Functions: A Fourier Approach. Universitat Rostock. Germany. Dedicated to Prof. L. Berg on the occasion of his 65th birthday.
|
|
- Brent Paul
- 5 years ago
- Views:
Transcription
1 Approximation Properties of Multi{Scaling Functions: A Fourier Approach Gerlind Plona Fachbereich Mathemati Universitat Rostoc 1851 Rostoc Germany Dedicated to Prof. L. Berg on the occasion of his 65th birthday Abstract In this paper, we consider approximation properties of a nite set of functions ( ::: r; 1) which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vector =( ) r;1 is renable, we setch a new way, how to derive necessary and sucient conditions for the renement mas in Fourier domain. 1. Introduction For applications of multi{wavelets in nite element methods, the problem occurs, how to construct renable vectors := ( ) r;1 (r 2 N) of functions with short support, such that algebraic polynomials of degree < m(m 2 N) can be exactly reproduced by a linear combination of integer translates of ( ::: r; 1). In Heil, Strang and Strela [9] and in Plona [13], the approximation properties of renable function vectors := ( ) r;1 were studied in some detail. In particular, new necessary and sucient conditions for the renement mas of could be derived. In [13], it could even be shown that the function vector can only provide approximation order m if its renement mas factorizes in a certain manner. For nding these results, [9] as well as [13] strongly used properties of doubly innite matrices determined by the matrix coecients occuring in the renement equation (in time domain). Now wewant tosetch away, how the necessary and sucient conditions for the renement mas of can completely be derived in the Fourier domain. 1
2 As in [13], the functions are allowed to have a noncompact support if they have a suitable decay rate. The main tool of our new approach is the so called superfunction, which iscontained in the span of the integer translates of ( ::: r; 1) and already provides the same approximation order as. The results are applied to some multi{scaling functions 1 rst considered by Donovan, Geronimo, Hardin and Massopust [6, 7]. 2. Notations Let us introduce some notations. Consider the Hilbert space L 2 = L 2 (R) ofall square integrable functions on R. The Fourier transform of f 2 L 2 (R) is dened by ^f := R 1 ;1 f(x)e;ix dx: The function vector with elements in L 2 (R) isrenable, if satises a renement equation of the form = l2z P l (2 ;l) (P l 2 R rr ) or equivalently, if satises the Fourier transformed renement equation with ^ := (^ ) r;1 ^ = P (=2) ^(=2) (1) and with the renement mas (two{scale symbol) P = P := 1 2 l2z P l e ;il : (2) Note that P is an (r r){matrix of 2{periodic functions. The components of a renable function vector are called multi{scaling functions. Let BV (R) be the set of all functions which are of bounded variation over R and normalized by lim f(x) f(x) jxj1 =1 2 lim (f(x + h)+f(x ; h)) h (;1 <x<1): If f 2 L 1 (R) \ BV (R), then the Poisson summation formula l2z f(l) e;iul = j2z ^f(u +2j) is satised (cf. Butzer and Nessel [3]). By C(R), we denote the set of continuous functions on R. For a measurable function f on R and m 2 N let f p := ( Z 1 ;1 jf(x)j p dx) 1=p jfj m p := D m f p f m p := 2 m D f p :
3 Here and in the following, D denotes the dierential operator with respect to x D:= d= dx. Let W m p (R) be the usual Sobolev space with the norm m p. The l p {norm of a sequence c := fc l g l2zis dened by c l p := ( P l2z jc lj p ) 1=p : For m 2 N, let E m (R) be the space of all functions f 2 C(R) with the decay property sup x2r fjf(x)j (1 + jxj)1+m+ g < 1 ( >): Let l 2 ;m := fc := (c ): P 1 =;1 (1 + jj2 ) ;m jc j 2 < 1g beaweighted sequence with the corresponding norm c l 2 ;m := 1 l=;1 (1 + jlj 2 ) ;m jc l j 2 1=2 Considering the functions 2 E m (R) ( ::: r; 1), we call the set B() := f (;l) : l 2 Z ::: r ; 1g L;m 2 {stable if there exist constants <A B<1 with A r;1 c 2 l 2 ;m r;1 r;1 l2z c l (;l) 2 B L 2 ;m : c 2 l 2 ;m for any sequences c = fc l g l2z2 l;m 2 ( ::: r; 1). Here L 2 ;m denotes the weighted Hilbert space L 2 = ;m ff : f L 2 := ;m (1+jj2 ) ;m=2 f 2 < 1g. Note that, if the functions are compactly supported, then the (algebraic) linear independence of the integer translates of ( ::: r; 1) yields the L 2 ;m{stability ofb(). For m,we obtain the well{nown L 2 {stability (Riesz stability). For 2 E m (R) ( ::: r; 1), we say that provides controlled L p {approximation order m (1 p 1), if the following three conditions are satised: For each f 2 Wp m (R) there are sequences c h = fc h g l l2z( ::: r; 1 h>) such that for a constant c independent ofh we have: r;1 P P ;1=p (1) f ; h l2z ch l (=h ; l) p ch m jfj m p. (2) Furthermore, c h l p c f p ( ::: r; 1): (3) There is a constant independent ofh such that for l 2 Z dist (lh supp f) > ) c h l ( ::: r; 1): This notation of controlled L p {approximation order, rst introduced in Jia and Lei [11], is a generalization of the well{nown denition of approximation order for compactly supported functions. In [11], the strong connection of controlled approximation order provided by and the Strang{Fix conditions for was shown. Note 3
