Relevant bibliographies by topics / Théorèmes de restriction de Fourier

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Théorèmes de restriction de Fourier'

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Published: 25 January 2025

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Journal articles on the topic "Théorèmes de restriction de Fourier"

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Kovač, Vjekoslav. "Fourier restriction implies maximal and variational Fourier restriction." Journal of Functional Analysis 277, no. 10 (November 2019): 3355–72. http://dx.doi.org/10.1016/j.jfa.2019.03.015.

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Demeter, Ciprian, and S. Zubin Gautam. "Bilinear Fourier Restriction Theorems." Journal of Fourier Analysis and Applications 18, no. 6 (June 6, 2012): 1265–90. http://dx.doi.org/10.1007/s00041-012-9230-9.

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Demeter, Ciprian. "Bourgain’s work in Fourier restriction." Bulletin of the American Mathematical Society 58, no. 2 (January 27, 2021): 191–204. http://dx.doi.org/10.1090/bull/1717.

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Kovač, Vjekoslav, and Diogo Oliveira e Silva. "A variational restriction theorem." Archiv der Mathematik 117, no. 1 (May 7, 2021): 65–78. http://dx.doi.org/10.1007/s00013-021-01604-1.

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AbstractWe establish variational estimates related to the problem of restricting the Fourier transform of a three-dimensional function to the two-dimensional Euclidean sphere. At the same time, we give a short survey of the recent field of maximal Fourier restriction theory.
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Shayya, Bassam. "Fourier restriction in low fractal dimensions." Proceedings of the Edinburgh Mathematical Society 64, no. 2 (April 30, 2021): 373–407. http://dx.doi.org/10.1017/s0013091521000201.

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AbstractLet $S \subset \mathbb {R}^{n}$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\mathbb {R}^{n}$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^{q}(X)} \lesssim \| f \|_{L^{p}(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \lesssim R^{\alpha }$ for all balls $B_R$ in $\mathbb {R}^{n}$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^{q}$ against the measure $\chi _X \,{\textrm {d}}x$. Our approach consists of replacing the characteristic function $\chi _X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted Hölder-type inequality that holds for general non-negative Lebesgue measurable functions on $\mathbb {R}^{n}$ and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du–Zhang theorem in the range $0 < \alpha < n/2$.
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Drury, S. W., and B. P. Marshall. "Fourier restriction theorems for degenerate curves." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 3 (May 1987): 541–53. http://dx.doi.org/10.1017/s0305004100066901.

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Fourier restriction theorems contain estimates of the formwhere σ is a measure on a smooth manifold M in ∝n. This paper is a continuation of [5], which considered this problem for certain degenerate curves in ∝n. Here estimates are obtained for all curves with degeneracies of finite order. References to previous work on this problem may be found in [5].
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Bruce, Benjamin Baker. "Fourier restriction to a hyperbolic cone." Journal of Functional Analysis 279, no. 3 (August 2020): 108554. http://dx.doi.org/10.1016/j.jfa.2020.108554.

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Carneiro, Emanuel, Diogo Oliveira e Silva, and Mateus Sousa. "Extremizers for Fourier restriction on hyperboloids." Annales de l'Institut Henri Poincaré C, Analyse non linéaire 36, no. 2 (March 2019): 389–415. http://dx.doi.org/10.1016/j.anihpc.2018.06.001.

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Nicola, Fabio. "Slicing surfaces and the Fourier restriction conjecture." Proceedings of the Edinburgh Mathematical Society 52, no. 2 (May 28, 2009): 515–27. http://dx.doi.org/10.1017/s0013091507000995.

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AbstractWe deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid and the elliptic hyperboloid in ℝn implies that for the cone in ℝn+1. We also prove a new restriction estimate for any surface in ℝ3 locally isometric to the plane and of finite type.
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Carbery, Anthony. "Restriction implies Bochner–Riesz for paraboloids." Mathematical Proceedings of the Cambridge Philosophical Society 111, no. 3 (May 1992): 525–29. http://dx.doi.org/10.1017/s0305004100075599.

