Sentence examples for minimum norm in from inspiring English sources

In this section, we study the problem of finding a minimizer of a continuously Fréchet-differentiable convex functional which has the minimum norm in Hilbert spaces.

Now, let us turn to the problem of minimizing a continuously Fréchet-differentiable convex functional with minimum norm in Hilbert spaces.

Then the sequence { x n } generated by algorithm (3.38) converges strongly to x ∗ = P Γ 2 ( 0 ) which is the minimum norm in Γ 2. Example 3.13 Let H 1 = H 2 = R with the inner product defined by 〈 x, y 〉 = x y for all x, y ∈ R and the standard norm | ⋅ |.

Then the sequence { x n } generated by algorithm (3.34) converges strongly to x ∗ = P Γ ( 0 ), which is the minimum norm in Γ. Algorithm 3.5 For fixed u ∈ H 1 and x 0 ∈ H 1 arbitrarily, let { x n } be a sequence defined by x n + 1 = P C [ α n u + ( 1 − α n ) ( x n − δ A ∗ ( I − P Q ) A x n ) ], n ∈ N, (3.35).

Then the sequence { x n } generated by algorithm (3.36) converges strongly to x ∗ = P Γ 1 ( 0 ), which is the minimum norm in Γ 1. Algorithm 3.9 For fixed u ∈ H 1 and x 0 ∈ H 1 arbitrarily, let { x n } be a sequence defined by { u n = α n u + ( 1 − α n ) ( x n − δ A ∗ ( I − S ) A x n ), x n + 1 = ( 1 − β n ) u n + β n T ( ( 1 − γ n ) u n + γ n T u n ), n ∈ N, (3.37).

It can, hence, be seen that this term acts as a minimum norm constraint in the LP criterion, in the sense that it penalizes the squared norm of the PEF impulse response coefficient vector: (24).

Finally, if we take f = 0, by a similar argument as that in Theorem 3.3, we deduce immediately that x ˜ is a minimum norm element in Γ.

In particular, if contains the origin 0, taking, the sequence converges strongly to the minimum norm element in.

In particular, if contains the origin 0, taking, then the sequence generated by (1.14) converges strongly to the minimum norm element in.

Then the sequence ({x_{n}}) generated by algorithm (3.36) converges strongly to (x^=P_{Gamma}(0)x^), which is the minimum norm element in Γ.

Then the sequence { x n } generated by (3.4) converges to a point x ∗ ∈ S which is the minimum norm element in S.