4 that, instead of using the denition of Jia and Lei [11], we also could tae the denition of local approximation order by Halton and Light [8]. For our considerations the equivalence to the Strang{Fix conditions is important. The theory of closed shift{invariant subspaces of L 2 (R), spanned by integer translates of a nite set of functions has been extensively studied (cf. e.g. de Boor, DeVore and Ron [1, 2] Jia [1]). In particular, it has been shown that the approximation order provided by a vector can already be realized by a nite linear combination f = r;1 We call f superfunction of. l2z a l (;l): (a l 2 R): 3. Approximation by renable function vectors In this section we shall give a new approach to necessary and sucient conditions for the renement mas of a renable vector ensuring controlled L p {approximation order m. In particular, we show, how a superfunction f of (providing the same approximation order as ) can be constructed by the coecients which occur in the linear combinations of reproducing the monomials. In the following, let r 2 N and m 2 N be xed. First we want to recall the result in [13] dealing with the connection between controlled L p {approximation order, reproduction of polynomials and Strang{Fix conditions. Theorem 1 (cf. [13]) Let =( ) r;1 beavector of functions 2 E m (R)\BV (R). Further, let B() be L 2 ;m{stable. Then the following conditions are equivalent: (a) The function vector provides controlled approximation order m (m 2 N). (b) Algebraic polynomials of degree <mcan be exactly reproduced by integer translates of,i.e., there arevectors y n l 2 R r (l 2 Z n ::: m; 1) such that the series P l2z (yn l )T (;l) are absolutely and uniformly convergent on any compact interval of R and l2z (yn l ) T (x ; l) =x n (x 2 R n ::: m; 1): (c) The function vector satises the Strang{Fix conditions of order m, i.e., there is a nitely supported sequence ofvectors fa l g l2z, such that f := l2z at l (;l) satises ^f() 6= D n ^f(2l) (l 2 Znfg n ::: m; 1): 4
5 The equivalence of (a) and (c) is already shown in Jia and Lei [11], Theorem 1.1. Further, (b) follows from (c) by [11], Corollary 2.3. For showing that (b) yields (c), in [13] the function f := m;1 a T ( + ) (3) is introduced. Here, the coecient vectors a are determined by (a ::: a m;1 ):=(y ::: ym;1 ) V ;1 with the Vandermonde matrix V := ( n ) m;1. Hence we have n y n = m;1 By Fourier transform of (3) we obtain ^f(u) =A(u) T ^(u) with n a (n ::: m; 1): (4) A(u) := m;1 a e iu : (5) That means, A(u) isan(rr){matrix of trigonometric polynomials. Observe that by (4) (D n A)() = m;1 (i) n a = i n y n (n ::: m; 1): Using the Poisson summation formula it can be shown that f satises the conditions (D ^f)(2l) l (l 2 Z ::: m; 1) and hence the Strang{Fix conditions of order m (cf. [13]). Observe that f in (3) is a superfunction of. In the new proof for the following theorem, this superfunction will be the main tool. Theorem 2 Let =( ) r;1 bearenable vector of functions 2 E m (R)\BV (R). Further, let B() be L 2 ;m {stable. Then the function vector provides Lp {controlled approximation order m if and only if the renement mas P of in (2) satises the following conditions: There arevectors y 2 Rr y 6= ( ::: m; 1) such that for n ::: m; 1 we have n n (y ) T (2i) ;n (D n; P )() = 2 ;n (y n ) T (6) n n (y ) T (2i) ;n (D n; P )() = T (7) where denotes the zero vector. 5
6 Proof: Note that the conditions (6){(7) can also be written in the form D n [A T (2u) P (u)]j u = (D n A) T () (n ::: m; 1) (8) D n [A T (2u) P (u)]j u= = T (n ::: m; 1) (9) where A, dened in (5), is the symbol of a superfunction f of in (3). From Theorem 1 we now that provides controlled approximation order m if and only if f satises the Strang{Fix conditions of order m. Hence we onlyhave to prove: The relations (8){(9) are satised if and only if f satises the Strang{Fix conditions of order m, i.e., (D n ^f)(2l) =cn l (n ::: m; 1) (1) with constants c n 2 R and c We show that the relations (8){(9) are satised if we have (1). Note that by (1) ^f(2u) =A T (2u) ^(2u) =A T (2u) P (u)^(u): Taing the derivatives, it follows on the one hand and on the other hand (D n ^f)(u) = D n [A T (u) n ^(u)] n = (D A T )(u)(d n; ^)(u) 2 n (D n ^f)(2u) = D n [A T (2u) P (u) n ^(u)] n = D [A T (2u) P (u)] (D n; ^)(u): 2. Let us rst show that the conditions (9) are satised. For all l 2 Zwe nd by (1) that = ^f(4l +2) =A T (4l +2) P (2l + ) ^(2l + ) = A T () P () ^(2l + ): Hence, linear independence of the sequences f^ ( +2l)g l2zfor ::: r; 1 gives A T () P () T : Note that the linear independence of the sequences f^ (u +2l)g l2zfor all u 2 R and for ::: r; 1 is satised if and only if the integer translates of form a L 2 {stable basis of their closed span (cf. Jia and Micchelli [12]). This was the rst step of the induction proof. 6
7 Let now D [A T (2u) P (u)]j u= = T be satised for ::: n; 1(n<m), and observe that by assumption (1) (D n ^f)(4l +2) = for all l 2 Z. Then, by the linear independence of f^ ( +2l)g l2zfor ::: r; 1 and by n n = 2 n (D n ^f)(4l +2) = D [A T (2u) P (u)]j u= (D n; ^)( +2l) = D n [A T (2u) P (u)]j u= ^( +2l) it follows that D n [A T (2u) P (u)]j u= = T : Thus, the relations (9) are satised. 3. Now we show that (1) yields (8). Let u =2l (l 2 Z). Then we have on the one hand by the Strang{Fix conditions and on the other hand ^f(4l) =A T () P () ^(2l) =c l ^f(2l) =A T () ^(2l) =c l : By linear independence of f^ (2l)g l2zfor ::: r; 1we obtain A T () P () = A T (): Again, we proceed by induction. Let now D [A T (2u) P (u)]j u =(D A T )() be satised for ::: n; 1(n<m) and observe that by assumption (D n ^f)(2l) = c n l (l 2 Z c 6= ). Then we nd for all l 2 Z n n 2 n (D n ^f)(4l) = D [A(2u) T P (u)]j u (D n; ^)(2l) n;1 n = (D A T )() (D n; ^)(2l) On the other hand, for l 2 Z (D n ^f)(2l) = n Hence, a comparison yields +D n [A T (2u) P (u)]j u ^(2l) =cn l n (D A T )() (D n; ^)(2l) =cn l : D n [A T (2u) P (u)]j u ^(2l) =(D n A T )() ^(2l) 7