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Let Σ ⊆ ℝn be a (compact) hypersurface with non-vanishing Gaussian curvature, with suitable parameterizations, also called Σ: U → ℝn (U open patches in ℝn−1). The restriction problem for Σ is the question of the a priori estimate (for f ∈ S(ℝ))(^denoting the Fourier transform). The Bochner-Riesz problem for Σ is the question of whether the functionsdefine Lp-bounded Fourier multiplier operators on ℝn in the range.
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Dissertations / Theses on the topic "Théorèmes de restriction de Fourier"

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Thabouti, Lotfi. "Estimées de Carleman L^p globales." Electronic Thesis or Diss., Bordeaux, 2024. http://www.theses.fr/2024BORD0491.

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Dans cette thèse, on étudie des inégalités de Carleman L^p pour des problèmes elliptiques et leurs applications à la quantification du prolongement unique par rapport aux perturbations du laplacien. On s'intéresse d'abord aux inégalités de Carleman L^p sur une bande de R^d (dgeq 3), notée mathcal{S}:= (0,1) imes R^{d-1}, pour le Laplacien. Grâce à la transformée de Fourier et une factorisation de l'opérateur conjugué, nous réduisons la démonstration de ces inégalités à la construction d'une paramétrice pour le problème du Laplacien avec des conditions au bord.En utilisant cette paramétrice, on redémontre d'abord des inégalités classiques de Carleman L^2 pour le Laplacien. Ensuite, en appliquant des techniques d'analyse harmonique, notamment le théorème de restriction de Fourier pour établir des résultats de continuité de type L^p-L^q , on obtient des estimations L^p - L^q sur cette paramétrice.On applique ensuite ces méthodes au cas qui nous intéresse, à savoir les inégalités de Carleman L^p pour le Laplacien défini sur Omega , un ouvert borné et régulier de R^d (dgeq 3) , avec un second membre f_2 + f_{2 *'} + div F , f_2 in L^2(Omega), , f_{2 *'} in L^{ frac{2d}{d+2}}(Omega), ,F in L^2(Omega; C^{d}), et une condition de Dirichlet g in H^{frac{1}{2}}(partial Omega) . On montre deux estimations de Carleman globales : une sur la norme H^1 de la solution et une sur sa norme L^{frac{2d}{d-2}} , en termes de normes L^2 à poids de f_2 et F , de la norme L^{frac{2d}{d+2}} de f_{2 *'} et de la norme H^{frac{1}{2}} de g . Cela nous permet, par exemple, d'obtenir une quantification du prolongement unique pour les solutions de Delta u = V u + W_1 cdotabla u + div(W_2 u) en fonction des normes de V dans L^{q_0}(Omega) , de W_1 dans L^{q_1}(Omega) et de W_2 dans L^{q_2}(Omega) pour q_0 in (d/2, infty] et q_1 et q_2 satisfaisant soit q_1, , q_2 > (3d-2)/2 et frac{1}{q_1} + frac{1}{q_2}< 4 (1-frac{1}{d})/(3d-2) , soit q_1, , q_2 > 3d/2 .Dans une troisième partie, on étudie une quantification du prolongement unique des solutions de l'équation Delta u = V u + W_1 cdotabla u + div(W_2 u) mais avec des potentiels d'ordre un plus singuliers dans la classe limite d'intégrabilité. En particulier, on considère le cas W_1 in L^{q_1} et W_2 in L^{q_2} , avec q_1>d et q_2 >d . En utilisant le lemme de T. Wolff sur les mesures euclidiennes et une version raffinée des estimations de Carleman, on obtient des résultats de quantification du prolongement unique pour les solutions u de Delta u = V u + W_1 cdotabla u + div (W_2 u) en fonction des normes des potentiels
In this thesis, we study L^p Carleman inequalities for elliptic problems and their applications to the quantification of unique continuation with respect to perturbations of the Laplacian. We first focus on L^p Carleman inequalities on a strip of R^d (dgeq 3) , denoted mathcal{S}:= (0,1) imes R^{d-1} , for the Laplacian. Using the Fourier transform and a factorisation of the conjugate operator, we reduce the proof of these inequalities to the construction of a parametrix for the Laplacian problem with boundary conditions. Utilising this parametrix, we first reprove classical L^2 Carleman inequalities for the Laplacian. Then, applying harmonic analysis techniques, particularly the Fourier restriction theorem to establish L^p-L^q type continuity results, we obtain L^p - L^q estimates for this parametrix.We then apply these methods to the case of interest, namely L^p Carleman inequalities for the Laplacian defined on Omega , a bounded and regular open subset of R^d (d geq 3) , with a right-hand side f_2 + f_{2 *'} + div F , f_2 in L^2(Omega), , f_{2 *'} in L^{ frac{2d}{d+2}}(Omega), ,F in L^2(Omega; C^{d}) , and a Dirichlet condition g in H^{frac{1}{2}}(partial Omega) . We establish two global Carleman estimates: one on the H^1 norm of the solution and another on its L^{frac{2d}{d-2}} norm, in terms of weighted L^2 norms of f_2 and F , the L^{frac{2d}{d+2}} norm of f_{2 *'} , and the H^{frac{1}{2}} norm of g . This allows us, for example, to obtain a quantification of unique continuation for solutions of Delta u = V u + W_1 cdotabla u + div(W_2 u) in terms of the norms of V in L^{q_0}(Omega) , W_1 in L^{q_1}(Omega) , and W_2 in L^{q_2}(Omega) for q_0 in (d/2, infty] and q_1 and q_2 satisfying either q_1, , q_2 > (3d-2)/2 and frac{1}{q_1} + frac{1}{q_2}< 4(1-frac{1}{d})/(3d-2) , or q_1, , q_2 > 3d/2 .In the third part, we study a quantification of unique continuation for solutions of the equation Delta u = V u + W_1 cdotabla u + div(W_2 u) but with first-order potentials that are more singular in the limit integrability class. In particular, we consider the case where W_1 in L^{q_1} and W_2 in L^{q_2} , with q_1 > d and q_2 > d . Using T. Wolff's lemma on Euclidean measures and a refined version of Carleman estimates, we obtain unique continuation quantification results for solutions u of Delta u = V u + W_1 cdotabla u + div(W_2 u) in terms of the norms of the potentials
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Papadimitropoulos, Christos. "Fourier restriction phenomenon in thin sets." Thesis, University of Edinburgh, 2010. http://hdl.handle.net/1842/4625.

