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## Stability estimates with a priori bound for the inverse local Radon transform (2014)

### Citations

2933 |
The analysis of linear partial differential operators. III
- Hörmander
- 1985
(Show Context)
Citation Context ...rder to improve our continuity estimate for analytic weights we need to choose a test function out of a particular sequence. The simple construction of this sequence can be found in Hörmander’s book =-=[16]-=-, or in Rodino’s book [25] (Proposition 1.4.10). Proposition 5.1. For any given neighborhood V of x0 ∈ Rn we can find a sequence ϕN ∈ C∞0 (V ) such that 0 ≤ ϕN and |∂αϕN | ≤ C |α|+1N |α|, |α| ≤ N, (5.... |

598 |
Problèmes aux limites non homogènes et applications
- Lions, Magenes
- 1968
(Show Context)
Citation Context ... for all j. Proposition 5.8. Suppose that χ : Ω → Λ is a Gσ-map and f ∈ Gσ(Λ), then f ◦ χ ∈ Gσ(Ω). The proposition is proved for analytic maps χ in [25], but the more general proposition is proved in =-=[20]-=-. 5.3 Legendre polynomials Recall that the Legendre polynomials {Pn}∞n=0 forms a complete orthogonal system over L2([−1, 1]). There are many equivalent definitions, for example: Pn(x) = 1 2n [n/2]∑ k=... |

53 |
An inversion formula for the attenuated X-ray transformation
- Novikov
(Show Context)
Citation Context ...= ξx + η}. I.e. we integrate f over the line with slope given by the ξ-parameter and intersecting the y-axis at y = η. The question of invertibility of Rm has been considered for a long time. Novikov =-=[23]-=- solved the problem when m is an attenuation, i.e. m(x, ξ, η) = exp ( − ∫ ∞ x µ(t, ξt+ η) dt ) , (1.3) 1Email: joelan@math.su.se 2Email: jabo@math.su.se 1 ar X iv :1 40 5. 39 22 v1s[ ma th. AP ]s15sM ... |

46 |
Continuous dependence on data for solutions of partial differential equations with a prescribed bound
- John
- 1960
(Show Context)
Citation Context ...xample on Hölder versus logarithmic continuity at the end of section 1.1) will hold without an a priori bound on the data, even for constant weights. This is also the case for analytic continuation, =-=[17]-=-. Neither will even weaker Sobolev estimates be valid in the local problem, such as ‖f‖Hs(R2) . ‖R[f ]‖H0,t(T×R), t > s. The same thing is true for the so-called exterior problem. Recently, Caro, Ferr... |

46 |
Linear partial differential operators in Gevrey spaces (World Scientific
- Rodino
- 1993
(Show Context)
Citation Context ...uity estimate for analytic weights we need to choose a test function out of a particular sequence. The simple construction of this sequence can be found in Hörmander’s book [16], or in Rodino’s book =-=[25]-=- (Proposition 1.4.10). Proposition 5.1. For any given neighborhood V of x0 ∈ Rn we can find a sequence ϕN ∈ C∞0 (V ) such that 0 ≤ ϕN and |∂αϕN | ≤ C |α|+1N |α|, |α| ≤ N, (5.1) where C > 0 is a consta... |

46 |
Orthogonal Functions
- Sansone
- 1991
(Show Context)
Citation Context ...] to n) in the sum and applying the binomial theorem. There are many results on convergence and the magnitude of an depending on the regularity of f . Good references are Sansone’s and Alexits’ books =-=[27, 1]-=-. Recall that the modulus of continuity of a function f on Rn is defined by the quantity ω(f ; r) = sup ‖x−y‖≤r |f(x)− f(y)|. In particular, f ∈ C0,α(R) implies that there are constant 0 < α ≤ 1, C0 >... |

32 |
Inversion of the attenuated radon transform
- Natterer
(Show Context)
Citation Context ...ted µ ∈ C0,α(R2). Independently, work by Arbuzov, Bukhgeim and Kazantsev could also be seen to imply an inversion formula [2]. After this was presented many related results and improvements followed, =-=[22, 10, 3, 4, 12, 18, 13, 14, 29]-=-. Strichartz, in [28], showed that the Radon transform is locally injective, in the sense that if supp f ⊂ {(x, y); y ≥ x2} and the line integrals as defined in (1.2) are zero in a (ξ, η)-neighborhood... |

22 |
Two-dimensional tomography problems and the theory of A-analytic functions”, pp. 6–20 in Algebra, geometri, analiz i matematiqeska fizika: 10- sib. xk
- Arbuzov, Bukhgeı̆m, et al.
- 1996
(Show Context)
Citation Context ...e 1 ar X iv :1 40 5. 39 22 v1s[ ma th. AP ]s15sM ays20 14 for compactly supported µ ∈ C0,α(R2). Independently, work by Arbuzov, Bukhgeim and Kazantsev could also be seen to imply an inversion formula =-=[2]-=-. After this was presented many related results and improvements followed, [22, 10, 3, 4, 12, 18, 13, 14, 29]. Strichartz, in [28], showed that the Radon transform is locally injective, in the sense t... |

