scholarly journals Beyond Dominated Convergence: Newer methods of Integration

2017 ◽  
Vol 0079 ◽  
pp. 53-74
Author(s):  
P. Muldowney
Keyword(s):  
Dominated Convergence

scholarly journals A dominated convergence theorem for Eisenstein series

2021 ◽  
Author(s):  
Johann Franke
Keyword(s):  
Fourier Series ◽  
Dirichlet Series ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Convergence Theorem ◽  
Rational Functions ◽  
New Approach ◽  
Series Representations ◽  
Dominated Convergence ◽  
Generalized Dirichlet

AbstractBased on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

scholarly journals On Fuzzy Henstock-Kurzweil-Stieltjes-♦-Double Integral on Time scales

2021 ◽  
Vol 2 (2) ◽  
pp. 38-49
Author(s):  
David AFARIOGUN ◽  
Adesanmi MOGBADEMU ◽  
Hallowed OLAOLUWA
Keyword(s):  
Uniform Convergence ◽  
Time Scales ◽  
Convergence Theorem ◽  
Monotone Convergence ◽  
Double Integral ◽  
Monotone Convergence Theorem ◽  
Integrable Functions ◽  
Dominated Convergence

We introduce and study some properties of fuzzy Henstock-Kurzweil-Stietljes-$ \Diamond $-double integral on time scales. Also, we state and prove the uniform convergence theorem, monotone convergence theorem and dominated convergence theorem for the fuzzy Henstock-Kurzweil Stieltjes-$\Diamond$-double integrable functions on time scales.

scholarly journals Convergence theorems for Banach space valued integrable multifunctions

1987 ◽  
Vol 10 (3) ◽  
pp. 433-442 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Convergent Sequence ◽  
Mosco Convergence ◽  
Convergence Theorems ◽  
Aumann Integral ◽  
Convergence Results ◽  
Measurable Multifunctions ◽  
Dominated Convergence ◽  
Convergence Of Sets

In this work we generalize a result of Kato on the pointwise behavior of a weakly convergent sequence in the Lebesgue-Bochner spacesLXP(Ω) (1≤p≤∞). Then we use that result to prove Fatou's type lemmata and dominated convergence theorems for the Aumann integral of Banach space valued measurable multifunctions. Analogous convergence results are also proved for the sets of integrable selectors of those multifunctions. In the process of proving those convergence theorems we make some useful observations concerning the Kuratowski-Mosco convergence of sets.

scholarly journals A DOMINATED CONVERGENCE THEOREM IN THE K-H INTEGRAL

2003 ◽  
Vol 7 (3) ◽  
pp. 507-512
Author(s):  
Jitan Lu ◽  
Peng-Yee Lee
Keyword(s):  
Convergence Theorem ◽  
Dominated Convergence

scholarly journals ALMOST-PERIODIC SOLUTIONS FOR AN ECOLOGICAL MODEL WITH INFINITE DELAYS

2007 ◽  
Vol 50 (1) ◽  
pp. 229-249 ◽  
Author(s):  
Yonghui Xia ◽  
Jinde Cao
Keyword(s):  
Periodic Solution ◽  
Lyapunov Functional ◽  
Ecological Model ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Volterra Model ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Infinite Delays ◽  
Dominated Convergence

AbstractBy using Lebesgue’s dominated convergence theorem and constructing a suitable Lyapunov functional, we study the following almost-periodic Lotka–Volterra model with $M$ predators and $N$ prey of the integro-differential equations\begin{alignat*}{2} \dot{x}_i(t)\amp=x_i(t)\biggl[b_i(t)-a_{ii}(t)x_i(t)-\sum_{k=1,k\neq i}^{N}a_{ik}(t)\int_{-\infty}^tH_{ik}(t-\sigma)x_k(\sigma)\,\mathrm{d}\sigma\\ \amp\hskip45mm-\sum_{l=1}^{M}c_{il}(t)\int_{-\infty}^tK_{il}(t-\sigma)y_l(\sigma)\,\mathrm{d}\sigma\biggr],\amp\quad i\amp=1,2,\dots,N,\\ \dot{y}_j(t)\amp=y_j(t)\biggl[-r_j(t)-e_{jj}(t)y_j(t) +\sum_{k=1}^{N}d_{jk}(t)\int_{-\infty}^tP_{jk}(t-\sigma)x_k(\sigma)\,\mathrm{d}\sigma \\ \amp\hskip45mm-\sum_{l=1,l\neq j}^{M} e_{jl}(t)\int_{-\infty}^tQ_{jl}(t-\sigma)y_l(\sigma)\,\mathrm{d}\sigma\biggr],\amp\quad j\amp=1,2,\dots,M. \end{alignat*}Some sufficient conditions are obtained for the existence of a unique almost-periodic solution of this model. Several examples show that the obtained criteria are new, general and easily verifiable.

scholarly journals Fatou's Lemma and Lebesgue's convergence theorem for measures

2000 ◽  
Vol 13 (2) ◽  
pp. 137-146 ◽  
Author(s):  
Onésimo Hernández-Lerma ◽  
Jean B. Lasserre
Keyword(s):  
Asymptotic Behavior ◽  
Banach Spaces ◽  
General Result ◽  
Convergence Theorem ◽  
Convergence Theorems ◽  
Vector Lattices ◽  
Fatou’S Lemma ◽  
Dominated Convergence ◽  
Special Case ◽  
Fatou's Lemma

Analogues of Fatou's Lemma and Lebesgue's convergence theorems are established for ∫fdμn when {μn} is a sequence of measures. A “generalized” Dominated Convergence Theorem is also proved for the asymptotic behavior of ∫fndμn and the latter is shown to be a special case of a more general result established in vector lattices and related to the Dunford-Pettis property in Banach spaces.

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