8 By linear independence of f^ (2l)g l2zfor ::: r; 1we obtain D n [A T (2u) P (u)]j u =(D n A T )(): Now the proof by induction is complete. 4. We are going to prove the reverse direction. Assume that the relations (8){(9) are satised. We show that then the conditions (D n ^f)(2l) =cn l (n ::: m; 1) hold, where c 6. For the -th derivative of ^f we nd 2 (D ^f)(4l) = D [A T (2u) P (u)]j u (D ; ^)(2l) = (D A)() (D ; ^)(2l) = (D ^f)(2l) and 2 (D ^f)(4l +2) = = : D [A T (2u) P (u)]j u= (D ; ^)(2l + ) Thus, we indeed obtain (D n ^f)(2l) =cn l. It only remains to show that c 6. By Poisson summation formula and using the L 2 {stability of we have ^() = A T () ^() = (y ^() )T = (y )T (;l) 6= : l2z Hence f satises the Strang{Fix conditions of order m. Remar 3 For proving the second direction in Theorem 2 we do not need any stability condition if we assume that (y ) T ^() 6=. Since y and ^() are a left and a right eigenvector of P (), respectively, this assumption is satised if the eigenvalue 1 of P () is simple. 4. The GHM{multi{scaling functions We consider the example of a vector of two multi{scaling functions := ( 1 ) T treated in Donovan, Geronimo, Hardin and Massopust ([6, 7]). In the special case s = s = s 1 (with s 2 [;1 1]) of their construction, the renement equation of is given by (x) =P (2x)+P 1 (2x ; 1) + P 2 (2x ; 2) + P 3 (2x ; 3) (11) 8
9 where P := P 2 := ; s2 ;4s;3 1 ; 3(s;1)(s+1)(s2 ;3s;1) 4(s+2) 2 ; 3(s;1)(s+1)(s2 ;s+3) 4(s+2) 2 3s 2 +s;1 3s 2 +s;1 P 1 := P 3 := ; s2 ;4s;3 ; 3(s;1)(s+1)(s2 ;s+3) 4(s+2) 2 1 ; 3(s;1)(s+1)(s2 ;3s;1) 4(s+2) 2 : For the renement mas P we have P (u) := 1 2 (P + P 1 e ;iu + P 2 e ;2iu + P 3 e ;3iu ): Applying the result of Theorem 2 we can show that provides the controlled L p { approximation order m = 2: Observing that P () = ;s 2 +4s+3 ;3(s;1) 3 (s+1) s 2 +2s+1 (DP )() = i s 2 ;4s;3 4(s+2) 9(s;1) 3 (s+1) ;(3s 2 +2s+1) 4(s+2) 2 and P () = 1 2 3(s2 ;1) (DP )() =i ;s 2 +4s+3 4(s+2) 3(s 2 ;1)(;s 2 +4s+3) ;3(s 2 ;1) 4(s+2) 2 we nd with the relations and y ;3(s2 ; 1) = s +2 1 y 1 ;3(s2 ; 1) = 2(s +2) 1 (y ) T P ()=(y ) T (y ) T P () T (2i) ;1 (y ) T (DP )() + (y 1 ) T P () = 2 ;1 (y 1 ) T (2i) ;1 (y ) T (DP )()+(y 1 ) T P () = T : Hence, (8){(9) are satised for m =2. Knowing y superfunction f of (as dened in (3)) by and y 1,we can construct a f(x) =(y ; y 1 ) T (x)+(y 1 ) T (x +1) obtaining f(x) = 3(1 ; s2 ) 2(s +2) ( (x)+ (x +1))+ 1 (x +1): 9
10 Application of the renement equation (11) on the right hand side yields f(x) = 9(1 ; s2 ) 4(s +2) ( (2x)+ (2x + 1)) + 3(1 ; s2 ) 4(s +2) ( (2x ; 1) + (2x + 2)) ( 1(2x +2)+ 1 (2x)) + 1 (2x +1) = 1 2 f(2x ; 1) + f(2x)+1 f(2x +1): 2 That means, f itself satises the renement equation of the hat{function h(x) := maxf(1 ;jxj) g. Hence, taing a proper normalization constant, the superfunction f coincides with the hat function h. Indeed, in [6] the approximation order 2 provided by was derived by showing that the hat{function h lies in the span of the integer translates of 1. References [1] de Boor, C., DeVore, R. A., Ron, A.: Approximation from shift{invariant subspaces of L 2 (R d ). Trans. Amer. Math. Soc. 341 (1994) 787{86. [2] de Boor, C., DeVore, R. A., Ron, A.: The structure of nitely generated shift{ invariant spaces in L 2 (R d ). J. Funct. Anal. 119(1) (1994) 37{78. [3] Butzer, P. L., Nessel, R. J.: Fourier Analysis and Approximation. Basel: Birhauser Verlag, [4] Chui, C. K.: An Introduction to Wavelets. Boston: Academic Press, [5] Daubechies, I.: Ten Lectures on Wavelets. Philadelphia: SIAM, [6] Donovan G., Geronimo, J. S., Hardin, D. P., Massopust, P. R.: Construction of orthogonal wavelets using fractal interpolation functions, preprint [7] Geronimo, J. S., Hardin, D. P., Massopust, P. R.: Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78, 373{41. [8] Halton, E. J., Light, W. A.: On local and controlled approximation order. J. Approx. Theory 72 (1993) 268{277. [9] Heil, C., Strang, G., Strela, V.: Approximation by translates of renable functions. Numer. Math. (to appear). [1] Jia, R. Q.: Shift{Invariant spaces on the real line, Proc. Amer. Math. Soc. (to appear). [11] Jia, R. Q., Lei, J. J.: Approximation by multiinteger translates of functions having global support. J. Approx. Theory 72 (1993) 2{23. 1
11 [12] Jia, R. Q., Micchelli, C. A.: Using the renement equations for the construction of pre{wavelets II: Powers of two, Curves and Surfaces (P.J. Laurent, A. Le Mehaute, L.L. Schumaer, eds.), pp. 29{246. [13] Plona, G.: Approximation order provided by renable function vectors. Constructive Approx. (to appear). [14] Strang, G., Strela, V.: Short wavelets and matrix dilation equations. IEEE Trans. on SP, vol. 43,
Moment Computation in Shift Invariant Spaces. Abstract. An algorithm is given for the computation of moments of f 2 S, where S is either
Moment Computation in Shift Invariant Spaces David A. Eubanks Patrick J.Van Fleet y Jianzhong Wang ẓ Abstract An algorithm is given for the computation of moments of f 2 S, where S is either a principal
More informationA NOTE ON MATRIX REFINEMENT EQUATIONS. Abstract. Renement equations involving matrix masks are receiving a lot of attention these days.