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We study the Fourier restriction phenomenon in settings where there is no underlying proper smooth subvariety. We prove an (Lp, L2) restriction theorem in general locally compact abelian groups and apply it in groups such as (Z/pLZ)n, R and locally compact ultrametric fields K. The problem of existence of Salem sets in a locally compact ultrametric field (K, | · |) is also considered. We prove that for every 0 < α < 1 and ǫ > 0 there exist a set E ⊂ K and a measure μ supported on E such that the Hausdorff dimension of E equals α and |bμ(x)| ≤ C|x|−α 2 +ǫ. We also establish the optimal extension of the Hausdorff-Young inequality in the compact ring of integers R of a locally compact ultrametric field K. We shall prove the following: For every 1 ≤ p ≤ 2 there is a Banach function space Fp(R) with σ-order continuous norm such that (i) Lp(R) ( Fp(R) ( L1(R) for every 1 < p < 2. (ii) The Fourier transform F maps Fp(R) to ℓp′ continuously. (iii) Lp(R) is continuously included in Fp(R) and Fp(R) is continuously included in L1(R). (iv) If Z is a Banach function space with the same properties as Fp(R) above, then Z is continuously included in Fp(R). (v) F1(R) = L1(R) and F2(R) = L2(R).
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Buschenhenke, Stefan [Verfasser]. "Restriction theorems for the Fourier transform / Stefan Buschenhenke." Kiel : Universitätsbibliothek Kiel, 2014. http://d-nb.info/1050388658/34.

