Duality and Biorthogonality for Weyl-Heisenberg Frames
Abstract Let $a>0, b>0, ab<1;$ and let $g\in L^2({\Bbb R}).$ In this paper we investigate the relation between the frame operator $S:f\in L^2({\Bbb R})\rightarrow \sum_{n,m}\,(f,g_{na,mb})\,g_{na,mb}$ and the matrix $H$ whose entries $H_{k,l\,;\,k',l'}$ are given by $(g_{k'/b...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Janssen, A.J.E.M. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
1994 |
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| Schlagwörter: |
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| Anmerkung: |
© Birkhäuser Boston 1994 |
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| Übergeordnetes Werk: |
Enthalten in: The journal of Fourier analysis and applications - Birkhäuser-Verlag, 1994, 1(1994), 4 vom: Nov., Seite 403-436 |
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| Übergeordnetes Werk: |
volume:1 ; year:1994 ; number:4 ; month:11 ; pages:403-436 |
| Links: |
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| DOI / URN: |
10.1007/s00041-001-4017-4 |
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OLC206293761X |
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| 245 | 1 | 0 | |a Duality and Biorthogonality for Weyl-Heisenberg Frames |
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| 520 | |a Abstract Let $a>0, b>0, ab<1;$ and let $g\in L^2({\Bbb R}).$ In this paper we investigate the relation between the frame operator $S:f\in L^2({\Bbb R})\rightarrow \sum_{n,m}\,(f,g_{na,mb})\,g_{na,mb}$ and the matrix $H$ whose entries $H_{k,l\,;\,k',l'}$ are given by $(g_{k'/b,l'/a},g_{k/b,l/a})$ for $k,l,k',l'\in{\Bbb Z}.$ Here $f_{x,y}(t)={\rm exp}(2\pi iyt)\,f(t-x),$$t\in{\Bbb R}$, for any $f\in L^2({\Bbb R}).$ We show that $S$ is bounded as a mapping of $L^2({\Bbb R})$ into $L^2({\Bbb R})$ if and only if $H$ is bounded as a mapping of $l^2({\Bbb Z}^2)$ into $l^2({\Bbb Z}^2).$ Also we show that $AI\leq S\leq BI$ if and only if $AI\leq\frac{1}{ab}\,H\leq BI,$ where $I$ denotes the identity operator of $L^2({\Bbb R})$ and $l^2({\Bbb Z}^2),$ respectively, and $A\geq 0,$$B<\infty.$ Next, when $g$ generates a frame, we have that $(g_{k/b,l/a})_{k,l}$ has an upper frame bound, and the minimal dual function $^{\circ}\gamma$ can be computed as $ab\,\sum_{k,l}\,(H^{-1})_{k,l\,;\,o,o}\,g_{k/b,l/a}.$ The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a $g$ generating a frame are inherited by $^{\circ}\gamma.$ In particular, we show that $^{\circ}\gamma\in{\cal S}$ when $g\in{\cal S}$ generates a frame $({\cal S}$ Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula. | ||
| 650 | 4 | |a Operator Composition | |
| 650 | 4 | |a Banach Algebra | |
| 650 | 4 | |a Tight Frame | |
| 650 | 4 | |a Shift Parameter | |
| 650 | 4 | |a Frame Operator | |
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10.1007/s00041-001-4017-4 doi (DE-627)OLC206293761X (DE-He213)s00041-001-4017-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Janssen, A.J.E.M. verfasserin aut Duality and Biorthogonality for Weyl-Heisenberg Frames 1994 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser Boston 1994 Abstract Let $a>0, b>0, ab<1;$ and let $g\in L^2({\Bbb R}).$ In this paper we investigate the relation between the frame operator $S:f\in L^2({\Bbb R})\rightarrow \sum_{n,m}\,(f,g_{na,mb})\,g_{na,mb}$ and the matrix $H$ whose entries $H_{k,l\,;\,k',l'}$ are given by $(g_{k'/b,l'/a},g_{k/b,l/a})$ for $k,l,k',l'\in{\Bbb Z}.$ Here $f_{x,y}(t)={\rm exp}(2\pi iyt)\,f(t-x),$$t\in{\Bbb R}$, for any $f\in L^2({\Bbb R}).$ We show that $S$ is bounded as a mapping of $L^2({\Bbb R})$ into $L^2({\Bbb R})$ if and only if $H$ is bounded as a mapping of $l^2({\Bbb Z}^2)$ into $l^2({\Bbb Z}^2).