scholarly journals Spectral Analysis of the Differential Operator in Wavelet Bases

1995 ◽  
Vol 2 (4) ◽  
pp. 382-391 ◽  
Author(s):  
Johan Waldén
Keyword(s):  
Spectral Analysis ◽  
Differential Operator ◽  
Wavelet Bases

scholarly journals On Singular Dissipative Fourth-Order Differential Operator in Lim-4 Case

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Ekin Uğurlu ◽  
Elgiz Bairamov
Keyword(s):  
Spectral Analysis ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Boundary Value ◽  
Inverse Operator ◽  
Order Differential Operator ◽  
Associated Functions ◽  
Fourth Order Differential Operator

A singular dissipative fourth-order differential operator in lim-4 case is considered. To investigate the spectral analysis of this operator, it is passed to the inverse operator with the help of Everitt's method. Finally, using Lidskiĭ's theorem, it is proved that the system of all eigen- and associated functions of this operator (also the boundary value problem) is complete.

Riesz basis of wavelets constructed from trigonometric B-splines

2015 ◽  
Vol 13 (04) ◽  
pp. 419-436
Author(s):  
Mahendra Kumar Jena
Keyword(s):  
Differential Operator ◽  
Sobolev Space ◽  
Sobolev Spaces ◽  
Riesz Basis ◽  
Scaling Function ◽  
Wavelet Bases ◽  
B Splines ◽  
Compactly Supported ◽  
The Scaling Function

In this paper, we construct a class of compactly supported wavelets by taking trigonometric B-splines as the scaling function. The duals of these wavelets are also constructed. With the help of these duals, we show that the collection of dilations and translations of such a wavelet forms a Riesz basis of 𝕃2(ℝ). Moreover, when a particular differential operator is applied to the wavelet, it also generates a Riesz basis for a particular generalized Sobolev space. Most of the proofs are based on three assumptions which are mild generalizations of three important lemmas of Jia et al. [Compactly supported wavelet bases for Sobolev spaces, Appl. Comput. Harmon. Anal. 15 (2003) 224–241].

scholarly journals On a Class of Non-Self-Adjoint Differential Operators

1960 ◽  
Vol 12 ◽  
pp. 641-659 ◽  
Author(s):  
R. R. D. Kemp
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Spectral Analysis ◽  
Differential Operator ◽  
Spectral Measure ◽  
Differential Operators ◽  
Expansion Theorem ◽  
Completely Continuous ◽  
Normal Operators ◽  
Uniformly Bounded

The problem of spectral analysis of non-self-adjoint (and non-normal) operators has received considerable attention recently. Livšic (5), and more recently Brodskii and Livšic (1) have considered operators on Hilbert space with completely continuous imaginary parts. Dunford (3) has generalized the notion of spectral measure and defined a class of spectral operators on Hilbert and Banach space. Schwartz (8) and Rota (7) have investigated conditions under which a differential operator will be spectral. The work of Naimark (6) and the author (4) on non-self-adjoint differential operators leads to an expansion theorem which implicitly defines a type of spectral measure. However the projections involved in this will not in general be bounded, much less uniformly bounded.

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