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Wilheim, Daniel. "Restriction and Kakeya problems of Fourier analysis in vector spaces over finite fields." Thesis, University of Edinburgh, 2007. http://hdl.handle.net/1842/13234.

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Schippa, Robert [Verfasser]. "Short-time Fourier transform restriction phenomena and applications to nonlinear dispersive equations / Robert Schippa." Bielefeld : Universitätsbibliothek Bielefeld, 2019. http://d-nb.info/1200097637/34.

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Garimella, Venkatalakshmi Gayatri. "Théorèmes de Paley-Wiener - opérateurs differentiels invariants sur les groupes de Lie nilpotents." Poitiers, 1997. http://www.theses.fr/1997POIT2277.

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Soient une fonction mesurable sur r#n, et sa transformee de fourier. Une caracterisation de la transformee de fourier de est donnee par le theoreme de paley-wiener pour r#n. Une version faible de ce theoreme dit que si la fonction est mesurable, bornee et a support compact, sa transformee de fourier se prolonge en une fonction holomorphe sur c#n, ce qui permet de conclure que = 0 si est nulle sur un ensemble dont la mesure de plancherel est strictement positive. On generalise cette version du theoreme de paley-wiener aux groupes de lie nilpotents simplement connexes. Cette propriete est conjecturee par d. Scott et a. Sitaram. On demontre cette conjecture par recurrence sur la dimension de g. Dans le chapitre ii on generalise la propriete ci-dessus aux groupes de lie completement resolubles. La demonstration, egalement par recurrence sur la dimension de g, utilise la mesure de plancherel explicite donnee par b. N. Currey. Dans le chapitre iii on etudie des operateurs differentiels sur un espace homogene nilpotent. Soit = ind#g#k#f une representation induite d'un groupe de lie nilpotent connexe et simplement connexe g, ou #f designe un caractere unitaire d'un sous-groupe connexe k = (exp t) et tel que les multiplicites des irreductibles de g apparaissant dans la decomposition de soient finies. Soit d# l'algebre des operateurs differentiels associee a. On demontre que cette algebre est isomorphe a l'algebre des fonctions polynomiales k-invariantes definies sur o# = f + t# g#*, lorsque t est un ideal de g, algebre de lie de g.
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Rodríguez, López Salvador. "Transference theory between quasi-Banach function spaces with applications to the restriction of Fourier multipliers." Doctoral thesis, Universitat de Barcelona, 2008. http://hdl.handle.net/10803/2118.

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In the early 1970 fs, R. Coifman and G. Weiss, generalizing the techniques introduced by A. Calderon, developed a method for transferring abstract convolution type operators, defined on general topological groups, and their respective bounds, to the so called gtransferred operators h, which are operators defined on general measure spaces. To be specific, if G is a topological group and R_x is a representation of G on some Banach space B and K is a convolution operator on G given by

Kf= çk(x-y) f(y) dy

with k an L^1 function, the transferred operator T is defined by letting

Tf= çk(x-y) R_xf(y) dy.

Transfer methods deal with the study of the preservation of properties of K that are still valid for T, mostly focusing on the preservation of boundedness on Lebesgue spaces Lp. These methods has been applied to several problems in Mathematical Analysis, and especially to the problem of restrict Fourier multipliers to closed subgroups. These techniques have been extended by other authors as N. Asmar, E. Berkson and A. Gillespie, among many others. It is worth noting however, that these prior developments have always been focused on inequalities for operators on Lebesgue spaces Lp.

In this thesis there are developed several transference techniques for quasi-Banach spaces more general than Lebesgue spaces Lp, as Lorentz spaces Lp, q, Orlicz-Lorentz, Lorentz-Zygmund spaces as well as for weighted Lebesgue spaces Lp(w). The most significant applications are obtained in the field of restriction of Fourier multipliers for rearrangement invariant spaces and weighted Lebesgue spaces Lp(w). Specifically, we get generalizations of the results obtained by K. De Leeuw for Fourier multipliers. There are also developed similar techniques in the context of multilinear operators of convolution type, where the basic example is the bilinear Hilbert transform, as well as for modular inequalities and inequalities arising in extrapolation
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Grünrock, Axel. "New applications of the Fourier restriction norm method to wellposedness problems for nonlinear evolution equations." [S.l.] : [s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=967445396.

