Riesz sequence - Alchetron, The Free Social Encyclopedia

In mathematics, a sequence of vectors (x_n) in a Hilbert space ( H , ⟨ ⋅ , ⋅ ⟩ ) is called a Riesz sequence if there exist constants 0 < c ≤ C < + ∞ such that

c ( ∑ n | a n | 2 ) ≤ ∥ ∑ n a n x n ∥ 2 ≤ C ( ∑ n | a n | 2 )

for all sequences of scalars (a_n) in the ℓ^p space ℓ². A Riesz sequence is called a Riesz basis if

s p a n ( x n ) ¯ = H .

Theorems

If H is a finite-dimensional space, then every basis of H is a Riesz basis.

Let φ be in the L^p space L²(R), let

φ n ( x ) = φ ( x − n )

and let φ ^ denote the Fourier transform of φ. Define constants c and C with 0 < c ≤ C < + ∞ . Then the following are equivalent:

1. ∀ ( a n ) ∈ ℓ 2 , c ( ∑ n | a n | 2 ) ≤ ∥ ∑ n a n φ n ∥ 2 ≤ C ( ∑ n | a n | 2 ) 2. c ≤ ∑ n | φ ^ ( ω + 2 π n ) | 2 ≤ C

The first of the above conditions is the definition for (φ_n) to form a Riesz basis for the space it spans.

References

Riesz sequence Wikipedia

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