The main contribution of this work is to show that any unity partition (where the sum of the partition is one) of the update law ensures convergence of the co-learning.
Denote the frequency-uniform decomposition operators □ k f = F − 1 ( ψ k F f ), k ∈ Z n, where ψ k = ψ ( ξ − k ) constitute a so-called unity partition with good properties, and ψ is a bump-like function.
Bernstein polynomials [1] have many useful properties, such as, the positivity, the continuity, and unity partition of the basis set over the interval.
Since the Bernstein polynomials have many good properties, such as the positivity continuity, recursion's relation, symmetry, unity partition of the basis set over ([a,b]), uniform approximation, differentiability and integrability, these polynomials are applied for the collocation methods [24 27].
During a speech in February 1904 in Dhaka, Curzon outlined the benefits that east Bengalis would receive from the partition, including a "unity which they have not enjoyed since the days of the old Mussalman viceroys and kings".
Let D 1, D 2, …, D N be rectangular regions with sides parallel to the coordinate planes covering G and let φ 1, φ 2, …, φ N be a corresponding partition of unity, i.e., φ j ∈ C 0 ∞ ( G ), σ j = supp φ j ⊂ D j and ∑ j = 1 N φ j ( x ) = 1, where C 0 ∞ ( G ) denotes the space of all infinitely differentiable functions on G with compact support.
Given 0 < δ < 1, let { ψ j p } 1 l be a finite partition of unity of Ω ¯ such that supp ψ j ⊆ B r j ( x j ) with x j ∈ Ω ¯ and 0 < r j ≤ δ.
For each φ ∈ C 0 ∞ , there is a subset {Ω1,..., Ω m } of λ∈Isuch that spt φ ⊂ ⋃ i = 1 m Ω i = D. Choose a partition of unity of D, {g1,..., g m }, subordinate to the covering Ω i, such that g i ∈ C 0 ∞ ( Ω i ), 0 ≤ g i ≤ 1 and ∑ i = 1 m g i ≡ 1 in D; see Lemma 2.3.1 in [11].
Herbig's proof of Theorem 7.6 also starts with the inequalities behind Theorem 3.3, though new arguments are required, including a careful analysis of a partition of unity in the phase variable.
Let { β 0, β 1, …, β n } be a continuous partition of unity on X subordinated to the covering { V 0, V 1, …, V n }.
In one variant, the key is a geometric fact that there exists a partition of unity ({J_{mathcal {C}}}_{{mathcal {C}}ne {mathcal {C}}_{min}}) indexed non-minimal partitions so that (sum _{{mathcal {C}}} J_{mathcal {C}}= {varvec{1}}) and so that on ({text {supp}}J_{mathcal {C}}cap {x,|, |x| > 1}), one has that, for some (Q>0), (|x_j-x_k| ge Q|x|) if ((jk) not subset {mathcal {C}}).
