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Inversion and extension of the finite Hilbert transform Guillermo P . Curbera Universidad de Sevilla September 11, 2019 Workshop on Banach spaces and Banach lattices ICMAT Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform

  1. Inversion and extension of the finite Hilbert transform Guillermo P . Curbera Universidad de Sevilla September 11, 2019 Workshop on Banach spaces and Banach lattices ICMAT Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 1 / 28

  2. Authorship Joint work with: Susumu Okada University of Tasmania Australia Werner J. Ricker Katholische Universität Eichstätt Germany Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 2 / 28

  3. The finite Hilbert transform The finite Hilbert transform 1 Inversion of the FHT 2 Extension of the FHT 3 Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 3 / 28

  4. The finite Hilbert transform The airfoil equation “The study of an ideal flow past a thin airfoil” lead in aerodynamics to the airfoil equation: ż 1 p . v . 1 f p x q p AE q x ´ t dx “ g p t q , a . e . t P p´ 1 , 1 q . π ´ 1 Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 4 / 28

  5. The finite Hilbert transform The airfoil equation “The study of an ideal flow past a thin airfoil” lead in aerodynamics to the airfoil equation: ż 1 p . v . 1 f p x q p AE q x ´ t dx “ g p t q , a . e . t P p´ 1 , 1 q . π ´ 1 Studied by: Birnbaum 1920’s; von Kármán 1930’s; Söhngen 1940’s; Tricomi 1950’s. Tricomi “Integral Equations” (1957) for the spaces L p p´ 1 , 1 q . Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 4 / 28

  6. The finite Hilbert transform The airfoil equation “The study of an ideal flow past a thin airfoil” lead in aerodynamics to the airfoil equation: ż 1 p . v . 1 f p x q p AE q x ´ t dx “ g p t q , a . e . t P p´ 1 , 1 q . π ´ 1 Studied by: Birnbaum 1920’s; von Kármán 1930’s; Söhngen 1940’s; Tricomi 1950’s. Tricomi “Integral Equations” (1957) for the spaces L p p´ 1 , 1 q . Nowadays is used in Tomography (image reconstruction). Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 4 / 28

  7. The finite Hilbert transform The finite Hilbert transform FHT The finite Hilbert transform is defined, for f P L 1 p´ 1 , 1 q , by the principal value integral: ż 1 Tf p t q : “ p . v . 1 f p x q x ´ t dx , t P p´ 1 , 1 q . π ´ 1 Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 5 / 28

  8. The finite Hilbert transform The finite Hilbert transform FHT The finite Hilbert transform is defined, for f P L 1 p´ 1 , 1 q , by the principal value integral: ż 1 Tf p t q : “ p . v . 1 f p x q x ´ t dx , t P p´ 1 , 1 q . π ´ 1 The setting of the L p -spaces is not the most adequate for studying the FHT, because: Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 5 / 28

  9. The finite Hilbert transform The finite Hilbert transform FHT The finite Hilbert transform is defined, for f P L 1 p´ 1 , 1 q , by the principal value integral: ż 1 Tf p t q : “ p . v . 1 f p x q x ´ t dx , t P p´ 1 , 1 q . π ´ 1 The setting of the L p -spaces is not the most adequate for studying the FHT, because: ñ L 2 , 8 p´ 1 , 1 q Ę X . T : X Ñ X is injective ð ñ X Ď L 2 , 1 p´ 1 , 1 q . T : X Ñ X has non-dense range ð Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 5 / 28

  10. The finite Hilbert transform Rearrangement invariant space (r.i.s.) Function space X on p´ 1 , 1 q such that: Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 6 / 28

  11. The finite Hilbert transform Rearrangement invariant space (r.i.s.) Function space X on p´ 1 , 1 q such that: X consists of measurable functions, X Ď L 0 p´ 1 , 1 q . X has a complete norm } ¨ } X . X in an ideal of measurable functions: | g | ď | f | a . e . & f P X ù ñ g P X & } g } X ď } f } X . X is rearrangement invariant: m pt x : | g p x q| ą λ u “ m pt x : | f p x q| ą λ u , for all λ ą 0 and f P X ù ñ g P X and } g } X “ } f } X . Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 6 / 28

  12. The finite Hilbert transform Rearrangement invariant space (r.i.s.) Function space X on p´ 1 , 1 q such that: X consists of measurable functions, X Ď L 0 p´ 1 , 1 q . X has a complete norm } ¨ } X . X in an ideal of measurable functions: | g | ď | f | a . e . & f P X ù ñ g P X & } g } X ď } f } X . X is rearrangement invariant: m pt x : | g p x q| ą λ u “ m pt x : | f p x q| ą λ u , for all λ ą 0 and f P X ù ñ g P X and } g } X “ } f } X . Examples: L p spaces, weak- L p spaces, Orlicz spaces, Lorentz L p , q spaces, Lorentz Λ φ spaces, Marcinkiewicz spaces,..... Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 6 / 28