$ Also we show that $AI\leq S\leq BI$ if and only if $AI\leq\frac{1}{ab}\,H\leq BI,$ where $I$ denotes the identity operator of $L^2({\Bbb R})$ and $l^2({\Bbb Z}^2),$ respectively, and $A\geq 0,$$B<\infty.$ Next, when $g$ generates a frame, we have that $(g_{k/b,l/a})_{k,l}$ has an upper frame bound, and the minimal dual function $^{\circ}\gamma$ can be computed as $ab\,\sum_{k,l}\,(H^{-1})_{k,l\,;\,o,o}\,g_{k/b,l/a}.$ The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a $g$ generating a frame are inherited by $^{\circ}\gamma.$ In particular, we show that $^{\circ}\gamma\in{\cal S}$ when $g\in{\cal S}$ generates a frame $({\cal S}$ Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula. Operator Composition Banach Algebra Tight Frame Shift Parameter Frame Operator Enthalten in The journal of Fourier analysis and applications Birkhäuser-Verlag, 1994 1(1994), 4 vom: Nov., Seite 403-436 (DE-627)185271340 (DE-600)1233179-X (DE-576)079875580 1069-5869 nnns volume:1 year:1994 number:4 month:11 pages:403-436 https://doi.org/10.1007/s00041-001-4017-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_105 GBV_ILN_2005 GBV_ILN_4314 AR 1 1994 4 11 403-436 |
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10.1007/s00041-001-4017-4 doi (DE-627)OLC206293761X (DE-He213)s00041-001-4017-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Janssen, A.J.E.M. verfasserin aut Duality and Biorthogonality for Weyl-Heisenberg Frames 1994 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser Boston 1994 Abstract Let $a>0, b>0, ab<1;$ and let $g\in L^2({\Bbb R}).$ In this paper we investigate the relation between the frame operator $S:f\in L^2({\Bbb R})\rightarrow \sum_{n,m}\,(f,g_{na,mb})\,g_{na,mb}$ and the matrix $H$ whose entries $H_{k,l\,;\,k',l'}$ are given by $(g_{k'/b,l'/a},g_{k/b,l/a})$ for $k,l,k',l'\in{\Bbb Z}.$ Here $f_{x,y}(t)={\rm exp}(2\pi iyt)\,f(t-x),$$t\in{\Bbb R}$, for any $f\in L^2({\Bbb R}).$ We show that $S$ is bounded as a mapping of $L^2({\Bbb R})$ into $L^2({\Bbb R})$ if and only if $H$ is bounded as a mapping of $l^2({\Bbb Z}^2)$ into $l^2({\Bbb Z}^2).$ Also we show that $AI\leq S\leq BI$ if and only if $AI\leq\frac{1}{ab}\,H\leq BI,$ where $I$ denotes the identity operator of $L^2({\Bbb R})$ and $l^2({\Bbb Z}^2),$ respectively, and $A\geq 0,$$B<\infty.$ Next, when $g$ generates a frame, we have that $(g_{k/b,l/a})_{k,l}$ has an upper frame bound, and the minimal dual function $^{\circ}\gamma$ can be computed as $ab\,\sum_{k,l}\,(H^{-1})_{k,l\,;\,o,o}\,g_{k/b,l/a}.$ The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a $g$ generating a frame are inherited by $^{\circ}\gamma.$ In particular, we show that $^{\circ}\gamma\in{\cal S}$ when $g\in{\cal S}$ generates a frame $({\cal S}$ Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula. Operator Composition Banach Algebra Tight Frame Shift Parameter Frame Operator Enthalten in The journal of Fourier analysis and applications Birkhäuser-Verlag, 1994 1(1994), 4 vom: Nov., Seite 403-436 (DE-627)185271340 (DE-600)1233179-X (DE-576)079875580 1069-5869 nnns volume:1 year:1994 number:4 month:11 pages:403-436 https://doi.org/10.1007/s00041-001-4017-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_105 GBV_ILN_2005 GBV_ILN_4314 AR 1 1994 4 11 403-436 |
