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1 ( ). Wierstrass Bernstein ([]) lim x xn w(x) = 0 ( n N) R w fw C 0 (R) lim (f P n)w L n (R) = 0 {P n } w Bernstein 950 ([5], [8] ) 970 Freud Freud weights w α (x) = exp( x α ) ([3] ) α Bernstein w α Christoffel infinite-finite range inequality x n w α (x) q n = (n/α) /α Freud number α > n P n (.) P n w α L p (R) C P n w α L p (J n) J n := [ C 2 q n, C 2 q n ] C, C 2 n N p w F(C 2 +) ( ) 984 Rakhmanov ([4]) Mhaskar-Saff ([0], []) (.) w F(C 2 +) Q(x) := log(/w(x)) w(x) = exp( Q(x)) 2 Q n N (.2) 2 π 0 a n tq (a n t) dt = n t 2 ( :5K04939) 200 Mathematics Subject Classification: 4A0, 4A7, 30E0, 42C05, 3A5 polynomial approximation, Freud type weights, Erdös type weights, Favard inequality, Markov-Bernstein inequaliy, de la Vallée Poussin mean suzukin@meijo-u.ac.jp web: breakthrough randam matrix P. Deift, T. Kriecherbauer and K.T-R. MacLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory, 95 (998), Q external field( ) log z u log( z u w(z)w(u)) weighted potential theory Q potential theory with external field

2 a n w Mhaskar-Rakhanov-Saff number (MRS number) {a n } n P n (.3) P n w L (R) P n w L ( x a n ) p < (.4) P n w L p (R) 2 P n w L p ( x a n ) Freud weight w α MRS number ( ) /α α 2 Γ(α/2)2 a n = 2 n /α Γ(α) F(C 2 +) w(x) = exp( Q(x)) (.5) T (x) := xq (x) Q(x) Freud Erdös Freud weight w α (α > ) Freud Erdös w(x) = exp( x α (e x )) P n n p fw L p (R) (.6) E p,n (w, f) := inf P P n (f P )w L p (R) Freud (.7) E p,n (w, f) C a n n E p,n (w, f ) (Jackson-Favard ) (.8) P w L p (R) C n a n P w L p ( x a n) ( P P n ) (Markov-Bernstein ) (.9) (f v n (f))w L p (R) CE p,n (w, f) (v n (f) de la Vallée Poussin ) ([9] ) Erdös T w Erdös (.7) (.8) (.9) P w C n P w L L p (R) a p ( x a n), (f v n(f)) w CE p,n (w, f). n L p (R) T /2 C C 0 {a n }, {b n } C a n b n Ca n a n b n f g R C f(x) g(x) Cf(x) T /4

3 2. R w(x) = exp( Q(x)) Q w F(C 2 +) (0) Q Q(0) = 0 () Q (2) Q (x) x 0 (3) lim x Q(x) =. (4) T (x) := xq (x)/q(x) (x 0) (0, ) quasi-increasing 3 T (x) Λ Λ > (5) C Q (x) Q (x) C Q (x) Q(x), a.e. x R (6) C 2 > 0 J Q (x) Q (x) C 2 R \ J Q (x) Q(x), a.e. x λ > 0 w F(C 2 +) Q C 3 (R) K x K Q (x) (2.) Q (x) C Q (x) Q (x), Q (x) Q(x) C λ w F λ (C 3 +) w {p n } 4 p n P n R p n (x)p m (x)w 2 (x)dx = δ nm fw L p (R) Fourier s m (f)(x) = R K m (x, t)f(t)w 2 (t)dt ( K m (x, t) := m k=0 p k (x)p k (t) f de la Vallée Poussin v n (f) 2n v n (f)(x) := n 2n m=n+ s m (f)(x). K n Christoffel-Darboux formura (2.2) K n (x, t) = γ n γ n ( ) pn (x)p n (t) p n (t)p n (x). x t γ n p n n p n (x) = γ n x n + /K(x, x) Christoffel (2.3) K n (x, x) = inf P (t)w(t) 2 dt P P n P (x) 2 R 3 f : (0, ) (0, ) c > 0 0 < x < y f(x) cf(y) quasi-increasing 4 Freud weight w 2 (x) = exp( x 2 ) Hermite )

