## On Resolvent Estimates For Abel Integral Operators And The Regularization Of Associated First Kind Integral Equations

Venue: | J. INT. EQS. APPL |

Citations: | 6 - 1 self |

### BibTeX

@ARTICLE{Plato_onresolvent,

author = {R. Plato},

title = {On Resolvent Estimates For Abel Integral Operators And The Regularization Of Associated First Kind Integral Equations},

journal = {J. INT. EQS. APPL},

year = {},

volume = {9},

pages = {253--278}

}

### OpenURL

### Abstract

In this paper resolvent estimates for Abel integral operators are provided. These estimates are applied to deduce regularizing properties of Lavrentiev's m-times iterated method as well as iterative schemes (with the discrepancy principle as corresponding parameter choice or stopping rule, respectively) for solving the corresponding Abel integral equations of the first kind.

### Citations

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(Show Context)
Citation Context ...��=2 \Gamma for 0 ! ff ! 1 \Delta . Similar resolvent estimates for V ff 2 i for the speci��c case fi = 1 and X = L 2 ([0; 1]), and in terms of the numerical range j can be found in Gohberg an=-=d Krein [5]-=-, Chapter V.6, and also in Gohberg and Krein [6], Appendix, Section 6. 2. A dioeerent proof of M 0 (V ff 2 )s2 i for 0 ! ff ! 1; fi = 1 and X = L p ([0; a]); 1sps1 j is given in GorenAEo and Yamamoto ... |

94 |
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Citation Context ...n approach for numerically solving linear ill-posed problems in Hilbert spaces is to regularize the normal equations, see the recent monographs and surveys by Engl [2], Groetsch [9], Hanke and Hansen =-=[11]-=-, Louis [13] and Murio [15] and their bibliographies. The approach in [18] \Gamma and in [19] \Delta is to avoid the normalization process, and also other than L 2 -spaces are admitted, and the main t... |

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(Show Context)
Citation Context ....2) is valid for M ` \Psi : If the linear operator B is sectorial with angle ` 0 and moreover has a dense domain, then B is of type \Gamma �� \Gamma ` 0 ; M 0 (B) \Delta \Gamma in the sense of Tan=-=abe [24], De-=-��nition 2.3.1 \Delta . As mentioned above, weakly sectorial operators are sectorial with some angle \Gamma this is e.g. Lemma 6.4.1 in Fattorini [3] \Delta : Lemma 2.3. Let the linear operator B ... |

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(Show Context)
Citation Context ...rial with angle ��=2, and one has M 0 (V j )s2; j = 1; 2. Remark. If X = L 2 \Gamma [0; a]; �� fi \Gamma1 d�� \Delta , then in fact one has M 0 (V 1 ) = M 0 (V 2 ) = 1; a reasoning is give=-=n in Halmos [10]-=-, Solution 150 \Gamma for the case a = 1; fi = 1 and for V 2 ; the general case follows similarly \Delta . Proof of Proposition 2.8. Throughout the proof, i=j in X means equality a.e. We again give th... |

79 |
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(Show Context)
Citation Context ...or numerically solving linear ill-posed problems in Hilbert spaces is to regularize the normal equations, see the recent monographs and surveys by Engl [2], Groetsch [9], Hanke and Hansen [11], Louis =-=[13]-=- and Murio [15] and their bibliographies. The approach in [18] \Gamma and in [19] \Delta is to avoid the normalization process, and also other than L 2 -spaces are admitted, and the main tool are reso... |

63 |
Convergence of iterations for linear equations
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(Show Context)
Citation Context ...Delta , and H r in (4.4) then takes the form H r = (I \Gamma ��A) r : (4.5) obviously is valid, and (4.6) with p 0 = 1 as well as (4.7) are easy consequences of Theorems 4.5.4 and 4.9.3 in Nevanli=-=nna [16] \Gamma for -=-�� ? 0 small enough \Delta . 4.1.3. An implicit iteration method. For strictly sectorial operators A 2 L(X) and for �� ? 0 we consider the implicit iteration method (I + ��A)u ffi r+1 = u ... |

37 |
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(Show Context)
Citation Context ...ods 4.1. A class of methods. A common approach for numerically solving linear ill-posed problems in Hilbert spaces is to regularize the normal equations, see the recent monographs and surveys by Engl =-=[2]-=-, Groetsch [9], Hanke and Hansen [11], Louis [13] and Murio [15] and their bibliographies. The approach in [18] \Gamma and in [19] \Delta is to avoid the normalization process, and also other than L 2... |