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10.1007/s00041-001-4017-4 doi (DE-627)OLC206293761X (DE-He213)s00041-001-4017-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Janssen, A.J.E.M. verfasserin aut Duality and Biorthogonality for Weyl-Heisenberg Frames 1994 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser Boston 1994 Abstract Let $a>0, b>0, ab<1;$ and let $g\in L^2({\Bbb R}).$ In this paper we investigate the relation between the frame operator $S:f\in L^2({\Bbb R})\rightarrow \sum_{n,m}\,(f,g_{na,mb})\,g_{na,mb}$ and the matrix $H$ whose entries $H_{k,l\,;\,k',l'}$ are given by $(g_{k'/b,l'/a},g_{k/b,l/a})$ for $k,l,k',l'\in{\Bbb Z}.$ Here $f_{x,y}(t)={\rm exp}(2\pi iyt)\,f(t-x),$$t\in{\Bbb R}$, for any $f\in L^2({\Bbb R}).$ We show that $S$ is bounded as a mapping of $L^2({\Bbb R})$ into $L^2({\Bbb R})$ if and only if $H$ is bounded as a mapping of $l^2({\Bbb Z}^2)$ into $l^2({\Bbb Z}^2).$ Also we show that $AI\leq S\leq BI$ if and only if $AI\leq\frac{1}{ab}\,H\leq BI,$ where $I$ denotes the identity operator of $L^2({\Bbb R})$ and $l^2({\Bbb Z}^2),$ respectively, and $A\geq 0,$$B<\infty.$ Next, when $g$ generates a frame, we have that $(g_{k/b,l/a})_{k,l}$ has an upper frame bound, and the minimal dual function $^{\circ}\gamma$ can be computed as $ab\,\sum_{k,l}\,(H^{-1})_{k,l\,;\,o,o}\,g_{k/b,l/a}.$ The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a $g$ generating a frame are inherited by $^{\circ}\gamma.$ In particular, we show that $^{\circ}\gamma\in{\cal S}$ when $g\in{\cal S}$ generates a frame $({\cal S}$ Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula. Operator Composition Banach Algebra Tight Frame Shift Parameter Frame Operator Enthalten in The journal of Fourier analysis and applications Birkhäuser-Verlag, 1994 1(1994), 4 vom: Nov., Seite 403-436 (DE-627)185271340 (DE-600)1233179-X (DE-576)079875580 1069-5869 nnns volume:1 year:1994 number:4 month:11 pages:403-436 https://doi.org/10.1007/s00041-001-4017-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_105 GBV_ILN_2005 GBV_ILN_4314 AR 1 1994 4 11 403-436 |
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10.1007/s00041-001-4017-4 doi (DE-627)OLC206293761X (DE-He213)s00041-001-4017-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Janssen, A.J.E.M. verfasserin aut Duality and Biorthogonality for Weyl-Heisenberg Frames 1994 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser Boston 1994 Abstract Let $a>0, b>0, ab<1;$ and let $g\in L^2({\Bbb R}).$ In this paper we investigate the relation between the frame operator $S:f\in L^2({\Bbb R})\rightarrow \sum_{n,m}\,(f,g_{na,mb})\,g_{na,mb}$ and the matrix $H$ whose entries $H_{k,l\,;\,k',l'}$ are given by $(g_{k'/b,l'/a},g_{k/b,l/a})$ for $k,l,k',l'\in{\Bbb Z}.$ Here $f_{x,y}(t)={\rm exp}(2\pi iyt)\,f(t-x),$$t\in{\Bbb R}$, for any $f\in L^2({\Bbb R}).$ We show that $S$ is bounded as a mapping of $L^2({\Bbb R})$ into $L^2({\Bbb R})$ if and only if $H$ is bounded as a mapping of $l^2({\Bbb Z}^2)$ into $l^2({\Bbb Z}^2).$ Also we show that $AI\leq S\leq BI$ if and only if $AI\leq\frac{1}{ab}\,H\leq BI,$ where $I$ denotes the identity operator of $L^2({\Bbb R})$ and $l^2({\Bbb Z}^2),$ respectively, and $A\geq 0,$$B<\infty.$ Next, when $g$ generates a frame, we have that $(g_{k/b,l/a})_{k,l}$ has an upper frame bound, and the minimal dual function $^{\circ}\gamma$ can be computed as $ab\,\sum_{k,l}\,(H^{-1})_{k,l\,;\,o,o}\,g_{k/b,l/a}.$ The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a $g$ generating a frame are inherited by $^{\circ}\gamma.$ In particular, we show that $^{\circ}\gamma\in{\cal S}$ when $g\in{\cal S}$ generates a frame $({\cal S}$ Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula. Operator Composition Banach Algebra Tight Frame Shift Parameter Frame Operator Enthalten in The journal of Fourier analysis and applications Birkhäuser-Verlag, 1994 1(1994), 4 vom: Nov., Seite 403-436 (DE-627)185271340 (DE-600)1233179-X (DE-576)079875580 1069-5869 nnns volume:1 year:1994 number:4 month:11 pages:403-436 https://doi.org/10.1007/s00041-001-4017-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_105 GBV_ILN_2005 GBV_ILN_4314 AR 1 1994 4 11 403-436 |
