Volterra operator

Volterra operator

Compact operators

Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017

where k is a continuous function on [0, 1]2 (equivalently, one may take a continuous function on {(x, t): 0 ≤ t ≤ x ≤ 1}). This is not exactly the type of operator treated in Example 9.1.6. The Volterra operator from Exercise 9.3.4 is a Volterra type operator, with k ≡ 1. Prove that every Volterra type operator is compact, and has the same spectrum as the Volterra operator.

Dirichlet problem for parabolic systems with Dini continuous coefficients

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Journal Information

Published in Applicable Analysis, 2021

E. A. Baderko, M. F. Cherepova

Next, we use the argument in [13]. K is a Volterra operator, and, by Lemmas 4.1, 4.2 and estimate (29), K is a linear bounded operator from into . Consequently, it follows from Lemma 4.3 that, for each function , Equation (31) has a unique solution and . Finally, since , and , it follows that the solution ϕ of Equation (31) belongs to . The proof of the theorem is complete.

A new regularization approach for numerical differentiation

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Journal Information

Published in Inverse Problems in Science and Engineering, 2020

Abinash Nayak

Note that although the Volterra operator is well defined on the space of integrable functions (), we will restrict it to the space of functions, that is, we shall consider to be the searching space for the solution of (1), since it is a Hilbert space and has a nice inner product . Hence the domain of the functional G is . It is not hard to see that the Volterra operator is linear and bounded on .

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Compact operators

Dirichlet problem for parabolic systems with Dini continuous coefficients

A new regularization approach for numerical differentiation