General Duality Principle by Means of Perturbed Problems and Conjugate Functionals

1985 ◽  
pp. 512-537
Author(s):  
Eberhard Zeidler
Keyword(s):  
Duality Principle ◽  
General Duality

scholarly journals A Duality Principle for Groups II: Multi-frames Meet Super-Frames

2020 ◽  
Vol 26 (6) ◽  
Author(s):  
R. Balan ◽  
D. Dutkay ◽  
D. Han ◽  
D. Larson ◽  
F. Luef
Keyword(s):  
Group Representation ◽  
Duality Principle ◽  
Group Representations ◽  
Fundamental Identity ◽  
Gabor Analysis ◽  
Time Frequency ◽  
Fundamental Properties ◽  
Super Frame ◽  
Funct Anal ◽  
General Duality

AbstractThe duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d .

A general duality principle in elasticity

1996 ◽  
Vol 24 (2) ◽  
pp. 87-123 ◽  
Author(s):  
Luqun Ni ◽  
Sia Nemat-Nasser
Keyword(s):  
Duality Principle ◽  
General Duality

On frames, dual frames, and the duality principle

2015 ◽  
Vol 45 (1) ◽  
pp. 183-200 ◽  
Author(s):  
Diana T. Stoeva
Keyword(s):  
Duality Principle ◽  
Dual Frames

scholarly journals On the prediction error for two-parameter stationary random fields

1992 ◽  
Vol 46 (1) ◽  
pp. 167-175
Author(s):  
R. Cheng
Keyword(s):  
Random Fields ◽  
Prediction Error ◽  
Duality Relation ◽  
Stationary Random Fields ◽  
Error Formulas ◽  
Two Parameter ◽  
General Duality

A number of Szegö-type prediction error formulas are given for two-parameter stationary random fields. These give rise to an array of elementary inequalities and illustrate a general duality relation.

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