On some extremal problems of different metrics for differentiable functions on the axis

Abstract

For an arbitrary fixed segment α β , [ ] ʚ R and given r ∈ N , A r , A 0 , and p > 0, we solve

Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function

“…We now prove (6). To this end, note that the proof of (5) is based on the statement that ' is a comparison function for the function x: Reasoning by analogy and using the condition L.x/ p Ä L.'/ p instead of (5), we prove that…”

“…If jx.t /j > 0 for t 2 .a; b/; then relation (14) follows from inequality (6). Therefore, we assume that x.t / has a zero t 0 2 .a; b/: Then, according to Lemma 2, we get…”

“…and on bounded subsets of the spaces T n and S n;r : Note that the solution of problems (3) and (4) on the classes r p was obtained earlier in [6].…”

“…Theorem 4 [6]. Let k; r 2 N; k < r; A 0 ; A r ; p > 0;ˆ2 W; and OE˛;ˇ R: Then Settingˆ.t / D t q=p ; q p; in the first relation of the theorem andˆ.t/ D t q ; q 1; in the second relation, we obtain sharp estimates for the norms kx .k/ k L q OE˛;ˇ ; k D 0; 1; : : : ; r 1; on the classes r p : For functions with zeros, the following statement is true:…”

Inequalities for Nonperiodic Splines on the Real Axis and Their Derivatives

“…In [12] (Corollary 1), it is proved that condition (7) for the function x 2 L r 1 yields the inequality…”