  13. The finite Hilbert transform Boundedness of T on r.i.s. Theorem (M. Riesz): For H the Hilbert transform on R H : L p p R q Ñ L p p R q ð ñ 1 ă p ă 8 . Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 7 / 28

  14. The finite Hilbert transform Boundedness of T on r.i.s. Theorem (M. Riesz): For H the Hilbert transform on R H : L p p R q Ñ L p p R q ð ñ 1 ă p ă 8 . Theorem (Boyd): For X r.i.s. on R H : X Ñ X ð ñ 0 ă α X ď α X ă 1 , where the Boyd indices of X (with E 1 { t the dilation operator f ÞÑ f p¨{ t q ): log } E 1 { t } log } E 1 { t } 0 ď α X : “ lim ď α X : “ lim ď 1 . log t log t t Ñ8 t Ñ 0 ` Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 7 / 28

  15. The finite Hilbert transform Boundedness of T on r.i.s. Theorem (M. Riesz): For H the Hilbert transform on R H : L p p R q Ñ L p p R q ð ñ 1 ă p ă 8 . Theorem (Boyd): For X r.i.s. on R H : X Ñ X ð ñ 0 ă α X ď α X ă 1 , where the Boyd indices of X (with E 1 { t the dilation operator f ÞÑ f p¨{ t q ): log } E 1 { t } log } E 1 { t } 0 ď α X : “ lim ď α X : “ lim ď 1 . log t log t t Ñ8 t Ñ 0 ` Theorem: For X r.i.s. on p´ 1 , 1 q T : X Ñ X ð ñ 0 ă α X ď α X ă 1 Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 7 / 28

  16. Inversion of the FHT The finite Hilbert transform 1 Inversion of the FHT 2 Extension of the FHT 3 Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 8 / 28

  17. Inversion of the FHT Inversion of the FHT Tricomi gave inversion formulae for L p p´ 1 , 1 q in two cases: Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 9 / 28

  18. Inversion of the FHT Inversion of the FHT Tricomi gave inversion formulae for L p p´ 1 , 1 q in two cases: When 1 ă p ă 2. When 2 ă p ă 8 . Not for p “ 2. Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 9 / 28

  19. Inversion of the FHT Inversion of the FHT Tricomi gave inversion formulae for L p p´ 1 , 1 q in two cases: When 1 ă p ă 2. When 2 ă p ă 8 . Not for p “ 2. We give inversion formulae for r.i.s. X on p´ 1 , 1 q in two cases : Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 9 / 28

  20. Inversion of the FHT Inversion of the FHT Tricomi gave inversion formulae for L p p´ 1 , 1 q in two cases: When 1 ă p ă 2. When 2 ă p ă 8 . Not for p “ 2. We give inversion formulae for r.i.s. X on p´ 1 , 1 q in two cases : When 1 { 2 ă α X ď α X ă 1. When 0 ă α X ď α X ă 1 { 2. Not when 1 { 2 P r α X , α X s . For example, X= L 2 , q for 1 ď q ď 8 . Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 9 / 28

  21. Inversion of the FHT The case 0 ă α X ď α X ă 1 { 2 Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´ 1 , 1 q satisfying 0 ă α X ď α X ă 1 { 2 . Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 10 / 28

  22. Inversion of the FHT The case 0 ă α X ď α X ă 1 { 2 Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´ 1 , 1 q satisfying 0 ă α X ď α X ă 1 { 2 . (a) T : X Ñ X is injective. (b) q T : X Ñ X and satisfies q TT “ I, for ˆ ˙ a f p t q q 1 ´ x 2 T ? T p f qp x q : “ ´ p x q . 1 ´ t 2 Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 10 / 28

  23. Inversion of the FHT The case 0 ă α X ď α X ă 1 { 2 Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´ 1 , 1 q satisfying 0 ă α X ď α X ă 1 { 2 . (a) T : X Ñ X is injective. (b) q T : X Ñ X and satisfies q TT “ I, for ˆ ˙ a f p t q q 1 ´ x 2 T ? T p f qp x q : “ ´ p x q . 1 ´ t 2 " * ż 1 f p x q ? (c) The range of T is R p T q “ f P X : 1 ´ x 2 dx “ 0 . ´ 1 (d) q T is an isomorphism from R p T q onto X. " * B F ż 1 f p x q ? (e) X “ f P X : 1 ´ x 2 dx “ 0 ‘ 1 . ´ 1 Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 10 / 28

  24. Inversion of the FHT The case 1 { 2 ă α X ď α X ă 1 Theorem (C., Okada & Ricker, 2018) Let X be a r.i.s. on p´ 1 , 1 q satisfying 1 { 2 ă α X ď α X ă 1 . Guillermo P . Curbera (Univ. Sevilla) The finite Hilbert transform 11.09.2019 11 / 28

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