix. In the proof, such a matrix is also shown to have an la-invertible LDU factorization.
Linear Algebra and its Applications, 1990
Journal of Approximation Theory, 1993
... It follows that ||/J,ißKn/U***(ß)) Page 12. APPROXIMATION BY TRANSLATES 13 with C< I (... more ... It follows that ||/J,ißKn/U***(ß)) Page 12. APPROXIMATION BY TRANSLATES 13 with C< I (*") = 2*-l. m = 1 m The proof of part (b) relies on Holder's inequality. Let q be the exponential conjugate to p, ie, \/q+l/p=l. Note that supp фьяВи. ...
Journal of Approximation Theory, 2004
The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching inf... more The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching influence in approximation theory since publication of their paper in 1973. In this paper, we further develop their idea and investigate quasi-projection operators. We give sharp estimates in terms of moduli of smoothness for approximation with scaled shift-invariant spaces by means of quasi-projection operators. In particular, we provide error analysis for approximation of quasi-projection operators with Lipschitz spaces. The study of quasi-projection operators has many applications to various areas related to approximation theory and wavelet analysis.
IEEE Transactions on Information Theory, 2005
In this paper we study a simple oversampled analog-to-digital (A/D) conversion in shift invariant... more In this paper we study a simple oversampled analog-to-digital (A/D) conversion in shift invariant spaces. The Beurling-Landau type theorem for bandlimited signal spaces is extended to shift invariant spaces, and then a non-uniform sampling theorem for shift invariant spaces is established, which says, a uniformly discrete set is a stable sampling set for a shift invariant space if its Beurling lower density is larger than a fixed density determined by the generator of the shift invariant space. Consequently, an oversampling theorem for shift invariant spaces is attained. These sampling theorems together with a theorem concerning the stability of stable sampling in shift invariant spaces shown by us, are used to build a simple oversampled A/D conversion scheme in shift invariant spaces. In such a scheme, the quantization error e is found to behave as e 2 = O(τ 2 ) with respect to the sampling interval τ , which is the same as that for bandlimited signal spaces derived very recently. Moreover, we demonstrate that the bitrate required to encode the converted digital signal only increases as logarithm of sampling ratio.
Constructive Approximation, 2000
Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator on ... more Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator on R is constructed that consists of continuous functions, has small support, and is exact of order 2. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual sides. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions [CDP] is then used to construct the corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to the Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolutions. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev norms · H s ([0,1]) for s ∈ (−0.824926, 2.5). In particular, the multiwavelets form Riesz bases for L 2 ([0, 1]). text, and local schemes for interpolating function and derivative data are available. Moreover, the interpolatory nature of the generators suggests convenient ways of adjoining different local tensor product bases by isoparametric mappings, thereby obtaining multiresolution analyses on more complex geometries. Therefore, we concentrate in this paper on the construction of biorthogonal multiwavelets on the interval [0, 1], generated by special C 1 piecewise Hermite cubics, with the following properties:
Canadian Journal of Mathematics, 1997
We consider the shift-invariant space, S(Φ), generated by a set Φ ≥ fû 1 , . . . , û r g of compa... more We consider the shift-invariant space, S(Φ), generated by a set Φ ≥ fû 1 , . . . , û r g of compactly supported distributions on R when the vector of distributions û :≥ (û 1 , . . . , û r ) T satisfies a system of refinement equations expressed in matrix form
Advances in Computational Mathematics, 2009
In this paper we investigate spline wavelets on the interval with homogeneous boundary conditions... more In this paper we investigate spline wavelets on the interval with homogeneous boundary conditions. Starting with a pair of families of B-splines on the unit interval, we give a general method to explicitly construct wavelets satisfying the desired homogeneous boundary conditions. On the basis of a new development of multiresolution analysis, we show that these wavelets form Riesz bases of certain Sobolev spaces. The wavelet bases investigated in this paper are suitable for numerical solutions of ordinary and partial differential equations.
