Sentence examples for basicity of the system from inspiring English sources
We will study the basicity of the system (4) in (L_{p cdot),rho} ) with respect to the parameters α and β.
The criterion of basicity of the system (1) in L p ≡ L p , 1 < p < + ∞, when λ n = n − α sign n, has been obtained earlier in [3, 4].
Under certain conditions on the parameters α and β equivalence of the basis properties (completeness, minimality, ω-linearly independence, basicity) of the system (2) in L p t are proved.
It is easy to see that Ker S = 0. Actually, let S f = 0. From the basicity of the system (3) in L p t and from (4) we obtain ( f, e i n x ) = 0, ∀ n ∈ Z. Also, from the basicity of system { e i n t } n ∈ Z in L p t it follows that f = 0. We show that for all g ∈ L p t, the equation S f = g in L p t is solved.
The dissolution ability of cellulose in the OES can also be interpreted by the hydrogen bond-accepting ability (basicity) of the system, as measured by the Kamlet-Taft parameter β [ 15, 24].
Stability of the basicity of this system in Lebesgue space with variable summability index is studied.
Under certain conditions on the weight function of the form of a power function, sufficient conditions for the basicity of this system are obtained in generalized weighted Lebesgue space.
Under certain conditions on the summability index (p (cdot )) and perturbation, the basicity of this system in Lebesgue spaces (L_{p (cdot )} (0,pi )) with variable summability index (p (cdot )) is proved.
Apparently, the study of basis properties (completeness, minimality, basicity) of these systems dates back to the well-known work of Paley and Wiener [1].
It is the aim of this paper to investigate basis properties (basicity, completeness, and minimality) of the system (1) in Lebesgue space L p t with variable summability index p ( t ), when { λ n } has the asymptotics λ n = n − α sign n + O ( | n | − β ), n → ∞, (2).
These investigations have allowed one to consider questions of basicity of some system of functions (for example, the classical system of exponents { e i n t } n ∈ Z ) in L p t.