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Guibourg, Denis. "Théorèmes de renouvellement pour des fonctionnelles additives associées à des chaînes de Markov fortement ergodiques." Phd thesis, Université Rennes 1, 2011. http://tel.archives-ouvertes.fr/tel-00583175.

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L'objectif de cette thèse s?inscrit dans une perspective d?extension des théorèmes de renouvellement du cas indépendant au cas de fonctionnelles additives markoviennes. Cette thèse prolonge les travaux de Yves Guivarc'h en dimension 1 et de Martine Babillot en dimension supérieure. Comme dans ces travaux, la chaîne de Markov qui génère la fonctionnelle additive est supposée fortement ergodique. Les preuves s?appuient sur la méthode spectrale de Nagaev-Guivarc'h, qui met en jeu des techniques de transformée de Fourier et de théorie de perturbation d'opérateurs. L'analyse de Fourier (Chapitre 2) s'inspire du travail de Martine Babillot, mais en remplaçant les arguments de distributions et le recours aux fonctions de Bessel modifiées par des calculs plus élémentaires. Les outils d'analyse fonctionnelle sont présentés au Chapitre 3. Dans le Chapitre 4, les théorèmes de renouvellement markoviens de M. Babillot et Y. Guivarc'h sont alors déduits des résultats des deux précédents chapitres. Dans les Chapitres 5 et 6, on applique la méthode spectrale en remplaçant la théorie usuelle de perturbation d'opérateurs par le théorème de Keller et Liverani. Cette nouvelle approche, inspirée des travaux récents de Hubert Hennion, Loïc Hervé et Françoise Pène, permet d'améliorer significativement les énoncés des théorèmes de renouvellement en termes de conditions de moment. En particulier, pour les modèles suivants - les chaînes de Markov V-géométriquement ergodiques, - les chaînes de Markov rho-mélangeantes, - les modèles itératifs lipschitziens, on démontre que les hypothèses se réduisent à des conditions de moment (presque) optimales (en comparaison avec le cas indépendant). Les applications aux modèles itératifs lipschitziens (chapitre 6) sont relatives aux fonctionnelles additives associées à une chaîne double prenant en compte les transformations lipschitziennes aléatoires sous-jacentes. Les résultats de ce chapitre sont obtenus en généralisant la définition des espaces de fonctions Lipschitz à poids introduits par Emile Le Page.
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Delage, Florian. "Théorèmes du type Ingham et fonctions orthogonales positives." Thesis, Strasbourg, 2016. http://www.theses.fr/2016STRAD031/document.

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Le travail de la thèse est constitué de deux parties indépendantes traitant toutes les deux du comportement de solutions d’équations différentielles partielles. On s’intéressera dans un premier temps aux fonctions orthogonales positives à certains espaces puis à quelques résultats de type « Ingham ». L’existence ou non de fonctions orthogonales positives à certains espaces de fonctions quasi-périodiques a d’importantes implications, en particulier pour l’étude du comportement oscillatoire des solutions d’équations de membranes vibrantes. On se propose ici de clarifier la situation d’un sous-espace défini par trois périodes et de donner des pistes de réflexion pour le cas de quatre périodes ou plus. On peut utiliser les séries de Fourier non harmoniques pour résoudre certains problèmes de contrôle en utilisant des variantes du théorème d’Ingham. On s’intéressera spécifiquement ici aux problèmes que pose la version vectorielle de ce théorème
The existence or non-existence of positive orthogonal functions for subspaces of almost periodical function has important applications in studying the oscillatory behavior of vibrations. Cazenave, Haraux and Komornik have obtained many theorems of this type. The purpose of this work is to answer an open question formulated in the 1980’s, and to completely clarify the situation for subspaces defined by three periods. We also give some results for subspaces defined by more periods than three periods. We also prove some vectorial result for Ingham type theorems
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Books on the topic "Théorèmes de restriction de Fourier"

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author, Müller Detlef 1954, ed. Fourier restriction for hypersurfaces in three dimensions and Newton polyhedra. Princeton: Princeton University Press, 2016.

