Article. On some inequalities for a trigonometric polynomial. Tatsuhiro HONDA*

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1 Bull iroshima Ist ech Research Vol 46 () 5 Article O some iequalities for a trigoometric polyomial atsuhiro ONDA* (Received Oct 7, ) Abstract We give a alterative proof of the Berstei iequalities ad the Segö iequalities for trigoometric polyomials or polyomials Later, we show that the th polariatio costat c(,) of the ilbert space equals Key Words: Berstei s iequality, Segö iequality, trigoometric polyomial Itroductio Let : R C be a trigoometric polyomial ( θ) ( α cos θ+ β si θ ) of degree with complex coefficiets α, β Let be the derivative of he, the followig estimate is called the Berstei iequality (cf []): ( θ) max ( θ) for θ θ [, π] For a trigoometric polyomial ( θ) ( a cos θ+ b si θ ) with real coefficiets a, b, the followig estimate is called the Segö iequality (cf [6]): ( ) θ ( θ) max ( θ) θ for all θ [, π] { } + I the preset paper, we show the Berstei iequality ad the Segö iequality for a trigoometric polyomial usig Lagrage s iterpolatio polyomial By usig this result, we show the Berstei iequalities for a polyomial o a ilbert space Moreover, we have that the th polariatio costat c(,) of the ilbert space equals Prelimiaries Let P : be a polyomial of degree We tae distict poits ζ,, ζ + If there exist such that P( ) ζ for,, +, the P is uique with degree I fact, the uique polyomial is give by + + q P( ) ( ) ζ ( ) q ( ), where q( ) ( ζ ) ( ζ ) ( ζ+ ) he the polyomial P() is called Lagrage s iterpolatio polyomial ad the poits ζ,, ζ + are called the iterpolatio odes Let f : be a cotiuous fuctio with distict poits f( ) for,, + he the polyomial () *Departmet of Architectural Egieerig, iroshima Istitute of echology, thoda@ccit-hiroshimaacp

2 atsuhiro ONDA + f q P( ) ( ) ( ) ( ) q ( ), is called Lagrage s iterpolatio polyomial for f with odes at,, + Usig Lagrage s iterpolatio polyomial, we obtai the followig propositio for trigoometric polyomials Propositio Let ( θ) ( α cos θ+ β si θ ) be a trigoometric polyomial of degree with complex coeffi- ciets α, β We put θ π for,,, he ( θ) θ+ θ + ( ) ( ) cosθ Proof We put e ad F( ) + ( θ ) Sice cos θ ad si θ he F() is a polyomial of degree () for all θ (3) F( ) { α( + ) + iβ( ) } i, we have F( ζ) F( ζ) We set Gζ ( ) for, ζ We may assume ζ he Gζ ( ) is a polyomial of degree i (-) with respect to We set e θ for,,, Sice Lagrage s iterpolatio polyomial for Gζ ( ) with odes at,, is uique, usig (), we have Gζ q Gζ ( ) ( ) ( ) ( ) q ( ) Let q( ) ( ) + he q ( ) So we obtai From (5) ad Gζ ( ) ζf ( ζ), we have F( ζ ) F( ζ) Gζ ( ) + F( ζ ) ζf ( ζ) F( ζ ) ( ) ( ) (4) (5) (6) for ζ \{ } Especially, i case ( θ) cosθ+ isi θ, we cosider F( ) he, from (6), ζ e Sice i i θ ( ) θ ( e ) e + e cosθ ece ζ ( ) ( ) ζ ad for every,,, we have ζ ζ ζ ( ) cosθ ( ) cosθ (7) From (6) ad (7), we have ζf ( ζ) F( ζ ) + F( ζ) ( ) (8)

3 O some iequalities for a trigoometric polyomial herefore Fe Fe ie + F ( ) ( ) ( e ) i ( θ) ie + ( e ) ( e ) ( e ) e ( ) So we have (3) sice we have for every,,,, + + i π Fe ( ) ( ) Fe ( ) F( e ) ( e ) ( θ+ θ ) e e ( θ+ θ ) e i( ) ( θ+ θ ) + Iequalities for a trigoometric polyomial We give the Berstei iequality for a trigoometric polyomial by Propositio without itegratio heorem For a trigoometric polyomial ( θ) ( α cos θ+ β si θ ) of degree with complex coefficiets α, β If max ( θ), the the followig estimate holds: θ [, π] ( θ) for θ Proof Sice the period of ( θ ) ad ( θ ) is π, by Propositio, we obtai where θ ( θ) max ( θ) max θ+ θ θ [, π] θ [, π] ( ) + ( ) cosθ max ( θ θ ) max [, ] cosθ [ π + ( θ), θ π θ, ] cosθ π for,,, Usig (7), we obtai ( θ ) for θ For a trigoometric polyomial ( θ) ( a cos θ+ b si θ) of degree with real coefficiets a, b, we ( d deote ) ( θ) ( θ) Usig heorem 4, we show the followig lemma dθ Lemma Let (θ) be a trigoometric polyomial of degree with real coefficiets ad max ( θ) If there exists a atural umber such that ( ) ( ) ( θ) ( θ) θ, + + for all + θ [, π] the ( ) + ( θ) ( ) ( θ) for all θ Proof By heorem 3, we have the estimate ( ) ( θ) (39) ( ) for all ad all θ We set F ( ) ( θ) ( θ) ( θ) + he, ( ) ( ) + ( ) F ( θ) ( θ) ( θ) + ( θ) (3) ( ) We assume that F( θ ) is the maximum value of F o the closed iterval [,π] he ( ) It follows from θ