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10.1007/s00041-001-4017-4 doi (DE-627)OLC206293761X (DE-He213)s00041-001-4017-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Janssen, A.J.E.M. verfasserin aut Duality and Biorthogonality for Weyl-Heisenberg Frames 1994 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser Boston 1994 Abstract Let $a>0, b>0, ab<1;$ and let $g\in L^2({\Bbb R}).$ In this paper we investigate the relation between the frame operator $S:f\in L^2({\Bbb R})\rightarrow \sum_{n,m}\,(f,g_{na,mb})\,g_{na,mb}$ and the matrix $H$ whose entries $H_{k,l\,;\,k',l'}$ are given by $(g_{k'/b,l'/a},g_{k/b,l/a})$ for $k,l,k',l'\in{\Bbb Z}.$ Here $f_{x,y}(t)={\rm exp}(2\pi iyt)\,f(t-x),$$t\in{\Bbb R}$, for any $f\in L^2({\Bbb R}).$ We show that $S$ is bounded as a mapping of $L^2({\Bbb R})$ into $L^2({\Bbb R})$ if and only if $H$ is bounded as a mapping of $l^2({\Bbb Z}^2)$ into $l^2({\Bbb Z}^2).$ Also we show that $AI\leq S\leq BI$ if and only if $AI\leq\frac{1}{ab}\,H\leq BI,$ where $I$ denotes the identity operator of $L^2({\Bbb R})$ and $l^2({\Bbb Z}^2),$ respectively, and $A\geq 0,$$B<\infty.$ Next, when $g$ generates a frame, we have that $(g_{k/b,l/a})_{k,l}$ has an upper frame bound, and the minimal dual function $^{\circ}\gamma$ can be computed as $ab\,\sum_{k,l}\,(H^{-1})_{k,l\,;\,o,o}\,g_{k/b,l/a}.$ The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a $g$ generating a frame are inherited by $^{\circ}\gamma.$ In particular, we show that $^{\circ}\gamma\in{\cal S}$ when $g\in{\cal S}$ generates a frame $({\cal S}$ Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula. Operator Composition Banach Algebra Tight Frame Shift Parameter Frame Operator Enthalten in The journal of Fourier analysis and applications Birkhäuser-Verlag, 1994 1(1994), 4 vom: Nov., Seite 403-436 (DE-627)185271340 (DE-600)1233179-X (DE-576)079875580 1069-5869 nnns volume:1 year:1994 number:4 month:11 pages:403-436 https://doi.org/10.1007/s00041-001-4017-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_105 GBV_ILN_2005 GBV_ILN_4314 AR 1 1994 4 11 403-436 |
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Enthalten in The journal of Fourier analysis and applications 1(1994), 4 vom: Nov., Seite 403-436 volume:1 year:1994 number:4 month:11 pages:403-436 |
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Janssen, A.J.E.M. |
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Janssen, A.J.E.M. ddc 510 ssgn 17,1 misc Operator Composition misc Banach Algebra misc Tight Frame misc Shift Parameter misc Frame Operator Duality and Biorthogonality for Weyl-Heisenberg Frames |
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510 VZ 17,1 ssgn Duality and Biorthogonality for Weyl-Heisenberg Frames Operator Composition Banach Algebra Tight Frame Shift Parameter Frame Operator |
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ddc 510 ssgn 17,1 misc Operator Composition misc Banach Algebra misc Tight Frame misc Shift Parameter misc Frame Operator |
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Duality and Biorthogonality for Weyl-Heisenberg Frames |
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Duality and Biorthogonality for Weyl-Heisenberg Frames |
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Janssen, A.J.E.M. |
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duality and biorthogonality for weyl-heisenberg frames |
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Duality and Biorthogonality for Weyl-Heisenberg Frames |
| abstract |
