Sharp inequalities of various metrics on the classes of functions with given comparison function

For any q > p > 0, ω > 0, d ≥ 2ω, we obtain the following sharp inequality of various metrics ‖x‖Lq(Id) ≤ ‖φ+ c‖Lq(I2ω) ‖φ+ c‖Lp(I2ω) ‖x‖Lp(Id) on the set Sφ(ω) of d-periodic functions x having zeros with given the sine-shaped 2ω-periodic comparison function φ, where c ∈ [−‖φ‖∞, ‖φ‖∞] is such that ‖x±‖Lp(Id) = ‖(φ+ c)±‖Lp(I2ω) . In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms ‖x+‖Lp(Id)/‖x−‖Lp(Id).

Sharp inequalities of various metrics on the classes of functions with given comparison function Abstract. For any q > p > 0, ω > 0, d ≥ 2ω, we obtain the following sharp inequality of various metrics on the set S ϕ (ω) of d-periodic functions x having zeros with given the sine-shaped 2ω-periodic comparison function ϕ, where c ∈ [− ϕ ∞ , ϕ ∞ ] is such that x ± Lp(I d ) = (ϕ + c) ± Lp(I2ω) .
In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms x + Lp(I d ) / x − Lp(I d ) .
por proving heorem I we require the following lemmsF Lemma 1. Under the conditions of Theorem 1 stisfying the onditions of heorem IF vet us suppose tht the sttement @PFQA is flseF ine ϕ is omprison funtion for funtion xD then o extly one of the inequlities @PFQA is flseF vet for exmple hen there is n a > 0 stiesfying st is ler tht x + a ∈ S ϕ (ω)F vet m e point of mximum of the spline ϕ + c nd let t 1 (t 2 ) e losest to the left @rightA of m zero of the splineF sn view of the seond reltion @PFSA there is n shift x(· + τ ) of the funtion x suh tht x(m + τ ) + a = ϕ(m) + cF ine ϕ + c is the omprison funtion for funtionxD then fy virtue of the inequlity a > 0D this implies the estimte tht ontrditing @PFIAF he inequlity @PFQA is provedF ine x ∈ S ϕ (ω)D the reltion @PFRA follows immeditely from @PFIA nd @PFQAF vemm I is provedF wherex is the restriction of the function x on I d , andφ is the restriction of the function ϕ + c on I 2ω . In particular, ProofF por proving @PFTA note tht y virtue of @PFQA for ny y ± ∈ (0, x ± ∞ ) there re the points 2ω] re hosen so tht |. sndeedD this immeditely follows from the theorem on the derivtive of the rerrngment @seeD for exmpleD RD the sttement IFQFPAF fy this theorem woreoverD @PFQA implies the reltion dtF henD I ± (0) = 0 nd sine the rerrngment is L p EnormE preservingD y @PFIA nd @PFRA we hve fesidesD I ± (ξ) = ∆ ± p (ξ) hnges sign on [0, ∞) t most one @from minus to plusAF hereforeD I(ξ) ≤ 0D ξ > 0F his is equivlent to @PFTAF prom @PFTA y virtue of the rrdyEvittlwoodEoly theorem @seeD for exmpleD RD theorem IFQFIA follows the inequlity @PFUAF vemm P is provedF Proof of Theorem 1F pix ny dEperiodi funtion x ∈ S ϕ (ω) nd numer c ∈ [− ϕ ∞ ϕ ∞ ] stisfying the onditions of heorem IF hen y virtue of vemm ID the inequlity @PFUA holds trueF he reltions @PFIA nd @PFUA immeditely implies the inequlity @PFPAF sts ury is oviousF heorem I is provedF 3. The inequality of various metric for the functions x ∈ L r ∞ (I 2π )F ell tht ϕ r (t)D r ∈ ND is the shift of the r th 2πEperiodi integrl with zero men vlue on period of the funtion ϕ 0 (t) = sgn sin t stisfying ϕ r (0) = 0D nd K r := ϕ r ∞ is the pvrd onstntF por λ > 0 set ϕ λ,r (t) := λ −r ϕ r (λt)F yviously the spline ϕ λ,r (t) is yviouslyD for given p, f p there is unique c ∈ [−K r , K r ] suh tht Theorem 2. Let r ∈ N; p , q > 0, q > p, f p ∈ [0, ∞]. Then for any function x ∈ f p L r ∞ (I 2π ) having zeros we have

(3.4)
hen @QFIA nd @QFRA in view of the de(nition of the lss f p L r ∞ (I 2π ) implies tht

