Geometry of Banach spaces and sharp versions of Jackson and Marchaud inequalities

Transcription

1 Geometry of Banach spaces and sharp versions of Jackson and Marchaud inequalities Andriy Prymak joint work with Zeev Ditzian January 2012 Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

2 Classical Jackson estimate for approximation by trigonometric polynomials on [ π, π] = T E n (f ) p cω r (f, n 1 ) p was sharpened by M. F. Timan (1966): { n n r k sr 1 E k (f ) s p k=1 } 1 s Here ω r (f, t) p = sup r h f L p(t ), h t cω r (f, n 1 ) p, 1 < p <, s = max(p, 2). h f (x) = f (x + h) f (x), r h f (x) = h r 1 h f (x) and E k (f ) p = min f T n Lp(T ). deg T n<k Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

3 F. Dai, Z. Ditzian, S. Tikhonov (2008) proved a general result for sharp Jackson estimates using a version of the Littlewood-Paley inequality. Examples of application include approximation by: algebraic polynomials on [ 1, 1], spherical harmonic polynomials on S d 1, functions of exponential type on R d, multivariate trigonometric polynomials on T d. Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

4 For 1 < p <, s = max(p, 2), the sharp Jackson inequality { n n r k sr 1 E k (f ) s p k=1 } 1 s cω r (f, n 1 ) p is essentially equivalent to the following sharp lower estimate of ω r (f, t) p by ω r+1 (f, u) p t r { 1 t } 1 u sr 1 ω r+1 (f, u) s s p du cω r (f, t) p. The well-known immediate (but much weaker) lower bound is ω r+1 (f, t) p 2ω r (f, t) p. Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

5 In the other direction, for 1 < p <, with q = min(p, 2), the sharp converse inequality { n ω r (f, n 1 ) p cn r k qr 1 E k (f ) q p k=1 is essentially equivalent to the sharp Marchaud inequality } 1 q { ω r (f, t) p ct r 1 u qr 1 ω r+1 (f, u) q p du t } 1 q. Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

6 Note that if p = 2, then q = s = 2, and we obtain an equivalence. ω r (f, t) 2 t r { 1 t u 2r 1 ω r+1 (f, u) 2 2 du } 1 2. Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

7 Z. Ditzian (1988) proved the sharp Marchaud inequality for Banach spaces B of functions on R d or T d for which translations are continuous isometries and for some 1 < q 2 and K > 1 f + g B + f g B ( f q 2 B + K g q B ) 1 q, f, g B, and showed this condition to be equivalent to ( f + g B + f g ) B 1 ct q, t > 0, 2 sup f B =1 g B =t which means that B has modulus of smoothness of power type q in terminology of geometry of Banach spaces. For L p spaces, 1 < p <, we have q = min{p, 2}. Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

8 Joint work with Z. Ditzian (2007): sharp Marchaud and converse inequalities in Orlicz spaces for which Φ(u 1 q ) is convex for some q, 1 < q 2, where Φ(u) is the Orlicz function. The condition f + g B + f g B 2 ( f q B + K g q B ) 1 q, f, g B, was obtained for an equivalent norm. Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

9 Joint work with Z. Ditzian (2011): sharp Jackson and lower estimates of ω r (f, t) B are achieved for Banach spaces B of functions on R d or T d (or S d 1 ) for which translations (rotations) are continuous isometries and for some s, 2 s < and k > 0 max( f + g B, f g B ) ( f s B + k g s B ) 1 s, f, g B. Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

10 For q 1 + s 1 = 1, the dual space B satisfies f + g B + f g B 2 if and only if ( f q B + K g q B ) 1 q, f, g B, max( f + g B, f g B ) ( f s B + k g s B ) 1 s, f, g B. This establishes that the last condition is equivalent to ( inf 1 f + g ) B cε s, ε > 0, f B = g B =1 2 f g B =ε which means that B has modulus of convexity of power type s in terminology of geometry of Banach spaces. Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

11 In summary, certain geometric property of an equivalent norm of a Banach space implies an approximation inequality in the space. Modulus of smoothness of power type q, 1 < q 2, implies the sharp Marchaud inequality (upper estimate of ω r in terms of ω r+1 ). Modulus of convexity of power type s, 2 s <, implies the sharp Jackson inequality (lower estimate of ω r in terms of ω r+1 ). Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

12 Joint work with Z. Ditzian (2011): sharp Jackson and lower estimates of ω r (f, t) O(Φ) are achieved for Orlicz spaces for which Φ(u 1 s ) is concave for some s, 2 s <, where Φ(u) is the Orlicz function. In fact, it is sufficient to require that Φ(u 1 s ) is concave on [0, a] and on [b, ) for some 0 < a < b. Examples: Φ(u) = max{u α, u β }, 1 < α < β, s max{2, β}; Φ(u) = u r (1 + ln u ), r (3 + 5)/2, s > r; Zygmund spaces L p (LogL) α : Φ(u) = u p (ln(2 + u)) αp, αp 1, p 1, s > p, s 2. Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

13 Theorem. Suppose B is a Banach space of functions on R +, R or T satisfying max( f + g B, f g B ) ( f s B + k g s B ) 1 s, f, g B for some 2 s < and f ( + ξ) B f ( ) B for ξ > 0. Then { 2 jrs ω r+1 (f, 2 j t) s B j=1 } 1 s cω r (f, t) B. Our results also cover: C 0 semigroups of contraction operators, sharp Jackson estimates for approximation by algebraic polynomials on a simplex in R d, sharp Jackson inequality on the sphere S d 1. Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14

14 References: [1] F. Dai, Z. Ditzian, S. Tikhonov, Sharp Jackson inequalities, J. Approx. Theory, 151 (2008), [2] Z. Ditzian, On the Marchaud inequality, Proc. Amer. Math. Soc., 103 (1988), [3] Z. Ditzian, A. Prymak, Sharp Marchaud and converse inequalities in Orlicz spaces, Proc. Amer. Math. Soc., 135 (2007), [4] Z. Ditzian, A. Prymak, Convexity, moduli of smoothness, and a Jackson type inequality, Acta Math. Hungar., 130 (2011), no. 3, [5] Y. Lindenstrauss, L. Tzafriri, Banach Spaces, Vol. II, Springer-Verlag (Berlin, 1979) [6] M. F. Timan, On Jackson s theorem in L p spaces, Ukrain. Mat. Zh., 18 (1966), no. 1, (in Russian) Andriy Prymak (University of Manitoba) Geometry of Banach spaces and approximation January / 14