4 atsuhiro ONDA this ad (3), that we cosider the followig two cases I case that ( ) ( θ ) Sice I case that ( ) ( θ ) from (39), we have ( ) F ( ) ( θ) ( θ ) ( θ ) + ( θ ) ( θ ) ( ) ( + ) ( ) ( θ ) for all θ ( ) F ( ) ( θ) ( θ ) ( θ ) + + ( + ) ( ) ( ) ( ) + θ θ for all θ his completes the proof Now we give a alterative proof of the Segö iequality for a trigoometric polyomial heorem Let ( θ) ( a cos θ+ b si θ) be a trigoometric polyomial of degree with real coefficiets a, b If max ( θ), the θ [, π] ( θ) ( θ) for all θ (3) { } + Proof We ca calculate the derivative of : ( ) ( ) a b cos θ θ π + si θ π a cos θ π b si θ π acos θ π bsi θ π, (3) + ( + ) ( + ) ( θ) a b + cos + θ π ( ) si θ + π + asi θ π + bcos θ π + Sice < for,,, for arbitrary ε >, there exists a atural umber such that So we have ad herefore + a b ( + ) < ε for > (33) a b + + cos θ π si θ π < ε (34) ( + ) a b + + ( + ) cos θ π si θ π < ε

5 O some iequalities for a trigoometric polyomial + ( ) a ( θ) ε b cos θ π < + + si θ π, + ( + By heorem 3, we have for all ad all θ From (3), we have It follows from (34) ad (35) that a ) ( ) si θ ε θ π < + + cos θ π + a b cos θ π + bsi θ π + ( ) ( θ) (35) ( ) ( θ) a + b + cos θ π si θ π he we have a cos θ π + bsi θ π ε ( θ ) + < + a + b < + ε (36) ece, from (33) ad (36), Lettig ε ted to, we obtai for ad all θ ( ) ( ) ( θ) ( θ) < ε+ a cos θ+ π + + b si θ π + ε+ asi + θ π b cos θ + π ε ε θ + π a b a cos bsi θ π + ε asi θ π + + bcos θ π + < ε + ( + ε) + 4ε( + ε) + 6ε+ 7ε ( ) ( ) ( θ) ( θ) Applyig Lemma 3 iductively, we have (3) his completes the proof Applicatios of the iequalities Let P : be a polyomial of degree o the complex plae ad P be the derivative of P We set P sup P( ) ad P sup P ( ) he we show the followig iequality called the Berstei iequality for a polyomial (cf [])

6 atsuhiro ONDA heorem For a polyomial P : of degree with complex coefficiets, the followig estimate holds: P P Proof We set ( ) P( e i θ i θ ) he, sice ( θ) P ( e ) ie θ, we have ( ) P ( e i θ θ ) By heorem 3, P sup P ( ) sup P ( e ) max ( θ) max ( θ) P θ [, π] θ [, π] θ [, π] Let be a ilbert space with the ier product, We set B { x ; x xx, } Let X be a ormed space with the orm X, ˇp: X be a cotiuous -liear mappig ad px ( ) pxx ˇ(,,, x) be the correspodig homogeeous polyomial of degree for We defie by Let Dp(x) deote the Fréchet derivative of p at x he for y p sup{ p( x) X; x B}, pˇ sup{ pˇ ( x,, x) X; x B,,, } Dp ( d x ) y dt px ( + ty) t { } I fact, px ( + ty) pˇ ( x+ ty,, x+ ty) pˇ ( x,, x) + tpˇ ( x,, x, y) + t px ˇ (,, x, y, y) ece Dp( x) y pˇ ( x,, x, y) (47) We deote by ( ; X ) the space of all homogeeous polyomial of degree Whe X, we write ( ) i place of ( ; X ) Propositio Let be a complex ilbert space ad X be a complex ormed space For p ( ; X ), Dp p Proof We put for x, y Sice if x, y, σ x, y if x, y x, y for x, y, we have xcosθ+ iσys B he B where a are some complex umbers xcosθ+ iσys xcosθ+ iσysi θ, xcosθ+ iσy s max{ x, y } ( ) ( ) xcosθ+ iσys x θ + iσy θ ( )! ( + )! + x+ iσyθ+ a θ, We tae a liear fuctioal ϕ : X with ϕ ad put ( θ ) ϕ p( xcos θ+ i σ y si θ ) he ( θ ) is a trigoometric polyomial of degree with complex coefficiets ad ( θ) ϕ p( x+ iσyθ+ a θ ) px ˇ ϕ ( + iσyθ+ a θ,, x+ iσyθ+ a θ ) ˇ ˇ ϕpx (,, x) + θϕpx (,, xi, σy) + bθ,

