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In general, there exists a real number R such that the series converges when −R < x < R but diverges if x < −R or x > R.
Due to the compactness of and boundness of, there exists a real number such that (3.1).
Otherwise, if, then there exists a real number such that when, which contradicts the condition that is integrable on.
If (Gsupset A), then there exists a real number (sigma>0) such that (Gsupset U sigma,A)).
where are constants, and satisfies the following hypothesis: is continuous, nondecreasing on, and nonincreasing on for each fixed there exists a real number such that for any, (1.3).
Since f is Δ-bounded there exists a real number A and (K_{2}= { tinmathbb{T}:vert f(t)vert leq A } ) with (delta_{Delta} ( K_{2} ) =1).
Then, since, there exists a real number such that (2.12).
if and only if there exists a real number with such that.
Since f is bounded, there exists a real number M such that supf(t) ≤ M.
Now we claim that there exists a real number r satisfying condition (2.2).
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Then, and are called, (5) -subweakly commuting on (see [9]) if for all, there exists a real number such that, (6 uniformly -subweakly commuting on (see [10]) if there exists a real number such that, for all and.
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