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Demeter, Ciprian. Fourier Restriction, Decoupling, and Applications. Cambridge University Press, 2019.

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Fourier Restriction, Decoupling and Applications. Cambridge University Press, 2020.

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Müller, Detlef, and Isroil A. Ikromov. Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (AM-194). Princeton University Press, 2016.

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Book chapters on the topic "Théorèmes de restriction de Fourier"

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Tao, Terence. "Some Recent Progress on the Restriction Conjecture." In Fourier Analysis and Convexity, 217–43. Boston, MA: Birkhäuser Boston, 2004. http://dx.doi.org/10.1007/978-0-8176-8172-2_10.

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Carton-Lebrun, C., and H. P. Heinig. "Weighted Extensions of Restriction Theorems for the Fourier Transform." In Recent Advances in Fourier Analysis and Its Applications, 579–96. Dordrecht: Springer Netherlands, 1990. http://dx.doi.org/10.1007/978-94-009-0665-5_32.

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Buschenhenke, Stefan, Detlef Müller, and Ana Vargas. "On Fourier Restriction for Finite-Type Perturbations of the Hyperbolic Paraboloid." In Geometric Aspects of Harmonic Analysis, 193–222. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-72058-2_5.

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Lou, Hongwei. "Weighted norm inequalities for the restriction of fourier transform to S n−1." In Harmonic Analysis, 130. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/bfb0087764.

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Larin, A. A. "Theorems on Restriction of Fourier–Bessel and Multidimensional Bessel Transforms to Spherical Surfaces." In Trends in Mathematics, 159–70. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-35914-0_8.

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Papadimitropoulos, Christos. "Salem Sets in the p-adics, the Fourier Restriction Phenomenon and Optimal Extension of the Hausdorff-Young Inequality." In Vector Measures, Integration and Related Topics, 327–38. Basel: Birkhäuser Basel, 2009. http://dx.doi.org/10.1007/978-3-0346-0211-2_30.

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"Linear Restriction Theory." In Fourier Restriction, Decoupling, and Applications, 1–25. Cambridge University Press, 2020. http://dx.doi.org/10.1017/9781108584401.003.

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"Bilinear Restriction Theory." In Fourier Restriction, Decoupling, and Applications, 35–78. Cambridge University Press, 2020. http://dx.doi.org/10.1017/9781108584401.005.

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"Multilinear Kakeya and Restriction Inequalities." In Fourier Restriction, Decoupling, and Applications, 105–29. Cambridge University Press, 2020. http://dx.doi.org/10.1017/9781108584401.008.

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"Wave Packets." In Fourier Restriction, Decoupling, and Applications, 26–34. Cambridge University Press, 2020. http://dx.doi.org/10.1017/9781108584401.004.

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Conference papers on the topic "Théorèmes de restriction de Fourier"

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Dmitriyev, Nickolay I. "The Phase Retrieval Method Using Optoelectronic Fourier Processor." In Spatial Light Modulators and Applications. Washington, D.C.: Optica Publishing Group, 1990. http://dx.doi.org/10.1364/slma.1990.tuc8.

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Recent increasing interest to the phase problem is due to adaptive optic systems design and research in the field of transparent objects computer tomography.The restriction,imposed on the phase of analysed field E(x,y)= = | K(x,y) | exp [i<£(x,y)] in the phase contrast devices, is widely known:<£(x,y) must be small,so that <£(x,y)« re .As for the iterative procedures of Gershberg -Saxton1 and Soathwell,2 phase distribution retrieval according to autocor- lation function 9 etc, ’s these algorithms are likely to be considered as a proof of existence of the phase problem solution rather then a methods which can be readily used in practice.
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Walther, A. "Quality of the Fourier transform produced by an imaging lens." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1989. http://dx.doi.org/10.1364/oam.1989.we3.

