An Operator Based Approach to Irregular Frames of Translates
We consider translates of functions in <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/5/449 |
| _version_ | 1828158887655636992 |
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| author | Peter Balazs Sigrid Heineken |
| author_facet | Peter Balazs Sigrid Heineken |
| author_sort | Peter Balazs |
| collection | DOAJ |
| description | We consider translates of functions in <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> along an irregular set of points, that is, <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo stretchy="false">{</mo> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo>−</mo> <msub> <mi>λ</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </msub> </semantics> </math> </inline-formula>—where <inline-formula> <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> </inline-formula> is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform. |
| first_indexed | 2024-04-11T23:52:33Z |
| format | Article |
| id | doaj.art-030d5faa4b4b4cf0a49ad2a0cb104e43 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-11T23:52:33Z |
| publishDate | 2019-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-030d5faa4b4b4cf0a49ad2a0cb104e432022-12-22T03:56:27ZengMDPI AGMathematics2227-73902019-05-017544910.3390/math7050449math7050449An Operator Based Approach to Irregular Frames of TranslatesPeter Balazs0Sigrid Heineken1Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, 1040 Wien, AustriaIMAS UBA-CONICET, Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, C1428EGA Buenos Aires, ArgentinaWe consider translates of functions in <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> along an irregular set of points, that is, <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo stretchy="false">{</mo> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo>−</mo> <msub> <mi>λ</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </msub> </semantics> </math> </inline-formula>—where <inline-formula> <math display="inline"> <semantics> <mi>ϕ</mi> </semantics> </math> </inline-formula> is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform.https://www.mdpi.com/2227-7390/7/5/449framesRiesz basesirregular translatescanonical dualsframe-related operators |
| spellingShingle | Peter Balazs Sigrid Heineken An Operator Based Approach to Irregular Frames of Translates Mathematics frames Riesz bases irregular translates canonical duals frame-related operators |
| title | An Operator Based Approach to Irregular Frames of Translates |
| title_full | An Operator Based Approach to Irregular Frames of Translates |
| title_fullStr | An Operator Based Approach to Irregular Frames of Translates |
| title_full_unstemmed | An Operator Based Approach to Irregular Frames of Translates |
| title_short | An Operator Based Approach to Irregular Frames of Translates |
| title_sort | operator based approach to irregular frames of translates |
| topic | frames Riesz bases irregular translates canonical duals frame-related operators |
| url | https://www.mdpi.com/2227-7390/7/5/449 |
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