(3.8)
gomining @QFRA nd @QFVAD nd lso tking into ount the ovious equlity his estimtes implies the inequlity @QFPA y virtue of @QFQAF sts ury is oviousF heorem P is provedF vet e the est oneEsided pproximtions y onstnts of the funtion f in L p (I 2π )F Corollary 1. Let r ∈ N; p , q > 0, q > p, α = r+1/q r+1/p , and let the numberc ∈ [0, K r ] implements the upper bound Then, for any function x ∈ L r ∞ (I 2π ), having zeros, the inequality holds true and, for any function x ∈ L r ∞ (I 2π ), we have Both inequalities are sharp on the respective classes and turn into equalities for the splines ϕ r (t) +c and ϕ r respectively. RemarkF heorem P nd gorollry I re proved in IF 4. The inequality of various metric for trigonometric polynomials. vet us rell tht T n is the spe of ll trigonometri polynomils of degree t most nF por For any trigonometric polynomial T ∈ f p T n with minimal period 2π/m, having zeros, the inequality holds true, where c ∈ [−1, 1] is the constant satisfying The inequality (4.1) is sharp for m = 1 in the sense Proof. pix polynomil T ∈ f p T n with miniml period 2π m hving zerosF et ϕ(t) := sin ntD ψ(t) := ϕ(t) + cD t ∈ RF sn view of homogeneity of the inequlity @RFIA we n ssume tht T Lp(I 2π/m ) = ψ Lp(I 2π/n ) . how tht vet us suppose tht the inequlity @RFSA is flseF henD there is γ ∈ (0, 1) suh tht γT ± ∞ ≤ ψ ± ∞ D moreoverD one of these inequlitiues turn into equlityF vetD for exmpleD henD the polynomil ψ is the omprison funtion for the polynomil γT @see the proof of the theorem VFIFI in TAF vet m e minimum point of the funtion ψ nd let t 1 (t 2 ) e losest to the left @rightA of m zero of the funtion ψF ssingD if neessryD to the shift of the polynomil γT D we n ssume tht ine the polynomil ψ is the omprison funtion for the polynomil γT D then his implies the estimte whih ontrdits @RFRAF he inequlity @RFSA is provedF st follows from the inequlity @RFSA nd the proof of the theorem VFIFI in T tht ϕ(t) = sin nt is the omprison funtion for the polynomil T F o T ∈ S ϕ ( π n )F hereforeD y virtue of @RFRAD the polynomil T stis(es the onditions of heorem ID nd heneD the onditions of vemms I nd PF eording to the inequlity @PFUA of vemm PD we hve T Lq(I 2π/m ) ≤ ϕ + c Lq(I 2π/n ) .
Then, for any trigonomrtic polynomial T ∈ T n with minimal period 2π/m, having zeros, the inequality sin(·) +c q sin(·) +c p · T p holds, and for any trigonomrtsc polynomial T ∈ T n with minimal period 2π/m, we have Both inequalities are sharp for m = 1 in the sense where T 0 n is the set of polynomials T ∈ T n having zeros, and RemarkF heorem Q nd gorollry P for m = 1 re proved in IF 5. The inequality of various metric for periodic polynomial splines. vet r, n ∈ NF ell tht S n,r stnds for the spe of polynomil splines of order r hving defet I with knots t the points kπ/n, k ∈ ZF st is ler tht S n,r ⊂ L r ∞ (R)F por p > 0D f p ∈ [0, ∞] set f p S n,r := s ∈ S n,r : Theorem 4. Let n, m ∈ N, m ≤ n; p , q > 0, q > p, f p ∈ [0, ∞]. For a spline s ∈ f p S n,r with minimal period 2π/m, having zeros, the inequality holds true, where c ∈ [−K r , K r ] is the constant satisfying ϕ n,r + n −r c ∈ f p S n,r .
The inequality (5.1) is sharp for m = 1 in the sense sup n∈N sup s∈fp Sn,r s q n 1/p−1/q s p = ϕ r + c q ϕ r + c p .
henD y virtue of @SFPA nd the de(nition of the lss f p S n,r D we hve vet us suppose tht the inequlity @SFSA is flseF henD there is γ ∈ (0, 1) suh tht γs ± ∞ ≤ ψ ± ∞ D moreoverD one of these inequlities turn into equlityF vetD for exmpleD henD E 0 (γs) ∞ ≤ E 0 (ψ) ∞ = ϕ n,r ∞ ndD y the ikhomirov inequlity UD where E 0 (x) ∞ is the est uniform pproximtion of the funtion x y onstntsF husD the spline γs stis(es the oditions of the uolmogorov omprison theorem SF fy this theorem the spline ϕ is the omprison funtion for the spline γsF vet m e minimum point of the funtion ψ nd let t 1 (t 2 ) e losest to the left @rightA of m zero of the funtion ψF ssingD if neessryD to the shift of the spline γsD we n ssume tht γs − ∞ = − γs(m).
husD the spline s stis(es the oditions of the uolmogorov omprison theorem SF fy this theorem the spline ϕ is the omprison funtion for the spline sF o s ∈ S ϕ ( π n )F hereforeD y virtue of @SFRAD the spline s stis(es the onditions of heorem ID nd heneD the onditions of vemms I nd PF eording to the inequlity @PFUA of vemm PD we hve s Lq(I 2π/m ) ≤ ϕ n,r + n −r c Lq(I 2π/n ) . prom this inequlityD due to the 2π/mEperiodi of the spline s nd 2π/nEperiodi of the spline ϕ n,r D we otin s q ≤ m n 1/q ϕ n,r + n −r c q . prom @SFTA nd @SFUA follows the inequlity @SFIAF he shrpness of @SFIA is oviousF heorem R is provedF Corollary 3. Let n, m ∈ N, m ≤ n; q , p > 0, q > p, and let the numberc ∈ [0, K r ] implements the upper bound sup c∈[0,Kr] ϕ r + c q ϕ r + c p .
Then, for any spines s ∈ S n,r with minimal period 2π/m, having zeros, the inequality s q ≤ n m 1 p − 1 q ϕ r +c q ϕ r +c p · s p holds, and for any splines s ∈ S n,r with minimal period 2π/m, we have E ± 0 (s) q ≤ n m 1 p − 1 q ϕ r + K r q ϕ r + K r p · E ± 0 (s) p .
Both inequalities are sharp for m = 1 in the sense sup n∈N sup s∈ S 0 n,r s q n 1/p−1/q s p = ϕ r +c q ϕ r +c p , where S 0 n,r is the set of splines s ∈ S n,r having zeros, and sup n∈N sup s∈ Sn,r E ± 0 (s) q n 1/p−1/q E ± 0 (s) p = sin(·) + K r q sin(·) + K r p .
RemarkF heorem R nd gorollry Q for m = 1 re proved in IF