Sentence examples for modular convergence in from inspiring English sources

We recall that a sequence (u_{n}) converges to u for the modular convergence in (W^{1}L_{M}(Omega)) if there exists (lambda>0) such that int_{Omega}M biggl(frac{|D^{alpha}u_{n}-D^{alpha}u|}{lambda} biggr),dxrightarrow0 quadmbox{as } nrightarrowinfty mbox{ for all } |alpha|leq1.

The space W 0 1 E M is defined as the (norm) closure of the Schwartz space D in W1E M and the space W 0 1 L M as the σ ( ∏ L M, ∏ E M ¯ ) closure of D in W1L M. We say that u n converges to u for the modular convergence in W1L M if for some λ > 0, ∫ Ω M D α u n - D α u λ d x → 0 for all |α| ≤ 1.

Then, there exists a smooth function (v j ) such that v j ≥ ψ, v j → v for the modular convergence in W 0 1, x L M ( Q ), ∂ v i ∂ t → ∂ v ∂ t for the modular convergence in W - 1, x L M ¯ ( Q ) + L 1 ( Q ).

Let us recall that for u n ∈ W 0 1, x L M ( Q ), there exists a smooth function u nσ (see [14]) such that u n σ → u n for the modular convergence in W 0 1, x L M ( Q ), ∂ u n σ ∂ t → ∂ u n ∂ t for the modular convergence in W - 1, x L M ¯ ( Q ) + L 1 ( Q ).

(3) Estimates for such a quantity can be useful to determine the degree of accuracy in the approximation by families of linear as well as nonlinear integral operators, in various settings, such as in the case of (L^{p}) convergence (see, e.g., [10]), or in the more general case of modular convergence in Orlicz spaces; see, e.g., [6, 11 15].

About (4): Since u ≥ ψ, then T k (u) ≥ T k and there exist a smooth function v j ≥ T k such that v j → T k (u) for the modular convergence in W 0 1, x L M ( Q ). ( 4 ) = n ∫ Q T n ( ( u n - ψ ) - ) ( T k ( u n ) - T k ( v j ) μ ) ρ m ( u n ) d x d t ≤ ε ( n, j, μ ).

Since the Δ 2 -condition implies equivalence of norm and modular convergence, { g 2 n } is modular convergent to h ∈ X ρ.

If the open set Ω has the segment property, then the space (mathcal{D}(Omega)) is dense in (W^{1}_{0}L_{M}(Omega)) for the modular convergence and thus for the topology (sigma(Pi L_{M},Pi L_{bar{M}})).

We will also use another type of convergence which is situated between norm and modular convergence.

It is well known that [6, 22] under the Δ 2 -condition the norm convergence and modular convergence are equivalent.

We know, by [15, 16] that under -condition the norm convergence and modular convergence are equivalent, which implies that the norm and modular convergence are also the same when we deal with the -type condition.