A NOTE ON MATRI REFINEMENT EQUATIONS THOMAS A. HOGAN y Abstract. Renement equations involving matrix masks are receiving a lot of attention these days. They can play a central role in the study of renable
More informationSTABILITY AND LINEAR INDEPENDENCE ASSOCIATED WITH SCALING VECTORS. JIANZHONG WANG y
STABILITY AND LINEAR INDEPENDENCE ASSOCIATED WITH SCALING VECTORS JIANZHONG WANG y Abstract. In this paper, we discuss stability and linear independence of the integer translates of a scaling vector =
More information2 W. LAWTON, S. L. LEE AND ZUOWEI SHEN is called the fundamental condition, and a sequence which satises the fundamental condition will be called a fu
CONVERGENCE OF MULTIDIMENSIONAL CASCADE ALGORITHM W. LAWTON, S. L. LEE AND ZUOWEI SHEN Abstract. Necessary and sucient conditions on the spectrum of the restricted transition operators are given for the
More informationMultiresolution analysis by infinitely differentiable compactly supported functions. N. Dyn A. Ron. September 1992 ABSTRACT
Multiresolution analysis by infinitely differentiable compactly supported functions N. Dyn A. Ron School of of Mathematical Sciences Tel-Aviv University Tel-Aviv, Israel Computer Sciences Department University
More information290 J.M. Carnicer, J.M. Pe~na basis (u 1 ; : : : ; u n ) consisting of minimally supported elements, yet also has a basis (v 1 ; : : : ; v n ) which f
Numer. Math. 67: 289{301 (1994) Numerische Mathematik c Springer-Verlag 1994 Electronic Edition Least supported bases and local linear independence J.M. Carnicer, J.M. Pe~na? Departamento de Matematica
More informationCENTER FOR THE MATHEMATICAL SCIENCES. Characterizations of linear independence and stability. of the shifts of a univariate renable function
UNIVERSITY OF WISCONSIN-MADISON CENTER FOR THE MATHEMATICAL SCIENCES Characterizations of linear independence and stability of the shifts of a univariate renable function in terms of its renement mask
More informationOF THE SHIFTS OF FINITELY. 27 February underlying renable function. Unfortunately, several desirable properties are not available with
STBILITY ND INDEPENDENCE OF THE SHIFTS OF FINITELY MNY REFINBLE FUNCTIONS Thomas. Hogan 7 February 1997 BSTRCT. Typical constructions of wavelets depend on the stability of the shifts of an underlying
More information. Introduction Let M be an s s dilation matrix, that is, an s s integer matrix whose eigenvalues lie outside the closed unit disk. The compactly suppo
Multivariate Compactly Supported Fundamental Renable Functions, Duals and Biorthogonal Wavelets y Hui Ji, Sherman D. Riemenschneider and Zuowei Shen Abstract: In areas of geometric modeling and wavelets,
More informationOctober 7, :8 WSPC/WS-IJWMIP paper. Polynomial functions are renable
International Journal of Wavelets, Multiresolution and Information Processing c World Scientic Publishing Company Polynomial functions are renable Henning Thielemann Institut für Informatik Martin-Luther-Universität
More informationPOINT VALUES AND NORMALIZATION OF TWO-DIRECTION MULTIWAVELETS AND THEIR DERIVATIVES
November 1, 1 POINT VALUES AND NORMALIZATION OF TWO-DIRECTION MULTIWAVELETS AND THEIR DERIVATIVES FRITZ KEINERT AND SOON-GEOL KWON,1 Abstract Two-direction multiscaling functions φ and two-direction multiwavelets
More informationConstruction of Orthogonal Wavelets Using. Fractal Interpolation Functions. Abstract
Construction of Orthogonal Wavelets Using Fractal Interpolation Functions George Donovan, Jerey S. Geronimo Douglas P. Hardin and Peter R. Massopust 3 Abstract Fractal interpolation functions are used
More information2 W. LAWTON, S. L. LEE AND ZUOWEI SHEN Denote by L 1 (R d ) and L 2 (R d ) the spaces of Lebesque integrable and modulus square integrable functions d
STABILITY AND ORTHONORMALITY OF MULTIVARIATE REFINABLE FUNCTIONS W. LAWTON, S. L. LEE AND ZUOWEI SHEN Abstract. This paper characterizes the stability and orthonormality of the shifts of a multidimensional
More informationApproximation from shift-invariant subspaces of L 2 (IR d ) Carl de Boor 1, Ronald A. DeVore 2, and Amos Ron 1 Abstract: A complete characterization i
Approximation from shift-invariant subspaces of L 2 (IR d ) Carl de Boor 1, Ronald A. DeVore 2, and Amos Ron 1 Abstract: A complete characterization is given of closed shift-invariant subspaces of L 2
More informationQUADRATURES INVOLVING POLYNOMIALS AND DAUBECHIES' WAVELETS *
QUADRATURES INVOLVING POLYNOMIALS AND DAUBECHIES' WAVELETS * WEI-CHANG SHANN AND JANN-CHANG YAN Abstract. Scaling equations are used to derive formulae of quadratures involving polynomials and scaling/wavelet
More informationApproximation by Multiple Refinable Functions
Approximation by Multiple Refinable Functions Rong-Qing Jia, S. D. Riemenschneider, and Ding-Xuan Zhou Department of Mathematical Sciences University of Alberta Edmonton, Canada T6G 2G1 Abstract We consider
More informationDecomposition of Riesz frames and wavelets into a finite union of linearly independent sets
Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets Ole Christensen, Alexander M. Lindner Abstract We characterize Riesz frames and prove that every Riesz frame
More informationThe organization of the paper is as follows. First, in sections 2-4 we give some necessary background on IFS fractals and wavelets. Then in section 5
Chaos Games for Wavelet Analysis and Wavelet Synthesis Franklin Mendivil Jason Silver Applied Mathematics University of Waterloo January 8, 999 Abstract In this paper, we consider Chaos Games to generate
More informationBiorthogonal Spline Type Wavelets
PERGAMON Computers and Mathematics with Applications 0 (00 1 0 www.elsevier.com/locate/camwa Biorthogonal Spline Type Wavelets Tian-Xiao He Department of Mathematics and Computer Science Illinois Wesleyan
More informationSpectral properties of the transition operator associated to a multivariate re nement equation q
Linear Algebra and its Applications 292 (1999) 155±178 Spectral properties of the transition operator associated to a multivariate re nement equation q Rong-Qing Jia *, Shurong Zhang 1 Department of Mathematical
More informationDensity results for frames of exponentials
Density results for frames of exponentials P. G. Casazza 1, O. Christensen 2, S. Li 3, and A. Lindner 4 1 Department of Mathematics, University of Missouri Columbia, Mo 65211 USA pete@math.missouri.edu
More informationAN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION
MATHEMATICS OF COMPUTATION Volume 65, Number 14 April 1996, Pages 73 737 AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION W LAWTON, S L LEE AND ZUOWEI SHEN Abstract This paper gives a practical
More informationOle Christensen 3. October 20, Abstract. We point out some connections between the existing theories for
Frames and pseudo-inverses. Ole Christensen 3 October 20, 1994 Abstract We point out some connections between the existing theories for frames and pseudo-inverses. In particular, using the pseudo-inverse
More informationQUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE. Denitizable operators in Krein spaces have spectral properties similar to those
QUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE BRANKO CURGUS and BRANKO NAJMAN Denitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces.
