Sentence examples for be two lacunary sequences from inspiring English sources

Theorem 3.8 Let β = ( l i ) and θ = ( k i ) be two lacunary sequences.

If there exists δ > 0 such that | I i j | ( h i + g j ) ≥ δ for every  i, j = 1, 2, 3, …  provided  I i, j ≠ ∅, then X k → ξ ( S θ F ( Δ m ) ) if and only if X k → ξ ( S β F ( Δ m ) ). Definition 3.1 Let β = ( l i ) and θ = ( k r ) be two lacunary sequences.

Theorem 4.4 Suppose β = { ℓ r }, θ = { k r } are two lacunary sequences.

Conversely, let for the two lacunary sequences β = ( l i ) and θ = ( k i ) there exist sequences ( s r ), ( t r ) ⊆ N and δ > 0 which satisfy the above three conditions.

Definition 3.3 Let θ be a lacunary sequence; two number sequences x = ( x k ) and y = ( y k ) are strong ℐ-asymptotically lacunary equivalent of multiple L provided that { r ∈ N : 1 h r ∑ k ∈ I r | x k y k − L | ≥ ε } ∈ I. (denoted by x ∼ N θ L ( I ) y ) and strong simply ℐ-asymptotically lacunary equivalent if L = 1.

Definition 1.8 Let θ = ( k r ) be a lacunary sequence, two number sequences x = ( x k ) and y = ( y k ) are said to be strong asymptotically lacunary equivalent of multiple L provided that lim r 1 h r ∑ k ∈ I r | x k y k − L | = 0.

Definition 1.7 Let θ = ( k r ) be a lacunary sequence, two nonnegative sequences [ x ] and [ y ] are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every ε > 0 lim r 1 h r | { k ∈ I r : | x k y k − L | ≥ ε } | = 0, where the vertical bars indicate the number elements in the enclosed set.

Let θ = ( k r ) be a lacunary sequence; the two nonnegative sequences x = ( x k ) and y = ( y k ) are said to be ℐ-asymptotically lacunary statistical equivalent of multiple L provided that for every ε > 0 and δ > 0, { r ∈ N : 1 h r | { k ∈ I r : | x k y k − L | ≥ ε } | ≥ δ } ∈ I.

Let θ be a lacunary sequence; the two nonnegative sequences x = ( x k ) and y = ( y k ) are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every ϵ > 0 lim r 1 h r | { k ∈ I r : | x k y k − L | ≥ ϵ } | = 0. (denoted by x ∼ S θ L y ) and simply asymptotically lacunary statistical equivalent if L = 1.

Let θ be a lacunary sequence; the two nonnegative sequences x = ( x k ) and y = ( y k ) are said to be ℐ-asymptotically lacunary statistical equivalent of multiple L provided that for every ϵ > 0, and δ > 0, { r ∈ N : 1 h r | { k ∈ I r : | x k y k − L | ≥ ε } | ≥ δ } ∈ I. (denoted by x ∼ S θ L ( I ) y ) and simply ℐ-asymptotically lacunary statistical equivalent if L = 1.

Definition 3.2 Let θ be a lacunary sequence; the two nonnegative sequences x = ( x k ) and y = ( y k ) are said to be ℐ-asymptotically lacunary statistical equivalent of multiple L provided that for every ϵ > 0 and δ > 0, { r ∈ N : 1 h r | { k ∈ I r : | x k y k − L | ≥ ε } | ≥ δ } ∈ I. (denoted by x ∼ S θ L ( I ) y ) and simply ℐ-asymptotically lacunary statistical equivalent if L = 1.