International Journal of Pure and Applied Mathematics Volume 25 No , BEST APPROXIMATION IN A HILBERT SPACE. E. Aghdassi 1, S.F.

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1 International Journal of Pure and Applied Mathematics Volume 5 No. 005, -8 BEST APPROXIMATION IN A HILBERT SPACE E. Aghdassi, S.F. Rzaev, Faculty of Mathematical Sciences University of Tabriz Tabriz, IRAN esaghdassi@tabrizu.ac.ir rzseymur@hotmail.com Abstract: In this paper we consider the shift operator on Hilbert spaces and by using this operator we define the modulus of continuity of fractional index, including relation to the K-functional and we prove the fractional analog we direct and inverse theorems of approximation theory. AMS Subject Classification: 4C0, 43A77, 43A90 Key Words: Hilbert space, self-adjoint operator, k-functional * Let A : DA H H be a self-adjoint operator on a separable Hilbert space H, which possesses a complete orthogonal system {w,w,...} of eigenvectors with Aw k = λ k w k for all n =,,..., where 0 < c λ λ λ n, as n. It is known that for example see [6] A α f := λ α k f,w k w k, α > 0, where f DA α iff λα k f,w k <. Received: August 0, 005 c 005, Academic Publications Ltd. Correspondence author

2 E. Aghdassi, S.F. Rzaev Since {w n } is complete, f = f,w k w k for all f H. Let H n =span{w,w,...,w n } which is an n-dimensional subspace of a real Hilbert space H and let S n f = n f,w k w k. It is well-known that for f H the element S n f is best approximation for f from H n Toepler s Best Approximation Theorem, see [5] or [3] p. 45, [] p. 0 given by E n f := E Hn f = inf f c,c,...,c k c k w k = f S n f = f a k = k=n+ where a k = f,w k. Due to specific properties of Hilbert spaces we have the following theorem. Theorem. For the element A α f Hα > 0 the element A α S n f is the best approximation for f from H n such that E n A α f = A α f A α S n f = A α f a k λα k = k=n+ a k a k λα k The proof of the case is similar to the proof of Toepler s Theorem see [5] and will not be repeated here. Corollary. Let w,w,...,w n H n and f DA α, Then a k λα k Aα f. Best approximations in Hilbert spaces for operator A α can be obtained easily, i.e. the best approximation A α S n of A α f from H n is given by the formula A α S n = λ α k f,w k w k.,.

3 BEST APPROXIMATION IN A HILBERT SPACE 3 Note. The best L -approximation of functions is a part of the general theory of best approximation in Hilbert spaces. It is known that if f L has Fourier series representation such as f = a 0 + a k cos kx + b k sin kx then s n is the best approximation to f and s n is the best approximation to f L. But this is not true for Fourier Legendre series in L. Theorem 3. Let f DA α, then the inequality E n f EnAα f λ n+ holds. α Indeed, by Theorem, E n A α f = k=n+ a k λα k From this the theorem is proved. λ α n+ a k k=n+ Corollary 4. If f DA α, then E n f Aα f λ n+ α. This inequality follows from properties E n f f. Theorem 5. If 0 < α < β, then A α f f α β A β f α β. = λ α n+ E nf. The proof is the same as Theorem in [] Landau inequality for derivative. From this and by Theorem, we have the following corollary. Corollary 6. If 0 < α < β, then E n A α f [E n f] α [ β En A β f ] α β. Using Theorem 3, and the above inequality we get Corollary 7. If 0 < α < β, then E n A α f EnAβ f λ n+ β α. Corollary 8. f H, then α > 0 E n f ν=0 Proof. By Corollary 4 we have On the other hand Let the polynomial P n f be the best approximation for A α P ν+ n λ v νn+ α. P n f P n P n f Aα P n f λ n+ α. P n f P n P n f = f P n P n f f P n f f P n P n f f P n f. Since the polynomial P n P n f is the best approximation of order n, then f P n P n f = E n f because 0 E n f E n f Aα P n f λ n+ α.