More informationSTABILITY OF INVARIANT SUBSPACES OF COMMUTING MATRICES We obtain some further results for pairs of commuting matrices. We show that a pair of commutin
On the stability of invariant subspaces of commuting matrices Tomaz Kosir and Bor Plestenjak September 18, 001 Abstract We study the stability of (joint) invariant subspaces of a nite set of commuting
More informationA new factorization technique of the matrix mask of univariate renable functions. Fachbereich Mathematik Computer Science Department
A new factorization technique of the matrix mask of univariate renable functions Gerlind Plonka Amos Ron Fachbereich Mathematik Computer Science Department Universitat Duisburg University of Wisconsin-Madison
More informationApproximation by Conditionally Positive Definite Functions with Finitely Many Centers
Approximation by Conditionally Positive Definite Functions with Finitely Many Centers Jungho Yoon Abstract. The theory of interpolation by using conditionally positive definite function provides optimal
More informationOn the Construction of Wavelets. on a Bounded Interval. Gerlind Plonka, Kathi Selig and Manfred Tasche. Universitat Rostock. Germany.
1 On the Construction of Wavelets on a Bounded Interval Gerlind Plonka, Kathi Selig and Manfred Tasche Fachbereich Mathematik Universitat Rostock D{18051 Rostock Germany Abstract This paper presents a
More informationINFINITUDE OF MINIMALLY SUPPORTED TOTALLY INTERPOLATING BIORTHOGONAL MULTIWAVELET SYSTEMS WITH LOW APPROXIMATION ORDERS. Youngwoo Choi and Jaewon Jung
Korean J. Math. (0) No. pp. 7 6 http://dx.doi.org/0.68/kjm.0...7 INFINITUDE OF MINIMALLY SUPPORTED TOTALLY INTERPOLATING BIORTHOGONAL MULTIWAVELET SYSTEMS WITH LOW APPROXIMATION ORDERS Youngwoo Choi and
More informationApplied and Computational Harmonic Analysis 11, (2001) doi: /acha , available online at
Applied and Computational Harmonic Analysis 11 305 31 (001 doi:10.1006/acha.001.0355 available online at http://www.idealibrary.com on LETTER TO THE EDITOR Construction of Multivariate Tight Frames via
More informationConstruction of Multivariate Compactly Supported Orthonormal Wavelets
Construction of Multivariate Compactly Supported Orthonormal Wavelets Ming-Jun Lai Department of Mathematics The University of Georgia Athens, GA 30602 April 30, 2004 Dedicated to Professor Charles A.
More informationON TRIVIAL GRADIENT YOUNG MEASURES BAISHENG YAN Abstract. We give a condition on a closed set K of real nm matrices which ensures that any W 1 p -grad
ON TRIVIAL GRAIENT YOUNG MEASURES BAISHENG YAN Abstract. We give a condition on a closed set K of real nm matrices which ensures that any W 1 p -gradient Young measure supported on K must be trivial the
More informationConstruction of orthogonal multiscaling functions and multiwavelets with higher approximation order based on the matrix extension algorithm
J. Math. Kyoto Univ. (JMKYAZ 6- (6, 75 9 Construction of orthogonal multiscaling functions and multiwavelets with higher approximation order based on the matrix extension algorithm By Shouzhi Yang and
More informationConstruction of Smooth Fractal Surfaces Using Hermite Fractal Interpolation Functions. P. Bouboulis, L. Dalla and M. Kostaki-Kosta
BULLETIN OF THE GREEK MATHEMATICAL SOCIETY Volume 54 27 (79 95) Construction of Smooth Fractal Surfaces Using Hermite Fractal Interpolation Functions P. Bouboulis L. Dalla and M. Kostaki-Kosta Received
More informationThe Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Common Homoclinic Points of Commuting Toral Automorphisms Anthony Manning Klaus Schmidt
More informationaxioms Construction of Multiwavelets on an Interval Axioms 2013, 2, ; doi: /axioms ISSN
Axioms 2013, 2, 122-141; doi:10.3390/axioms2020122 Article OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Construction of Multiwavelets on an Interval Ahmet Altürk 1 and Fritz Keinert 2,
More informationMRA Frame Wavelets with Certain Regularities Associated with the Refinable Generators of Shift Invariant Spaces
Chapter 6 MRA Frame Wavelets with Certain Regularities Associated with the Refinable Generators of Shift Invariant Spaces Tian-Xiao He Department of Mathematics and Computer Science Illinois Wesleyan University,
More informationEstimating the support of a scaling vector
Estimating the support of a scaling vector Wasin So Jianzhong Wang October, 00 Abstract An estimate is given for the support of each component function of a compactly supported scaling vector satisfying
More information1 Ordinary points and singular points
Math 70 honors, Fall, 008 Notes 8 More on series solutions, and an introduction to \orthogonal polynomials" Homework at end Revised, /4. Some changes and additions starting on page 7. Ordinary points and
More informationHigh-Order Balanced Multiwavelets: Theory, Factorization, and Design
1918 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 49, NO 9, SEPTEMBER 2001 High-Order Balanced Multiwavelets: Theory, Factorization, Design Jérôme Lebrun, Associate Member, IEEE, Martin Vetterli, Fellow,
More informationARCHIVUM MATHEMATICUM (BRNO) Tomus 32 (1996), 13 { 27. ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR DIFFERENCE EQUATION
ARCHIVUM MATHEMATICUM (BRNO) Tomus 32 (996), 3 { 27 ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR DIFFERENCE EQUATION P. J. Y. Wong and R. P. Agarwal Abstract. We oer sucient conditions for the
More informationLINEAR INDEPENDENCE OF PSEUDO-SPLINES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 9, September 006, Pages 685 694 S 000-9939(06)08316-X Article electronically published on March 3, 006 LINEAR INDEPENDENCE OF PSEUDO-SPLINES