4 w MRS number a n a n γ n /γ n (2.4) K n (x, x) C n a n T /2 (x) w 2 (x) (2.5) a 2n a n, T (a 2n ) T (a n ), a 2n a n a n T (a n ) a n (2.6) lim n n = 0 a n = O(n /Λ ) w Erdös η > 0 a n = O(n η ) 3. (.3) ([7], [9], [7] ) C µ Uµ(z) := log z t dµ(t) w = exp( Q) F(C 2 +) I w (µ) I w (µ) := log z t w(t)w(t) dµ(z)dµ(t) = log z t dµ(z)dµ(t) + 2 Q(z)dµ(z) R M(R) (3.) V w := inf{i w (µ); µ M(R)} Q(x) log x ( x ) V w = I w (µ w ) < µ w M(R) supp(µ w ) w Uµ w R c w := V w Qdµ w (3.2) { Uµw (x) + Q(x) c w ( x R) Uµ w (x) + Q(x) = c w ( x supp(µ w )) µ w w c w w Robin 5 I := supp(µ w ) 6 Uµ w I (a, b) R \ I (a, b I) (Uµ w (x)) = (x 5 (w ) R Frostman 6 K K outer boundary R R

5 t) 2 dµ w (t) > 0 Uµ w I Q (a, b) Uµ w (x) + Q(x) < Uµ w (a) + Q(a) = c w (3.2) Q I = [ a, a] Mhaskar-Saff x > 0 F (x) := log x 2 2 π x 0 Q(t) x2 t 2 dt x = a 7 F (a) = 0 a 2 atq (at) (3.3) π dt = 0 t 2 (.3) ω n (x) := exp( Q(x) n ) ω n R µ ωn (.2) (3.3) supp(µ ωn ) = [ a n, a n ] P P n P monic P (z) = z n + P (3.4) w(x)p (x) = ω n n(x)p (x) M, x [ a n, a n ] (3.5) P (z) M exp(n( Uµ ωn (z) + c ωn )) C log(/ P (z) ) = nuν(z) ν 8 (3.4) µ ωn (3.2) Uµ ωn (x) c ωn = Q(x) log M Uν(x) + n n supp(µ ωn ) 9 (3.5) (3.2) R Q(x)/n Uµ ωn (x)+c ωn (3.5) w(x)p (x) M ( x R) M = P w L ( x a n) (.3) Fekete Chebyshev ( [5]) E R ( E C w = exp( Q) E ). µ E,w w E cap(e, w) := e Vw(E), V w (E) = I w (µ E,w ) δ n (E, w) := max z i z j w(z i )w(z j ) z,,z n E i<j n t n (E, w) := min p P n w n (z)(z n p(z)) L (E) 2/n(n ) 7 K = [ x, x] F (x) = log cap(k) Q(t)dν K (t) cap(k) ν K (w ) K R Uµ w (t) Q(t) + c w ν K ( ) Uν K (t) log(/cap(k)) F (x) c w = F (a) 8 P (z) = (z z ) (z z n ) nν = δ z + + δ zn (Dirac ) log(/ P (z) ) 9 µ, ν dν dµ µ Uµ Uν + c (c ) supp(µ) C

6 cap(e, w) w δ(e, w) = lim n δ n (E, w), t(e, w) := lim n t n (E, w) Chebyshev ( ) (3.6) cap(e, w) = δ(e, w) = t(e, w) exp Q(x)dµ E,w (x) R ([6], [0]) δ n (E, w) {z (n),, z n (n) } Fekete monic Fekete n δ (n) z + + δ (n) z n µ E,w ( ) n 4. w F(C 2 +), p. ([8]) C f f w L p (R) (4.) E p,n (w, f) C a n n E p,n (w, f ) ( n N) (Jackson-Favard ) Freud Mhasker [9] Erdös Lubinsky [7] E p,n (w, f) η n E p,n (w, f ), lim n η n = 0 {η n } w (4.) w F(C 2 +) η n = Ca n /n 2. ([9]) C n N P P n (4.2) P w T /2 C n P w L L p (R) a p ( x a n ) n (Markov-Bernstein ) 0 < λ < 3/2 w F λ (C 3 +) C 0 (4.3) P w L p (R) C n a n P T /2 w L p ( x a C0 n) w Freud T (4.2) (.8) 0 (4.3) 3 w (4.2) (4.3) [6, p.294] P w L p (R) CnT /2 (a n )/a n P w L p (R) 0 < x a n T (x) CT (a n ) (4.3) T /2 3 w F λ (C 3 +) P (k) w L p (R) C(n/a n ) k P T k/2 w L p (R) ([4, Lemma 2.5]) 0 (4.3) C 0 = T /2 w (4.2) T /2 w F(C 2 +)