Transactions of the American Mathematical Society, 1995
Let be a triangulation of some polygonal domain in R 2 and S r k ( ), the space of all bivariate ... more Let be a triangulation of some polygonal domain in R 2 and S r k ( ), the space of all bivariate C r piecewise polynomials of total degree k on . In this paper, we construct a local basis of some subspace of the space S r k ( ), where k 3r+2, that can be used to provide the highest order of approximation, with the property that the approximation constant of this order is independent of the geometry of with the exception of the smallest angle in the partition. This result is obtained by means of a careful choice of locally supported basis functions which, however, require a very technical proof to justify their stability in optimalorder approximation. A new formulation of smoothness conditions for piecewise polynomials in terms of their B-net representations is derived for this purpose.
Refinement equations play an important role in computer graphics and wavelet analysis.In this pap... more Refinement equations play an important role in computer graphics and wavelet analysis.In this paper we investigate multivariate refinement equations associated with a dilationmatrix and a finitely supported refinement mask. We characterize the L p -convergenceof a subdivision scheme in terms of the p-norm joint spectral radius of a collection of matricesassociated with the refinement mask. In particular, the 2-norm joint
Advances in Computational Mathematics, 2006
In this paper a pair of wavelets are constructed on the basis of Hermite cubic splines. These wav... more In this paper a pair of wavelets are constructed on the basis of Hermite cubic splines. These wavelets are in C1 and supported on (−1, 1). Moreover, one wavelet is symmetric, and the other is antisymmetric. These spline wavelets are then adapted to the interval (0, 1) .T he construction of boundary wavelets is remarkably simple. Furthermore, global stability of
Mathematics of Computation, 1998
We consider solutions of a system of refinement equations written in the form
This paper studies the spectrum of continuous renement opera- tors and relates their spectral pro... more This paper studies the spectrum of continuous renement opera- tors and relates their spectral properties with the solutions of the corresponding continuous renement equations.
SIAM Journal on Matrix Analysis and Applications, 1999
We consider the smoothness of solutions of a system of re nement equations written in the form as... more We consider the smoothness of solutions of a system of re nement equations written in the form as = X 2Z Z a( ) (2 ? )
Proceedings of the American Mathematical Society, 1994
ABSTRACT Let G be a semigroup of commuting linear operators on a linear space S with the group op... more ABSTRACT Let G be a semigroup of commuting linear operators on a linear space S with the group operation of composition. The solvability of the system of equations l i f=φ i , i=1,⋯,r, where l i ∈G and φ i ∈S, was considered by Dahmen and Micchelli in their studies of the dimension of the kernel space of certain linear operators. The compatibility conditions l j φ i =l i φ j , i≠j, are necessary for the system to have a solution in S. However, in general, they do not provide sufficient conditions. We discuss what kinds of conditions on operators will make the compatibility sufficient for such systems to be solvable in S.
Proceedings of the American Mathematical Society, 1985
The local approximation order from a scale (S h ) of approximating functions on IR m is character... more The local approximation order from a scale (S h ) of approximating functions on IR m is characterized in terms of the linear spac (and its Fourier transform) of the finitely many compactly supported functions ϕ whose integer translates ϕ(· − j), j ∈ Z Z m , span the space S = S 1 from which the scale is derived. This provides a correction of similar results stated and proved, in part, by Strang and Fix.
Mathematics of Computation, 1998
Wavelets are generated from refinable functions by using multiresolution analysis. In this paper ... more Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in W k−1 1 (R s ) provides approximation order k.
Linear Algebra and its Applications, 1982
nonfinite totally positive matrix A is shown to have one and only one "main diagonal." This means... more nonfinite totally positive matrix A is shown to have one and only one "main diagonal." This means that exactly one diagonal of A has the property that all finite sections of A principal with respect to this diagonal are invertible and their inverses converge boundedly and entrywise to A I. This is shown to imply restrictions on the possible shapes of such a matrix. In the proof, such a matrix is also shown to have an la-invertible LDU factorization.
Linear Algebra and its Applications, 1990
Journal of Approximation Theory, 1993
... It follows that ||/J,ißKn/U***(ß)) Page 12. APPROXIMATION BY TRANSLATES 13 with C< I (... more ... It follows that ||/J,ißKn/U***(ß)) Page 12. APPROXIMATION BY TRANSLATES 13 with C< I (*") = 2*-l. m = 1 m The proof of part (b) relies on Holder's inequality. Let q be the exponential conjugate to p, ie, \/q+l/p=l. Note that supp фьяВи. ...