More informationA NOTE ON FUNCTION SPACES GENERATED BY RADEMACHER SERIES
Proceedings of the Edinburgh Mathematical Society (1997) 40, 119-126 A NOTE ON FUNCTION SPACES GENERATED BY RADEMACHER SERIES by GUILLERMO P. CURBERA* (Received 29th March 1995) Let X be a rearrangement
More informationOverview of normed linear spaces
20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural
More informationc 2000 Society for Industrial and Applied Mathematics
SIAM J MATH ANAL Vol 32, No 2, pp 420 434 c 2000 Society for Industrial and Applied Mathematics DISTRIBUTIONAL SOLUTIONS OF NONHOMOGENEOUS DISCRETE AND CONTINUOUS REFINEMENT EQUATIONS RONG-QING JIA, QINGTANG
More informationViewed From Cubic Splines. further clarication and improvement. This can be described by applying the
Radial Basis Functions Viewed From Cubic Splines Robert Schaback Abstract. In the context of radial basis function interpolation, the construction of native spaces and the techniques for proving error
More informationConstruction of Biorthogonal Wavelets from Pseudo-splines
Construction of Biorthogonal Wavelets from Pseudo-splines Bin Dong a, Zuowei Shen b, a Department of Mathematics, National University of Singapore, Science Drive 2, Singapore, 117543. b Department of Mathematics,
More informationAffine and Quasi-Affine Frames on Positive Half Line
Journal of Mathematical Extension Vol. 10, No. 3, (2016), 47-61 ISSN: 1735-8299 URL: http://www.ijmex.com Affine and Quasi-Affine Frames on Positive Half Line Abdullah Zakir Husain Delhi College-Delhi
More informationThe Sobolev regularity of renable functions Amos Ron and Zuowei Shen 1. Introduction 1.1. General Let be a function in L (IR d ). We say that is (dyad
UNIVERSITY OF WISCONSIN-MADISON CENTER FOR THE MATHEMATICAL SCIENCES The Sobolev regularity of renable functions Amos Ron Zuowei Shen Computer Science Department Department of Mathematics University of
More informationApproximation of Integrable Functions by Wavelet Expansions
Results Math 72 27, 23 2 c 26 The Authors. This article is published with open access at Springerlink.com 422-6383/7/323-9 published online October 25, 26 DOI.7/s25-6-64-z Results in Mathematics Approximation
More informationConstrained Leja points and the numerical solution of the constrained energy problem
Journal of Computational and Applied Mathematics 131 (2001) 427 444 www.elsevier.nl/locate/cam Constrained Leja points and the numerical solution of the constrained energy problem Dan I. Coroian, Peter
More informationON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES
ON SAMPLING RELATED PROPERTIES OF B-SPLINE RIESZ SEQUENCES SHIDONG LI, ZHENGQING TONG AND DUNYAN YAN Abstract. For B-splineRiesz sequencesubspacesx span{β k ( n) : n Z}, thereis an exact sampling formula
More informationEXAMPLES OF REFINABLE COMPONENTWISE POLYNOMIALS
EXAMPLES OF REFINABLE COMPONENTWISE POLYNOMIALS NING BI, BIN HAN, AND ZUOWEI SHEN Abstract. This short note presents four examples of compactly supported symmetric refinable componentwise polynomial functions:
More informationUNIVERSITY OF WISCONSIN-MADISON CENTER FOR THE MATHEMATICAL SCIENCES. On the construction of multivariate (pre)wavelets
UNIVERSITY OF WISCONSIN-MADISON CENTER FOR THE MATHEMATICAL SCIENCES On the construction of multivariate (pre)wavelets Carl de Boor 1, Ronald A. DeVore 2, and Amos Ron 1 Technical Summary Report #92-09
More information2 BAISHENG YAN When L =,it is easily seen that the set K = coincides with the set of conformal matrices, that is, K = = fr j 0 R 2 SO(n)g: Weakly L-qu
RELAXATION AND ATTAINMENT RESULTS FOR AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL BAISHENG YAN Abstract. Consider functional I(u) = R jjdujn ; L det Duj dx whose energy-well consists of matrices
More informationHow smooth is the smoothest function in a given refinable space? Albert Cohen, Ingrid Daubechies, Amos Ron
How smooth is the smoothest function in a given refinable space? Albert Cohen, Ingrid Daubechies, Amos Ron A closed subspace V of L 2 := L 2 (IR d ) is called PSI (principal shift-invariant) if it is the
More informationShift-Invariant Spaces and Linear Operator Equations. Rong-Qing Jia Department of Mathematics University of Alberta Edmonton, Canada T6G 2G1.
Shift-Invariant Spaces and Linear Operator Equations Rong-Qing Jia Department of Mathematics University of Alberta Edmonton, Canada T6G 2G1 Abstract In this paper we investigate the structure of finitely
More information2 A. Ron subdivision and the cascade. This second part is written as a short monograph, aiming to single out the few underlying principles, and to dem
Wavelets and Their Associated Operators Amos Ron Abstract. This article is devoted to the study of wavelets based on the theory of shift-invariant spaces. It consists of two, essentially disjoint, parts.
More informationGarrett: `Bernstein's analytic continuation of complex powers' 2 Let f be a polynomial in x 1 ; : : : ; x n with real coecients. For complex s, let f
1 Bernstein's analytic continuation of complex powers c1995, Paul Garrett, garrettmath.umn.edu version January 27, 1998 Analytic continuation of distributions Statement of the theorems on analytic continuation
More informationLinear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space
Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) Contents 1 Vector Spaces 1 1.1 The Formal Denition of a Vector Space.................................. 1 1.2 Subspaces...................................................