7 3. ([9]) 0 < λ < 3/2 w F λ (C 3 +) α R w F(C 2 ) w T α w T T a n/c0 a n a C0 n T a n w (.2) (.5) 4. ([3]) w F(C 2 +) T (a n ) C 0 (n/a n ) 2/3 C fw L p (R) n N (4.4) (f v n(f)) w CE p,n (w, f). L p (R) ft /4 w L p (R) T /4 (4.5) (f v n (f))w L p (R) CE p,n(t /4 w, f). f f w L p (R) (4.6) (f v n(f)) w T /4 L p (R) C a n n f w L p (R) Christoffel (2.3) (2.4) p = v n (f)w/t /4 L p (R) C fw L p (R) v n (f) L p (R) C ft /4 w L p (R) p = p = Riesz-Torin p v n (P ) = P ( P P n ) (4.4) (4.5) (4.6) (4.) (4.4) 2 v n(f)w/t /2 L p (R) Cn/a n ft /4 w L p (R) 5. ([20]) 0 < p fw L p (R) (4.7) ρ p (f) := lim sup n n log n log(/e p,n (f, w)) f λ a.e. ρ p (f) < ρ p (f) = 0 λ = 0 ρ p (f) 0 (4.8) λ A ρ p (f) λ B A := lim inf x T (x), B = lim sup x T (x) [2] [2] Mhaskar Freud Freud weight w α (x) = exp( x α ) /λ /α = /ρ p (f) ([9, p.77]). Erdös A = B = λ = ρ p (f) Cesaro de la Vallée Poussin

8 [] S. N. Bernstein, Le problem de la approximation des fonctions continus sur tout l axe reel et l une de ses applications, Bull. Math. Soc. France, 52 (924), [2] S. N. Bernstein, Leçon sur les propriétés extrémales et la meilleure approximation des function analytiques d une variable réell, Gauthier-Villars, Paris, 926. [3] K. Itoh, R. Sakai and N. Suzuki, The de la Vallée Poussin mean and polynomial approximation for exponential weights, International J. of Analysis, 205, ID , 8p. [4] K. Itoh, R. Sakai and N. Suzuki, An estimate for derivative of the de la Vallée Poussin mean, to appear. [5] P. Koosis, The logarithmic integral I, Cambridge Univ. Press, Cambridge 988. [6] A. L. Levin and D. S. Lubinsky, Orthogonal polynomials for exponential weights, Springer, New York, 200. [7] D. S. Lubinsky, Which weights on R admit Jackson theorem?, Israel J. of Math. 55 (2006), [8] D. S. Lubinsky, A survey of weights polynomial approximation with exponential weights, Surv. Approx. Theory, 3 (2007), -05. [9] H. N. Mhaskar, Introduction to the theory of weighted polynomial approximation, World Scientific, Singapore, 996. [0] H. N. Mhaskar and E. B. Saff, Extremal problems for polynomials with exponential weights, Trans. Amer. Math. Soc., 285 (984), [] H. N. Mhaskar and E. B. Saff, Where does the sup norm of a weighted polynomial live?, Constr. Approx., (985), 7-9. [2] H. N. Mhaskar and E. B. Saff, Where does the L p norm of a weighted polynomial live?, Trans. Amer. Math. Soc., 303 (987), [3] P. Navai, Geza Freud, orthgonal polynomials and Christoffel functions: a case study, J. Approx. Theory, 48 (986), [4] E. A. Rakhmanov, On asymptotic properties of polynomials orthogonal on the real axis, Math. USSR. Sbornik, 47 (984), [5] T. Ransford, Potential theory in the complex plane, Cambridge Univ. Press, Cambridge, 995. [6] E. B. Saff, Logarithmic potential theory with applications to approximation theory, Surv. Approx. Theory, 5 (200), [7] E. B. Saff and V. Totik, Logarithmic potentials with external fields, Springer, New York, 997. [8] R. Sakai and N. Suzuki, Favard-type inequalities for exponential weights, Pioneer J. Math. and Math. Sci., 3 (20), -6. [9] R. Sakai and N. Suzuki, Mollification of exponential weights and its application to Markov-Bernstein inequality, Pioneer J. Math. and Math. Sci., 7 (203), [20] R. Sakai and N. Suzuki, A characterization of real entire functions by polynomial approximation for exponential weights,, B43 (203), [2] R. S. Varga, On an extension of a result of S.N.Bernstein, J. Approx. Theory, (968),

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