Abstract Let $a>0, b>0, ab<1;$ and let $g\in L^2({\Bbb R}).$ In this paper we investigate the relation between the frame operator $S:f\in L^2({\Bbb R})\rightarrow \sum_{n,m}\,(f,g_{na,mb})\,g_{na,mb}$ and the matrix $H$ whose entries $H_{k,l\,;\,k',l'}$ are given by $(g_{k'/b,l'/a},g_{k/b,l/a})$ for $k,l,k',l'\in{\Bbb Z}.$ Here $f_{x,y}(t)={\rm exp}(2\pi iyt)\,f(t-x),$$t\in{\Bbb R}$, for any $f\in L^2({\Bbb R}).$ We show that $S$ is bounded as a mapping of $L^2({\Bbb R})$ into $L^2({\Bbb R})$ if and only if $H$ is bounded as a mapping of $l^2({\Bbb Z}^2)$ into $l^2({\Bbb Z}^2).$ Also we show that $AI\leq S\leq BI$ if and only if $AI\leq\frac{1}{ab}\,H\leq BI,$ where $I$ denotes the identity operator of $L^2({\Bbb R})$ and $l^2({\Bbb Z}^2),$ respectively, and $A\geq 0,$$B<\infty.$ Next, when $g$ generates a frame, we have that $(g_{k/b,l/a})_{k,l}$ has an upper frame bound, and the minimal dual function $^{\circ}\gamma$ can be computed as $ab\,\sum_{k,l}\,(H^{-1})_{k,l\,;\,o,o}\,g_{k/b,l/a}.$ The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a $g$ generating a frame are inherited by $^{\circ}\gamma.$ In particular, we show that $^{\circ}\gamma\in{\cal S}$ when $g\in{\cal S}$ generates a frame $({\cal S}$ Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula. © Birkhäuser Boston 1994 |
| abstractGer |
Abstract Let $a>0, b>0, ab<1;$ and let $g\in L^2({\Bbb R}).$ In this paper we investigate the relation between the frame operator $S:f\in L^2({\Bbb R})\rightarrow \sum_{n,m}\,(f,g_{na,mb})\,g_{na,mb}$ and the matrix $H$ whose entries $H_{k,l\,;\,k',l'}$ are given by $(g_{k'/b,l'/a},g_{k/b,l/a})$ for $k,l,k',l'\in{\Bbb Z}.$ Here $f_{x,y}(t)={\rm exp}(2\pi iyt)\,f(t-x),$$t\in{\Bbb R}$, for any $f\in L^2({\Bbb R}).$ We show that $S$ is bounded as a mapping of $L^2({\Bbb R})$ into $L^2({\Bbb R})$ if and only if $H$ is bounded as a mapping of $l^2({\Bbb Z}^2)$ into $l^2({\Bbb Z}^2).$ Also we show that $AI\leq S\leq BI$ if and only if $AI\leq\frac{1}{ab}\,H\leq BI,$ where $I$ denotes the identity operator of $L^2({\Bbb R})$ and $l^2({\Bbb Z}^2),$ respectively, and $A\geq 0,$$B<\infty.$ Next, when $g$ generates a frame, we have that $(g_{k/b,l/a})_{k,l}$ has an upper frame bound, and the minimal dual function $^{\circ}\gamma$ can be computed as $ab\,\sum_{k,l}\,(H^{-1})_{k,l\,;\,o,o}\,g_{k/b,l/a}.$ The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a $g$ generating a frame are inherited by $^{\circ}\gamma.$ In particular, we show that $^{\circ}\gamma\in{\cal S}$ when $g\in{\cal S}$ generates a frame $({\cal S}$ Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula. © Birkhäuser Boston 1994 |
| abstract_unstemmed |
Abstract Let $a>0, b>0, ab<1;$ and let $g\in L^2({\Bbb R}).$ In this paper we investigate the relation between the frame operator $S:f\in L^2({\Bbb R})\rightarrow \sum_{n,m}\,(f,g_{na,mb})\,g_{na,mb}$ and the matrix $H$ whose entries $H_{k,l\,;\,k',l'}$ are given by $(g_{k'/b,l'/a},g_{k/b,l/a})$ for $k,l,k',l'\in{\Bbb Z}.$ Here $f_{x,y}(t)={\rm exp}(2\pi iyt)\,f(t-x),$$t\in{\Bbb R}$, for any $f\in L^2({\Bbb R}).$ We show that $S$ is bounded as a mapping of $L^2({\Bbb R})$ into $L^2({\Bbb R})$ if and only if $H$ is bounded as a mapping of $l^2({\Bbb Z}^2)$ into $l^2({\Bbb Z}^2).$ Also we show that $AI\leq S\leq BI$ if and only if $AI\leq\frac{1}{ab}\,H\leq BI,$ where $I$ denotes the identity operator of $L^2({\Bbb R})$ and $l^2({\Bbb Z}^2),$ respectively, and $A\geq 0,$$B<\infty.$ Next, when $g$ generates a frame, we have that $(g_{k/b,l/a})_{k,l}$ has an upper frame bound, and the minimal dual function $^{\circ}\gamma$ can be computed as $ab\,\sum_{k,l}\,(H^{-1})_{k,l\,;\,o,o}\,g_{k/b,l/a}.$ The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a $g$ generating a frame are inherited by $^{\circ}\gamma.$ In particular, we show that $^{\circ}\gamma\in{\cal S}$ when $g\in{\cal S}$ generates a frame $({\cal S}$ Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula. © Birkhäuser Boston 1994 |
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