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Many users of coherent imaging techniques would be delighted with a lens that produces not only a high quality Fourier transform but also a high resolution image of a specified object plane. Unfortunately there is a theorem in geometrical optics that prohibits the design of lens systems that can perform both these tasks flawlessly. To shed some light on the nature of this limitation several proofs of this theorem are presented. One of these proofs, based on a straightforward application of the wave equation without any reference to geometrical optics, shows clearly that this design restriction has a firm foundation in the basic laws of physics.
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3

Haque, Tariqul, and Robert A. Meyer. "Recovery of images embedded in periodic interference by a Fourier transform phase." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1986. http://dx.doi.org/10.1364/oam.1986.tue6.

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If an unwanted image is superimposed on an original image, the Fourier transform phase of the degraded image highly depends on the Fourier transform magnitude of the unwanted image. The signals which produce negligible magnitudes do not significantly deviate the phase of the original image on superimposition. The degraded phase, which is approximately the original phase, can then be employed to reconstruct the original image. The class of signals which satisfy the above restriction on the magnitudes are generally periodic.
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4

Shen, Lixin, and Yunlong Sheng. "Distortion invariant pattern recognition using the orthogonal Fourier–Mellin moments." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/oam.1992.ws2.

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Scale and rotation by invariant pattern recognition is achieved using the orthogonal Fourier–Mellin moments, Q n (r)exp(jmθ), where Q n (r) are the polynomials on r and are generated by orthogonalizing the set of powers {r0,r1,…, r n }. The Q n (r) are in fact the modified Zernike radial polynomials without the restrictions that the radial moment orders must be even and greater than the circular moment orders |m|. The orthogonal Fourier–Mellin moments may be also calculated in the Cartesian coordinate system as the modified complex moments without the restriction that the moments order p and q must be positive integers. The Zernike moments have been shown to have the best overall performance among the various image moments. The performance of the new moments for pattern recognition is compared with that of the Zernike moments in terms of image representation, class separability, and noise sensitivity as functions of the number of the total moment features and of the radial orders of the moments. Experimental results show that for a given class of objects the orthogonal Fourier–Mellin moments use features of much lower radial orders for image classification, so it is then less sensitive to noise.
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Aragão, Dunfrey P., Davi H. dos Santos, Adriano Mondini, Cosimo Distante, and Luiz M. G. Gonçalves. "Analysis with SARIMA and FFT Between Two Neighboring Cities Regarding the Implementation of Restrictive Lockdown Measures." In Anais Estendidos da Conference on Graphics, Patterns and Images. Sociedade Brasileira de Computação - SBC, 2022. http://dx.doi.org/10.5753/sibgrapi.est.2022.23280.

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In this article, we perform the SARIMA model to regress the curve of daily COVID-19 deaths and the impact of implementing restrictive measures like lockdown. For comparison, we adopt two neighboring Brazilian cities with similar characteristics and decompose the original curve of cases to extract the seasonal curve. Using Fast Fourier Transform, we noticed that restriction of human circulation had a direct impact on COVID-19 cases and deaths in Araraquara by identifying the frequencies that compose the seasonal curves during the disease’s transmission period, which for future work allows analysis and identification of events and actions through the approach.
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Cowart, Jim, Patrick Moore, Harrison Yosten, Leonard Hamilton, and Dianne Luning Prak. "Diesel Engine Acoustic Emission Airflow Clogging Diagnostics With Machine Learning." In ASME 2018 Internal Combustion Engine Division Fall Technical Conference. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/icef2018-9601.