4 4 E. Aghdassi, S.F. Rzaev In this relation instead of n we write ν n and we consider ν=0 {E ν n E ν+ n} = E n f then we have Corollary 8. Theorem 9. Let f DA α, α > 0, then E n f λ n+ α K α A α f; where K α f;t = inf g DA α { f g + t α A α g }, λ n+ Proof. E n f E n f g+e n g f g +E n g. Let g DA α, then by Corollary 4 E n f f g + E na α g λ n+ { α E n f inf f g + E na α g g DA α λ n+ α A α f; λ n+ E na α f λ n+ α K α λ n+ α E n f λ n+ α K α A α f; } = K α f; λ n+. λ n+ Lemma 0. Inequality of Bernstein s Type Let P n f = n c kw k, then A α P n λ α n P n. Proof. Indeed P n f = n c k and A α P n f = n c kλ α k w k. Then A α P n f = c k λα k λα n P n A α P n f λ α n P n. Let the function ψ k h, 0 < h h 0 k = 0,,,... satisfies the following condition: number m > 0, m 0 and constant c > 0, c > 0, such that all the k, and h; and also ψ k h c kh m. ψ k h c kh m as 0 kh µ 0, 3 from and 3 we have ψ 0 h, also lim h 0 ψ k h =.

5 BEST APPROXIMATION IN A HILBERT SPACE 5 Now we introduce in H a family of bounded linear operator {T h } defined as follows T h f f h = ψ k hf;w k w k. 4 then From this definition we have: T h f f, 0 < h < h 0. f T h f 0, h 0. 3 AT h f = T h Af. Now we define the modulus of continuity of fractional index: Let r h := E T h r = k r k k=0 r h f = ψ k h r f;w k w k = k=0 T h k r > 0 ψ k h r f;w k w k. Definition. ω r f;τ := sup { r h f : 0 < h τ}, then ω rf,τ is called the modulus of continuity of fractional index. Lemma. Let the function ψ k h satisfies the condition, then { n ν=n a ν f } where α max, µ 0 h 0, Cα = Proof. From and 4 we have Cαω r f, α µ 0 rm c r. µ 0, απ r h f = [ψ k h ] r f,ω k ω k, and by virtue of Parseval s equality From this and 3 r h f = ψ k h r a k f. 5

6 6 E. Aghdassi, S.F. Rzaev n k=n a k f αµ 0 rm Thus the proof is complete. is true. n k=n k µ 0 rm a αn k f n αµ 0 rm C r µ0 ψ n r a k αn f k=n C α r µ f C µ 0 αω r f, αh αn Theorem. Under the condition in Lemma the inequality Proof. Indeed, a m f = m=n E nf Cα k=0 k+ n m= k n m=n+ m ω rf, µ 0 αm a m C α ω r f, µ 0 k+ nα, from this and by virtue of Toepler s Theorem we have E n f C α Since, by virtue of properties ω r f,τ, for any k ωr f, µ kn 0 k nα m= k n+ ωk r f, µ 0 k. 6 nα m ω r f, µ 0 mα. Thus ωrf, µ 0 k nα m=n+ m ω rf, µ 0. 7 mα From 6 and 7 we have Theorem. The following theorem is analogous to lemma of S.B. Stechkin in [4], Lemma.

7 BEST APPROXIMATION IN A HILBERT SPACE 7 Theorem 3. Let the function {ψ k n} is satisfies the condition then for r m and h < τ 0 < τ ω r f,τ c 3n rm where c 3 = rm c r + rm +r. k rm E k f, Proof. From 5 and the condition ψ k n and we have for all n and h > 0 r h f c r h rm k rm a k f + r k=n+ c k f. If n τ, then rm n rm. From this and by Toepler s Theorem for n τ. We have ω rf,τ c r n rm Now by Toepler s Theorem we get k rm a k f + r E n+ f. 8 k rm a k f = k rm [Ek f E k+ f] Also [k rm k rm ]Ek f rm E n+ f rm n rm k rm Ek f. 9 k rm Ek f. 0 From 8, 9 and 0 we have the theorem. Finally, we note that under some additional conditions all the theorems can be proved in a normed space. References [] I.K. Dauravet, Introduction to the Approximation Theory of Functions, Leningrad 977, In Russian.

8 8 E. Aghdassi, S.F. Rzaev [] I.Y. Lyubich, On inequality between ordered linear operators, Izvestiya AN SSSR, Ser. Matem., 4 960, [3] I.P. Natanson, Constractive Function Theory, Volume : Approximation in Mean, Frederie Unqcor Publishing Co [4] S.B. Stechkin, About the Kolmogorov-Seliverstov Theorem s, Izvestiya AN SSSR, Ser. Matem., [5] A. Toepler, Bemekenswerte Eigenschaften der Periodisches Reihen, Remarkable Properties of Periodic Series, Wiener Akad, Anz [6] E. Zeidler, Applied Functional Analysis: Application to Mathematical Physics, Applied Mathematical Sciences, Volume 08, Springer-Verlag, New York 995.