Journal of Approximation Theory, 2004
The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching inf... more The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching influence in approximation theory since publication of their paper in 1973. In this paper, we further develop their idea and investigate quasi-projection operators. We give sharp estimates in terms of moduli of smoothness for approximation with scaled shift-invariant spaces by means of quasi-projection operators. In particular, we provide error analysis for approximation of quasi-projection operators with Lipschitz spaces. The study of quasi-projection operators has many applications to various areas related to approximation theory and wavelet analysis.
IEEE Transactions on Information Theory, 2005
In this paper we study a simple oversampled analog-to-digital (A/D) conversion in shift invariant... more In this paper we study a simple oversampled analog-to-digital (A/D) conversion in shift invariant spaces. The Beurling-Landau type theorem for bandlimited signal spaces is extended to shift invariant spaces, and then a non-uniform sampling theorem for shift invariant spaces is established, which says, a uniformly discrete set is a stable sampling set for a shift invariant space if its Beurling lower density is larger than a fixed density determined by the generator of the shift invariant space. Consequently, an oversampling theorem for shift invariant spaces is attained. These sampling theorems together with a theorem concerning the stability of stable sampling in shift invariant spaces shown by us, are used to build a simple oversampled A/D conversion scheme in shift invariant spaces. In such a scheme, the quantization error e is found to behave as e 2 = O(τ 2 ) with respect to the sampling interval τ , which is the same as that for bandlimited signal spaces derived very recently. Moreover, we demonstrate that the bitrate required to encode the converted digital signal only increases as logarithm of sampling ratio.
Constructive Approximation, 2000
Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator on ... more Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator on R is constructed that consists of continuous functions, has small support, and is exact of order 2. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual sides. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions [CDP] is then used to construct the corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to the Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolutions. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev norms · H s ([0,1]) for s ∈ (−0.824926, 2.5). In particular, the multiwavelets form Riesz bases for L 2 ([0, 1]). text, and local schemes for interpolating function and derivative data are available. Moreover, the interpolatory nature of the generators suggests convenient ways of adjoining different local tensor product bases by isoparametric mappings, thereby obtaining multiresolution analyses on more complex geometries. Therefore, we concentrate in this paper on the construction of biorthogonal multiwavelets on the interval [0, 1], generated by special C 1 piecewise Hermite cubics, with the following properties:
Canadian Journal of Mathematics, 1997
We consider the shift-invariant space, S(Φ), generated by a set Φ ≥ fû 1 , . . . , û r g of compa... more We consider the shift-invariant space, S(Φ), generated by a set Φ ≥ fû 1 , . . . , û r g of compactly supported distributions on R when the vector of distributions û :≥ (û 1 , . . . , û r ) T satisfies a system of refinement equations expressed in matrix form
Advances in Computational Mathematics, 2009
In this paper we investigate spline wavelets on the interval with homogeneous boundary conditions... more In this paper we investigate spline wavelets on the interval with homogeneous boundary conditions. Starting with a pair of families of B-splines on the unit interval, we give a general method to explicitly construct wavelets satisfying the desired homogeneous boundary conditions. On the basis of a new development of multiresolution analysis, we show that these wavelets form Riesz bases of certain Sobolev spaces. The wavelet bases investigated in this paper are suitable for numerical solutions of ordinary and partial differential equations.
Transactions of the American Mathematical Society, 1995
Let be a triangulation of some polygonal domain in R 2 and S r k ( ), the space of all bivariate ... more Let be a triangulation of some polygonal domain in R 2 and S r k ( ), the space of all bivariate C r piecewise polynomials of total degree k on . In this paper, we construct a local basis of some subspace of the space S r k ( ), where k 3r+2, that can be used to provide the highest order of approximation, with the property that the approximation constant of this order is independent of the geometry of with the exception of the smallest angle in the partition. This result is obtained by means of a careful choice of locally supported basis functions which, however, require a very technical proof to justify their stability in optimalorder approximation. A new formulation of smoothness conditions for piecewise polynomials in terms of their B-net representations is derived for this purpose.