More informationThe Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-19 Wien, Austria The Negative Discrete Spectrum of a Class of Two{Dimentional Schrodinger Operators with Magnetic
More information2 Abstract The research summarized in this report is the result of a two-year eort that has focused on evaluating the viability of wavelet bases for t
SAND98-2456 Unlimited Release Printed November 1998 An Investigation of Wavelet Bases for Grid-Based Multi-Scale Simulations 1 Final Report Mar A. Christon, Timothy G. Trucano and Joe R. Weatherby Computational
More informationTheorem 1. Suppose that is a Fuchsian group of the second kind. Then S( ) n J 6= and J( ) n T 6= : Corollary. When is of the second kind, the Bers con
ON THE BERS CONJECTURE FOR FUCHSIAN GROUPS OF THE SECOND KIND DEDICATED TO PROFESSOR TATSUO FUJI'I'E ON HIS SIXTIETH BIRTHDAY Toshiyuki Sugawa x1. Introduction. Supposebthat D is a simply connected domain
More informationNonstationary Subdivision Schemes and Totally Positive Refinable Functions
Nonstationary Subdivision Schemes and Totally Positive Refinable Functions Laura Gori and Francesca Pitolli December, 2007 Abstract In this paper we construct a class of totally positive refinable functions,
More informationON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS. G. Makay
ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS G. Makay Student in Mathematics, University of Szeged, Szeged, H-6726, Hungary Key words and phrases: Lyapunov
More informationApplied and Computational Harmonic Analysis
Appl. Comput. Harmon. Anal. 32 (2012) 139 144 Contents lists available at ScienceDirect Applied and Computational Harmonic Analysis www.elsevier.com/locate/acha Letter to the Editor Frames for operators
More informationAbstract Minimal degree interpolation spaces with respect to a nite set of
Numerische Mathematik Manuscript-Nr. (will be inserted by hand later) Polynomial interpolation of minimal degree Thomas Sauer Mathematical Institute, University Erlangen{Nuremberg, Bismarckstr. 1 1, 90537
More informationA Finite Element Method for an Ill-Posed Problem. Martin-Luther-Universitat, Fachbereich Mathematik/Informatik,Postfach 8, D Halle, Abstract
A Finite Element Method for an Ill-Posed Problem W. Lucht Martin-Luther-Universitat, Fachbereich Mathematik/Informatik,Postfach 8, D-699 Halle, Germany Abstract For an ill-posed problem which has its origin
More informationFrame Wavelet Sets in R d
Frame Wavelet Sets in R d X. DAI, Y. DIAO Department of Mathematics University of North Carolina at Charlotte Charlotte, NC 28223 xdai@uncc.edu Q. GU Department of Mathematics Each China Normal University
More information1 Solutions to selected problems
1 Solutions to selected problems 1. Let A B R n. Show that int A int B but in general bd A bd B. Solution. Let x int A. Then there is ɛ > 0 such that B ɛ (x) A B. This shows x int B. If A = [0, 1] and
More information8 Singular Integral Operators and L p -Regularity Theory
8 Singular Integral Operators and L p -Regularity Theory 8. Motivation See hand-written notes! 8.2 Mikhlin Multiplier Theorem Recall that the Fourier transformation F and the inverse Fourier transformation
More informationFourier-like Transforms
L 2 (R) Solutions of Dilation Equations and Fourier-like Transforms David Malone December 6, 2000 Abstract We state a novel construction of the Fourier transform on L 2 (R) based on translation and dilation
More informationApplications of Polyspline Wavelets to Astronomical Image Analysis
VIRTUAL OBSERVATORY: Plate Content Digitization, Archive Mining & Image Sequence Processing edited by M. Tsvetkov, V. Golev, F. Murtagh, and R. Molina, Heron Press, Sofia, 25 Applications of Polyspline
More informationIntroduction Wavelet shrinage methods have been very successful in nonparametric regression. But so far most of the wavelet regression methods have be
Wavelet Estimation For Samples With Random Uniform Design T. Tony Cai Department of Statistics, Purdue University Lawrence D. Brown Department of Statistics, University of Pennsylvania Abstract We show
More informationStability of Kernel Based Interpolation
Stability of Kernel Based Interpolation Stefano De Marchi Department of Computer Science, University of Verona (Italy) Robert Schaback Institut für Numerische und Angewandte Mathematik, University of Göttingen
More informationarxiv: v2 [math.fa] 27 Sep 2016
Decomposition of Integral Self-Affine Multi-Tiles Xiaoye Fu and Jean-Pierre Gabardo arxiv:43.335v2 [math.fa] 27 Sep 26 Abstract. In this paper we propose a method to decompose an integral self-affine Z
More informationTHE PRINCIPLE OF LIMITING ABSORPTION FOR THE NON-SELFADJOINT. Hideo Nakazawa. Department of Mathematics, Tokyo Metropolitan University
THE PRINCIPLE OF LIMITING ABSORPTION FOR THE NON-SELFADJOINT SCHR ODINGER OPERATOR WITH ENERGY DEPENDENT POTENTIAL Hideo Nakazaa Department of Mathematics, Tokyo Metropolitan University. 1.9 1.Introduction
More information/00 $ $.25 per page
Contemporary Mathematics Volume 00, 0000 Domain Decomposition For Linear And Nonlinear Elliptic Problems Via Function Or Space Decomposition UE-CHENG TAI Abstract. In this article, we use a function decomposition
More informationA Tilt at TILFs. Rod Nillsen University of Wollongong. This talk is dedicated to Gary H. Meisters
A Tilt at TILFs Rod Nillsen University of Wollongong This talk is dedicated to Gary H. Meisters Abstract In this talk I will endeavour to give an overview of some aspects of the theory of Translation Invariant
More information2 Garrett: `A Good Spectral Theorem' 1. von Neumann algebras, density theorem The commutant of a subring S of a ring R is S 0 = fr 2 R : rs = sr; 8s 2
1 A Good Spectral Theorem c1996, Paul Garrett, garrett@math.umn.edu version February 12, 1996 1 Measurable Hilbert bundles Measurable Banach bundles Direct integrals of Hilbert spaces Trivializing Hilbert
More informationMATRIX VALUED WAVELETS AND MULTIRESOLUTION ANALYSIS
MATRIX VALUED WAVELETS AND MULTIRESOLUTION ANALYSIS A. SAN ANTOLÍN AND R. A. ZALIK Abstract. We introduce the notions of matrix valued wavelet set and matrix valued multiresolution analysis (A-MMRA) associated