22 | V 2003 The attenuated x-ray transform: recent developments Inside Out
- Finch
(Show Context)
Citation Context ...ted µ ∈ C0,α(R2). Independently, work by Arbuzov, Bukhgeim and Kazantsev could also be seen to imply an inversion formula [2]. After this was presented many related results and improvements followed, =-=[22, 10, 3, 4, 12, 18, 13, 14, 29]-=-. Strichartz, in [28], showed that the Radon transform is locally injective, in the sense that if supp f ⊂ {(x, y); y ≥ x2} and the line integrals as defined in (1.2) are zero in a (ξ, η)-neighborhood... |

16 |
2004 On the attenuated Radon transform with full and partial measurements Inverse Problems
- Bal
(Show Context)
Citation Context ...ted µ ∈ C0,α(R2). Independently, work by Arbuzov, Bukhgeim and Kazantsev could also be seen to imply an inversion formula [2]. After this was presented many related results and improvements followed, =-=[22, 10, 3, 4, 12, 18, 13, 14, 29]-=-. Strichartz, in [28], showed that the Radon transform is locally injective, in the sense that if supp f ⊂ {(x, y); y ≥ x2} and the line integrals as defined in (1.2) are zero in a (ξ, η)-neighborhood... |

12 |
An example of nonuniqueness for a generalized Radon transform
- Boman
- 1993
(Show Context)
Citation Context ...-neighborhood of the origin, then f = 0 in some neighborhood of the origin in the (x, y)-plane. The corresponding statement unfortunately does not hold in general for the weighted Radon transform. In =-=[5]-=-, an example of a smooth m ≥ 0 is given for which Rm is not locally injective. However, in the case when m is real-analytic it is still true that the weighted Radon transform is locally injective, [9]... |

10 |
Dévissage Gevrey,” in Journées Singulières de Dijon (Univ
- Ramis
- 1978
(Show Context)
Citation Context ...ace of estimates (5.2) one sometimes use the equivalent |∂αf(x)| ≤ C |α|+1|α|σ|α|. (5.3) Also observe that we may assume f ≥ 0 which is illustrated by the following example of a Gσ0 (R)-function from =-=[24]-=-: φ(x) = exp ( − 1 ((1−t)t) 1 σ−1 ) , t ∈ (0, 1) 0 , t /∈ (0, 1). Remark 5.4. Observe that G1(Ω) is the set of all analytic functions on Ω. 28 Definition 5.5. For σ > 1 we define Gσ0 (Ω) to be the ... |

8 |
A remark on the weighted Radon transform on the plane. Inverse Probl
- Gindikin
(Show Context)
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6 |
Convergence problems of orthogonal series, Translated from German by I Folder
- Alexits
- 1961
(Show Context)
Citation Context ...] to n) in the sum and applying the binomial theorem. There are many results on convergence and the magnitude of an depending on the regularity of f . Good references are Sansone’s and Alexits’ books =-=[27, 1]-=-. Recall that the modulus of continuity of a function f on Rn is defined by the quantity ω(f ; r) = sup ‖x−y‖≤r |f(x)− f(y)|. In particular, f ∈ C0,α(R) implies that there are constant 0 < α ≤ 1, C0 >... |

5 |
Local non-injectivity for weighted radon transforms
- Boman
(Show Context)
Citation Context ...cally injective, [9]. In [6] the class of weights for which the same conclusion holds was extended to smooth weights that satisfy an additional condition, first introduced by Gindikin, [14]. See also =-=[7, 8, 21, 15]-=- for more discussions on the local injectivity problem. It is however still not known how large the subspace of smooth weights is for which local injectivity holds. In practice it is hard to verify lo... |

5 | Stability estimates for the Radon transform with restricted data and applications, preprint arXiv:1211.1887
- Caro, Ferreira, et al.
- 2012
(Show Context)
Citation Context ...en weaker Sobolev estimates be valid in the local problem, such as ‖f‖Hs(R2) . ‖R[f ]‖H0,t(T×R), t > s. The same thing is true for the so-called exterior problem. Recently, Caro, Ferreira and Ruiz in =-=[11]-=-, Theorem 2.5(c), proved an estimate of very similar type as we are about to achieve, but only for the ordinary Radon transform. Bukhgeim has also made a contribution in this direction when m is analy... |

4 |
Inversion of the scalar and vector attenuated x-ray transforms in a unit disc
- Karantsev, Bukhgeim
(Show Context)
Citation Context |