7 O some iequalities for a trigoometric polyomial where b are some complex umbers ece By heorem 3, (47) ad (48), d ( ) d ( θ) i σ ϕ p ˇ ( x,, x, y ) (48) θ θ ( ) ϕ Dp( x) y ϕpˇ ( x,, x, y) ( ) max ( θ) θ [, π] By the ah-baach theorem, we ca choose ϕ satisfyig ϕ ( Dp( x) y) Dp( x) y X herefore Dp( x) y p (49) for x, y B For p ( ), we set c(, ) if{ M; pˇ M pfor all p ( )} his c(, ) is called the th polariatio costat of the ilbert space ad c( ) lim sup c (, ) is called the polariatio costat of the space It is clear that c(, ), sice p pˇ X Usig Propositio 4, we have the Berstei iequality for a polyomial ad the polariatio costat of the complex ilbert space heorem Let be a complex ilbert space ad X be a complex ormed space For p ( ; X ), pˇ p Proof As i the proof of Propositio 4, it follows from (47) ad (49) that px ˇ(,, x, y ) p for x, y B We fix y ad put p x ˇ ( ) p ( x,, x, y ) he we ca loo upo p ( x) as homogeeous polyomial of degree ( ) ad p sup{ pˇ ( x,, x, y ) ; x B } p X As i the above, we get px ˇ(,, x, y, y) X p It follows from the above aalogue by iductio that we have X p ece pxy ˇ(,,, y, y ) p X pˇ p Corollary If is a complex ilbert space, the c(, ) c() Usig the Segö iequality for trigoometric polyomials with real coefficiets, we obtai we have the Berstei iequality for a polyomial ad the polariatio costat of the real ilbert space Propositio Let be a real ilbert space ad X be a real ormed space For p ( ; X ), Dp p Proof We may assume that p without loss of geerality y x x, y We put w for x, y B y x x, y, x y he xcosθ+ ws We tae a liear fuctioal ϕ : X with ϕ ad put ( θ ) ϕ p( xcos θ+ wsi θ ) he ( ) ϕ p( x) As i the proof of Propositio 4, ( ) ϕ ( Dpxw ( ) ) From (47),

8 atsuhiro ONDA ( ) ϕ Dp( xx ) ϕ p( x) ( ) herefore, by heorem 33, ( ( )) ( ) + ϕ Dp( x) y ϕ Dp( x) x x, y w y xxy, x, y ϕ ( Dp( xx ) )+ y xxy, ϕ Dpxw ( ) x, y ( ) + y x x, y ( ) ( ) x, y + y xxy, ( ) + ( ) ( ) herefore By the ah-baach theorem, we ca choose ϕ satisfyig ϕ Dp( x) y Dp( x) y X Dp( x) y X p for x, y B heorem Let be a real ilbert space ad X be a real ormed space For p ( ; X ), pˇ p Proof As i the proof of heorem 43, usig iductively Propositio 45, we show this theorem Corollary If is a real ilbert space, the c(, ) c( ) Refereces [] S Berstei, Leços sur les Propriétés Extrémales et la Meilleure Approximatio des Foctios Aalytiques d ue Variable Réele, Gauthier-Villars, Paris, 96 [] S Diee, Complex Aalysis o Ifiite Dimesioal Spaces, Spriger Moograph Math, 999 [3] L A arris, Berstei s Polyomial Iequalities ad Fuctioal Aalysis, Irish Math Soc Bull (996), 9 33 [4] oda, M Miyagi, M Nishihara, S Ohgai ad M Yoshida, Some Estimates for Polyomials, Asia-Europea Joural of Mathematics, Vol, No 3 (9), [5] V V Prasolov, Polyomials, Spriger, 999 [6] G Segö, Über eie Sats des err Serge Berstei, Schrifte der Köigsberger Gelehrte Gesellschaft, (98), 59 7 [7] M Yoshida, Berstei s Iequality ad its Applicatios, Fuuoa Uiv Sci Rep - (4), 7 [8] A Zigmud, rigoometric Series, Cambridge Press, 959