Necessary Conditions for the Uniform Convergence and Abel-Summability of Eigenfunction Expansions with Irregular Ordinary Differential Bundles

1986 ◽  
Vol 5 (6) ◽  
pp. 543-553 ◽  
Author(s):  
Gerhard Freiling
Keyword(s):  
Uniform Convergence ◽  
Necessary Conditions ◽  
Eigenfunction Expansions ◽  
Abel Summability

scholarly journals A Necessary and Sufficient Condition for the Equivalence of the Topologies of Uniform and Compact Convergence

1979 ◽  
Vol 22 (4) ◽  
pp. 467-470 ◽  
Author(s):  
Michael H. Clapp ◽  
Ray C. Shiflett
Keyword(s):  
Uniform Convergence ◽  
Metric Spaces ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Introductory Courses ◽  
Necessary And Sufficient ◽  
Compact Convergence

Let (X, p) and (Y, d) be metric spaces with at least two points. It is usual for introductory courses in topology to study the set Yx of all functions mapping X to Y with the pointwise, compact-open, uniform convergence, and uniform convergence on compacta topologies. Some care is taken to show sufficient conditions for these topologies to be equivalent [1, 2]. However, the question of necessary conditions are dismissed with examples showing that the topologies are not in general equivalent.

On the behaviour of eigenfunction expansions in the complex domain

1986 ◽  
Vol 104 (1-2) ◽  
pp. 73-91 ◽  
Author(s):  
Gerhard Freiling
Keyword(s):  
Boundary Value Problems ◽  
Pointwise Convergence ◽  
Necessary Conditions ◽  
Boundary Value ◽  
Complex Domain ◽  
Asymptotic Estimates ◽  
Eigenvalues And Eigenfunctions ◽  
Eigenfunction Expansions ◽  
Irregular Boundary

SynopsisBy using asymptotic estimates for the eigenvalues and eigenfunctions or irregular boundary value problems, we state necessary conditions for the pointwise convergence and for the divergence of the corresponding eigenfunction expansions.

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