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A diesel engine electrical generator set (’gen-set’) was instrumented with in-cylinder indicating sensors as well as acoustic emission microphones near the engine. Air filter clogging was emulated by progressive restriction of the engine’s inlet air flow path during which comprehensive engine and acoustic data were collected. Fast Fourier Transforms (FFTs) were analyzed on the acoustic data. Dominant FFT peaks were then applied to supervised machine learning neural network analysis with MATLAB based tools. The progressive detection of the air path clogging was audibly determined with correlation coefficients greater than 95% on test data sets for various FFT minimum intensity thresholds. Further, unsupervised machine learning Self Organizing Maps (SOMs) were produced during normal-baseline operation of the engine. Application of the degrading air flow engine sound data was then applied to the normal-baseline operation SOM. The quantization error of the degraded engine data showed clear statistical differentiation from the normal operation data map. This unsupervised SOM based approach does not know the engine degradation behavior in advance, yet shows clear promise as a method to monitor and detect changing engine operation. Companion in-cylinder combustion data additionally shows the degrading nature of the engine’s combustion with progressive airflow restriction (richer and lower density combustion).
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Gagnon, L. "Similarity properties of optical precursors." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1990. http://dx.doi.org/10.1364/oam.1990.thy6.

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Optical precursors determine how a wave that is initially well-defined builds up when it propagates in a dispersive medium. A rigorous uniform description can be accomplished by means of a modern asymptotic theory.1 In a first analysis, that is, for time-domain local precursors, these waves are described in the Fourier-integral representation in terms of one Bessel function (Sommerfeld precursor) and the Airy integral (Brillouin precursor).2 In this work we determine, by using an operatorial equivalence based on the dispersion relation and under the same local restriction, the evolution equations satisfied by the precursors. Because of their asymptotic nature, precursors appear naturally as self-similar solutions of these equations. The above operatorial equivalence further suggests a simple way of extending the problem to take into account a perturbative Kerr nonlinearity,ßI, where I is the intensity of the field and ß is a perturbation parameter. Although such effect is negligible in practice, it is instructive to see that the equations still remain tractable. In fact, similarity properties still exist and permit us to perform a simple perturbation scheme. Our results show that nonlinearity increases (decreases) the Sommerfeld precursor spreading for ß 0 (ß < 0) and yields a self-phase modulation of the Brillouin precursor.
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Orynyak, Igor, and Andrii Oryniak. "Efficient Solution for Cylindrical Shell Based on Short and Long (Enhanced Vlasov’s) Solutions on Example of Concentrated Radial Force." In ASME 2018 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/pvp2018-85032.

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There is the general feeling among the scientists that everything what could be performed by theoretical analysis for cylindrical shell was already done in last century, or at least, would require so tremendous efforts, that it will have a little practical significance in our era of domination of powerful and simple to use commercial software. Present authors partly support this point of view. Nevertheless there is one significant mission of theory which is not exhausted yet, but conversely is increasingly required for engineering community. We mean the educational one, which would provide by rather simple means the general understanding of the patterns of deformational behavior, the load transmission mechanisms, and the dimensionless combinations of physical and geometrical parameters which governs these patterns. From practical consideration it is important for avoiding of unnecessary duplicate calculations, for reasonable restriction of the geometrical computer model for long structures, for choosing the correct boundary conditions, for quick evaluation of the correctness of results obtained. The main idea of work is expansion of solution in Fourier series in circumferential direction and subsequent consideration of two simplified differential equations of 4th order (biquadratic ones) instead of one equation of 8th order. The first equation is derived in assumption that all variables change more quickly in axial direction than in circumferential one (short solution), and the second solution is based on the opposite assumption (long solution). One of the most novelties of the work consists in modification of long solution which in fact is well known Vlasov’s semi-membrane theory. Two principal distinctions are suggested: a) hypothesis of inextensibility in circumferential direction is applied only after the elimination of axial force; b) instead of hypothesis zero shear deformation the differential dependence between circumferential displacement and axial one is obtained from equilibrium equation of circumferential forces by neglecting the forth order derivative. The axial force is transmitted to shell by means of short solution which gives rise (as main variables in it) to a radial displacement, its angle of rotation, bending radial moment and radial force. The shear force is also generated by it. The latter one is equilibrated by long solution, which operates by circumferential displacement, axial one, axial force and shear force. The comparison of simplified approach consisted from short solution and enhanced Vlasov’s (long) solution with FEA results for a variety of radius to wall thickness ratio from big values and up to 20 shows a good accuracy of this approach. So, this rather simple approach can be used for solution of different problems for cylindrical shells.
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