More informationA RECONSTRUCTION FORMULA FOR BAND LIMITED FUNCTIONS IN L 2 (R d )
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 12, Pages 3593 3600 S 0002-9939(99)04938-2 Article electronically published on May 6, 1999 A RECONSTRUCTION FORMULA FOR AND LIMITED FUNCTIONS
More informationINVARIANCE OF A SHIFT-INVARIANT SPACE
INVARIANCE OF A SHIFT-INVARIANT SPACE AKRAM ALDROUBI, CARLOS CABRELLI, CHRISTOPHER HEIL, KERI KORNELSON, AND URSULA MOLTER Abstract. A shift-invariant space is a space of functions that is invariant under
More information446 SCIENCE IN CHINA (Series F) Vol. 46 introduced in refs. [6, ]. Based on this inequality, we add normalization condition, symmetric conditions and
Vol. 46 No. 6 SCIENCE IN CHINA (Series F) December 003 Construction for a class of smooth wavelet tight frames PENG Lizhong (Λ Π) & WANG Haihui (Ξ ) LMAM, School of Mathematical Sciences, Peking University,
More informationFURTHER STUDIES OF STRONGLY AMENABLE -REPRESENTATIONS OF LAU -ALGEBRAS
FURTHER STUDIES OF STRONGLY AMENABLE -REPRESENTATIONS OF LAU -ALGEBRAS FATEMEH AKHTARI and RASOUL NASR-ISFAHANI Communicated by Dan Timotin The new notion of strong amenability for a -representation of
More informationAN ALGORITHM FOR MATRIX EXTENSION AND WAVELET CONSTRUCTION W. LAWTON, S. L. LEE AND ZUOWEI SHEN Abstract. This paper gives a practical method of exten
AN ALGORITHM FOR MATRIX EXTENSION AND WAVELET ONSTRUTION W. LAWTON, S. L. LEE AND ZUOWEI SHEN Abstract. This aer gives a ractical method of extending an nr matrix P (z), r n, with Laurent olynomial entries
More informationSpurious Chaotic Solutions of Dierential. Equations. Sigitas Keras. September Department of Applied Mathematics and Theoretical Physics
UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports Spurious Chaotic Solutions of Dierential Equations Sigitas Keras DAMTP 994/NA6 September 994 Department of Applied Mathematics and Theoretical Physics
More informationINDUSTRIAL MATHEMATICS INSTITUTE 2000:15 Symmetric framelets A. Petukhov IMI Department of Mathematics Preprint Series University of South Carolina
INDUSTRIAL MATHEMATICS INSTITUTE 2000:5 Symmetric framelets A. Petukhov IMI Preprint Series Department of Mathematics University of South Carolina Symmetric framelets Alexander Petukhov June 27, 2000 Abstract
More informationON COMPACTNESS OF THE DIFFERENCE OF COMPOSITION OPERATORS. C φ 2 e = lim sup w 1
ON COMPACTNESS OF THE DIFFERENCE OF COMPOSITION OPERATORS PEKKA NIEMINEN AND EERO SAKSMAN Abstract. We give a negative answer to a conjecture of J. E. Shapiro concerning compactness of the dierence of
More informationBiorthogonal Spline-Wavelets on the Interval Stability and. RWTH Aachen Aachen. Germany. e{mail:
Biorthogonal Spline-Wavelets on the Interval Stability and Moment Conditions Wolfgang Dahmen Institut fur Geometrie und Praktische Mathematik RWTH Aachen 556 Aachen Germany e{mail: dahmen@igpm.rwth-aachen.de
More informationAN INEQUALITY FOR THE NORM OF A POLYNOMIAL FACTOR IGOR E. PRITSKER. (Communicated by Albert Baernstein II)
PROCDINGS OF TH AMRICAN MATHMATICAL SOCITY Volume 9, Number 8, Pages 83{9 S -9939()588-4 Article electronically published on November 3, AN INQUALITY FOR TH NORM OF A POLYNOMIAL FACTOR IGOR. PRITSKR (Communicated
More informationY. Liu and A. Mohammed. L p (R) BOUNDEDNESS AND COMPACTNESS OF LOCALIZATION OPERATORS ASSOCIATED WITH THE STOCKWELL TRANSFORM
Rend. Sem. Mat. Univ. Pol. Torino Vol. 67, 2 (2009), 203 24 Second Conf. Pseudo-Differential Operators Y. Liu and A. Mohammed L p (R) BOUNDEDNESS AND COMPACTNESS OF LOCALIZATION OPERATORS ASSOCIATED WITH
More informationApplied Mathematics &Optimization
Appl Math Optim 29: 211-222 (1994) Applied Mathematics &Optimization c 1994 Springer-Verlag New Yor Inc. An Algorithm for Finding the Chebyshev Center of a Convex Polyhedron 1 N.D.Botin and V.L.Turova-Botina
More informationA Riesz basis of wavelets and its dual with quintic deficient splines
Note di Matematica 25, n. 1, 2005/2006, 55 62. A Riesz basis of wavelets and its dual with quintic deficient splines F. Bastin Department of Mathematics B37, University of Liège, B-4000 Liège, Belgium
More informationApproximately dual frames in Hilbert spaces and applications to Gabor frames
Approximately dual frames in Hilbert spaces and applications to Gabor frames Ole Christensen and Richard S. Laugesen October 22, 200 Abstract Approximately dual frames are studied in the Hilbert space
More informationMath 497 R1 Winter 2018 Navier-Stokes Regularity
Math 497 R Winter 208 Navier-Stokes Regularity Lecture : Sobolev Spaces and Newtonian Potentials Xinwei Yu Jan. 0, 208 Based on..2 of []. Some ne properties of Sobolev spaces, and basics of Newtonian potentials.
More informationBin Han Department of Mathematical Sciences University of Alberta Edmonton, Canada T6G 2G1
SYMMETRIC ORTHONORMAL SCALING FUNCTIONS AND WAVELETS WITH DILATION FACTOR d = Bin Han Department of Mathematical Sciences University of Alberta Edmonton, Canada T6G 2G1 email: bhan@math.ualberta.ca Abstract.
More informationShayne Waldron ABSTRACT. It is shown that a linear functional on a space of functions can be described by G, a
UNIVERSITY OF WISCONSIN-MADISON CENTER FOR THE MATHEMATICAL SCIENCES Symmetries of linear functionals Shayne Waldron (waldron@math.wisc.edu) CMS Technical Summary Report #95-04 October 1994 ABSTRACT It
More informationA Stable Finite Dierence Ansatz for Higher Order Dierentiation of Non-Exact. Data. Bob Anderssen and Frank de Hoog,
A Stable Finite Dierence Ansatz for Higher Order Dierentiation of Non-Exact Data Bob Anderssen and Frank de Hoog, CSIRO Division of Mathematics and Statistics, GPO Box 1965, Canberra, ACT 2601, Australia
More informationSo reconstruction requires inverting the frame operator which is often difficult or impossible in practice. It follows that for all ϕ H we have
CONSTRUCTING INFINITE TIGHT FRAMES PETER G. CASAZZA, MATT FICKUS, MANUEL LEON AND JANET C. TREMAIN Abstract. For finite and infinite dimensional Hilbert spaces H we classify the sequences of positive real
More information