4 |
An Elementary Proof of Local Invertibility for Generalized and Attenuated Radon Transforms
- Markoe, Quinto
- 1985
(Show Context)
Citation Context ...cally injective, [9]. In [6] the class of weights for which the same conclusion holds was extended to smooth weights that satisfy an additional condition, first introduced by Gindikin, [14]. See also =-=[7, 8, 21, 15]-=- for more discussions on the local injectivity problem. It is however still not known how large the subspace of smooth weights is for which local injectivity holds. In practice it is hard to verify lo... |

4 |
inversion—variations on a theme
- Strichartz
- 1982
(Show Context)
Citation Context ...ukhgeim and Kazantsev could also be seen to imply an inversion formula [2]. After this was presented many related results and improvements followed, [22, 10, 3, 4, 12, 18, 13, 14, 29]. Strichartz, in =-=[28]-=-, showed that the Radon transform is locally injective, in the sense that if supp f ⊂ {(x, y); y ≥ x2} and the line integrals as defined in (1.2) are zero in a (ξ, η)-neighborhood of the origin, then ... |

4 |
Orthogonal Polynomials. Number v. 23
- Szegő
- 1967
(Show Context)
Citation Context ...m 4.6.6 in [1]) it then follows that ‖f(x)− pN (x)‖∞ ≤ A0ω ( f ; 2 N ) ≤ A0C0 ( 2 N )α . Thus ‖f − SN [f ]‖2 ≤ A0C0 ( 2 N )α√∫ 1 −1 dx = √ 2A0C0 ( 2 N )α . In Szegö’s book on orthogonal polynomials, =-=[30]-=- one can also find better and more explicit bounds on Pn for n > 0, e.g. (1− x2)1/4|Pn(x)| ≤ √ 2 pin . (5.5) Using (5.5) together with ‖Pn‖2 = √ 2 2n+1 (and P0(x) = 1) one can easily verify Lemma 5.11... |

3 |
and Eric Todd Quinto. Support theorems for real-analytic Radon transforms
- Boman
- 1987
(Show Context)
Citation Context ...[5], an example of a smooth m ≥ 0 is given for which Rm is not locally injective. However, in the case when m is real-analytic it is still true that the weighted Radon transform is locally injective, =-=[9]-=-. In [6] the class of weights for which the same conclusion holds was extended to smooth weights that satisfy an additional condition, first introduced by Gindikin, [14]. See also [7, 8, 21, 15] for m... |

3 |
and Jan-Olov Strömberg, Novikov’s inversion formula for the attenuated Radon transform—a new approach
- Boman
(Show Context)
Citation Context |

2 |
Local injectivity for weighted radon transforms
- Boman
- 2012
(Show Context)
Citation Context ...cally injective, [9]. In [6] the class of weights for which the same conclusion holds was extended to smooth weights that satisfy an additional condition, first introduced by Gindikin, [14]. See also =-=[7, 8, 21, 15]-=- for more discussions on the local injectivity problem. It is however still not known how large the subspace of smooth weights is for which local injectivity holds. In practice it is hard to verify lo... |

2 |
On the injectivity of the attenuated Radon transform
- Hertle
- 1984
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1 |
and Alexandru Tamasan. Inverse source problems in transport equations
- Bal
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1 | A local uniqueness theorem for weighted Radon transforms
- Boman
(Show Context)
Citation Context ...example of a smooth m ≥ 0 is given for which Rm is not locally injective. However, in the case when m is real-analytic it is still true that the weighted Radon transform is locally injective, [9]. In =-=[6]-=- the class of weights for which the same conclusion holds was extended to smooth weights that satisfy an additional condition, first introduced by Gindikin, [14]. See also [7, 8, 21, 15] for more disc... |

1 |
Generalized Fourier transforms, inverse problems, and integrability in 4 + 2
- Fokas
- 2009
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1 |
Linear operators and ill-posed problems. Consultants Bureau
- Savel′ev
- 1995
(Show Context)
Citation Context ...orem 2.5(c), proved an estimate of very similar type as we are about to achieve, but only for the ordinary Radon transform. Bukhgeim has also made a contribution in this direction when m is analytic, =-=[19]-=-. Finally we mention [26] where Rullg̊ard and Quinto presented quantitative Sobolev-type estimates with a remainder term. 1.1 Main results We will consider the local stability problem for the weighted... |

1 |
and Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Probl
- Rullg̊ard
(Show Context)
Citation Context ...timate of very similar type as we are about to achieve, but only for the ordinary Radon transform. Bukhgeim has also made a contribution in this direction when m is analytic, [19]. Finally we mention =-=[26]-=- where Rullg̊ard and Quinto presented quantitative Sobolev-type estimates with a remainder term. 1.1 Main results We will consider the local stability problem for the weighted Radon transform as defin... |

1 |
An identity for triplets of double Hilbert transforms, with applications to the attenuated Radon transform
- Strömberg, Andersson
(Show Context)
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