31 |
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- 1993
(Show Context)
Citation Context ...solving linear ill-posed problems in Hilbert spaces is to regularize the normal equations, see the recent monographs and surveys by Engl [2], Groetsch [9], Hanke and Hansen [11], Louis [13] and Murio =-=[15]-=- and their bibliographies. The approach in [18] \Gamma and in [19] \Delta is to avoid the normalization process, and also other than L 2 -spaces are admitted, and the main tool are resolvent condition... |

24 |
The Cauchy problem
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(Show Context)
Citation Context ... M 0 (B) \Delta \Gamma in the sense of Tanabe [24], De��nition 2.3.1 \Delta . As mentioned above, weakly sectorial operators are sectorial with some angle \Gamma this is e.g. Lemma 6.4.1 in Fattor=-=ini [3]-=- \Delta : Lemma 2.3. Let the linear operator B : X oe D(B) ! X be weakly sectorial. Then B is sectorial with angle ` 0 := arcsin(1=M 0 (B)). For the proof of Lemma 2.3 we refer to [3]. In the proofs o... |

9 |
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(Show Context)
Citation Context ...ented. 1.2. The Radon transform for radially symmetric functions. In this subsection we present an application where an Abel integral equation arises. This example is taken from GorenAEo and Vessella =-=[7]. Th-=-e two-dimensional Radon transform R maps a function / : R 2 ! R into the set of integrals of / along the lines L `;s ; ` 2 [0; 2��]; ss0, i.e., (R/)(`; s) := Z L `;s /(x) dx = Z 1 \Gamma1 / \Gamma... |

9 |
The discrepancy principle for a class of regularization methods
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(Show Context)
Citation Context ...depend also on b, and R(A) denotes the closure of R(A). Remarks. 1. Theorem 4.1 generalizes results obtained for Hilbert spaces X and symmetric, positive semide��nite operators A 2 L(X), see Vaini=-=kko [25]-=-. Note that Theorem 4.1 is even important for the numerical solution of nonsymmetric equations Au = fsin Hilbert spaces X since methods of type (4.3) avoid the normalization A ? Au = A ? fs, where A ?... |

6 |
On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations
- Plato
- 1996
(Show Context)
Citation Context ...aces is to regularize the normal equations, see the recent monographs and surveys by Engl [2], Groetsch [9], Hanke and Hansen [11], Louis [13] and Murio [15] and their bibliographies. The approach in =-=[18]-=- \Gamma and in [19] \Delta is to avoid the normalization process, and also other than L 2 -spaces are admitted, and the main tool are resolvent conditions. In the next two subsections we review some b... |

3 |
Similarity of Volterra operators and related problems in the theory of differential equations of fractional order
- Malamud
- 1994
(Show Context)
Citation Context ...eans that V ff 1 and V ff 2 are accretive; cf. also Gerlach and v.Wolfersdorf [4] and the literature cited therein \Gamma where V 1=2 2 for fi = 1 and a = 1 is considered \Delta . 4. Recently Malamud =-=[14]-=- has shown that certain analytic perturbations of the kernel associated with V ff 2 lead to integral operators ~ V ff 2 that are similar to V ff 2 and which are therefore also strictly sectorial. 4. R... |

3 |
Iterative and parametric methods for linear ill-posed equations
- Plato
- 1995
(Show Context)
Citation Context ...0+ kt(tI + V j ) \Gamma1 k 1 = 2; j = 1; 2: (2.4) Proof. We present the proof for V 1 only, since the same technique applies to prove the assertion for V 2 \Gamma this proof in fact is carried out in =-=[19] \Delta . V 1 ob-=-viously is well-de��ned and in L(X), with kV 1 k 1 = a fi =fi. Moreover V 1 is inverse to the \Gamma unbounded \Delta operator B : X oe D(B) ! X de��ned by (Bf)(��) := \Gamma�� \Gamma(... |

2 |
Inverses of generators
- deLaubenfels
- 1988
(Show Context)
Citation Context ...lications. In this paper we provide resolvent estimates for Abel integral operators (1.1) and (1.2) \Gamma operating from X into X for the spaces X = L p \Gamma [0; a]; �� fi \Gamma1 d�� \Delt=-=a ; p 2 [1; 1], an-=-d X = C[0; a], respectively \Delta , i.e., we provide norm estimates of (I +A) \Gamma1 for speci��cs2 C , with I denoting the identity operator in the underlying space X. As a preparation, in Sect... |

2 |
On approximate computation of the values of the normal derivative of solutions to linear partial differential equations of second order with application to Abel's integral equation
- Gerlach, Wolfersdorf
- 1986
(Show Context)
Citation Context ...r X = L 2 \Gamma [0; a]; �� fi \Gamma1 d�� \Delta and 0 ! ffs1 we have in fact M 0 (V ff j ) = 1; j = 1; 2. This in fact means that V ff 1 and V ff 2 are accretive; cf. also Gerlach and v.Wolf=-=ersdorf [4]-=- and the literature cited therein \Gamma where V 1=2 2 for fi = 1 and a = 1 is considered \Delta . 4. Recently Malamud [14] has shown that certain analytic perturbations of the kernel associated with ... |

2 |
Introduction to the Theory and
- Gohberg, Krein
- 1970
(Show Context)
Citation Context ...lvent estimates for V ff 2 i for the speci��c case fi = 1 and X = L 2 ([0; 1]), and in terms of the numerical range j can be found in Gohberg and Krein [5], Chapter V.6, and also in Gohberg and Kr=-=ein [6]-=-, Appendix, Section 6. 2. A dioeerent proof of M 0 (V ff 2 )s2 i for 0 ! ff ! 1; fi = 1 and X = L p ([0; a]); 1sps1 j is given in GorenAEo and Yamamoto [8]. 3. It follows immediately from the remark t... |

1 |
On regularized inversion of Abel integral equations
- GorenAEo, Yamamoto
- 1995
(Show Context)
Citation Context ..., Chapter V.6, and also in Gohberg and Krein [6], Appendix, Section 6. 2. A dioeerent proof of M 0 (V ff 2 )s2 i for 0 ! ff ! 1; fi = 1 and X = L p ([0; a]); 1sps1 j is given in GorenAEo and Yamamoto =-=[8]. 3. It -=-follows immediately from the remark that follows Proposition 2.8 and from (3.2) that for X = L 2 \Gamma [0; a]; �� fi \Gamma1 d�� \Delta and 0 ! ffs1 we have in fact M 0 (V ff j ) = 1; j = 1; ... |

1 |
Semigroups and Applications to Partial Dioeerential Operators
- Pazy
- 1983
(Show Context)
Citation Context ... H r in (4.4) one has H r = (I + ��A) \Gammar : The commutativity condition (4.5) is satis��ed trivially, and (4.6) with p 0 = 1 can be derived by standard results in semigroup theory, see e.g=-=., Pazy [17]-=-, Theorem 1.7.7, for the case p = 0, and see [17], Theorem 2.5.5, for the case p = 1; the general case p ? 0 follows quite similar as for the case p = 1, see [18] or [19] for the details. Finally, (4.... |

1 |
On functions of a positive operator
- Pustyl'nik
- 1982
(Show Context)
Citation Context ...n underlying weakly sectorial operator we can consider Lavrentiev 's method \Gamma cf. Section 4 \Delta . Weakly sectorial operators coincide with `weakly positive' operators introduced in Pustyl'nik =-=[20]. Our no-=-tation is justi��ed by the fact that weakly sectorial operators A ful��l a resolvent condition over a (small) sector, cf. Lemma 2.3. First, however, we introduce the sector \Sigma ` ae C , \Si... |

1 |
Pseudospectra of the convection-dioeusion operator
- Reddy, Trefethen
- 1994
(Show Context)
Citation Context ...= \Phis2 ae(A) : k(I \Gamma A) \Gamma1 ks" \Gamma1 \Psi [ oe(A); " ? 0; where oe(A) := C nae(A) denotes the spectrum of A. "-pseudospectra are considered in various papers, see e.g., Re=-=ddy, Trefethen [21] for very rec-=-ent results on that topic; the "-pseudospectra for \GammaV 1 are illustrated in Figure 2. 2.2.2. The case X = L p \Gamma [0; a]; �� fi \Gamma1 d�� \Delta . For real fi ? 0 and 1sp ! 1 let... |

1 |
and V u Qu#c Ph#ng. Regularization of ill-posed problems involving unbounded operators in Banach spaces
- Schock
- 1991
(Show Context)
Citation Context ...2.3.3 and its proof for the latter estimate. Lavrentiev's iterated method \Gamma for unbounded weakly sectorial operators A in Banach spaces, in general \Delta is considered e.g., in Schock and Ph#ng =-=[23] whe-=-re certain a priori parameter choices are provided. 4.1.2. The Richardson iteration. For strictly sectorial operators A 2 L(X), a ��rst iteration method of type (4.3) is the Richardson iteration w... |