On approximation processes defined by the cosine operator function - PowerPoint PPT Presentation

On approximation processes defined by the cosine operator function in a Banach space Andi Kivinukk, Anna Saksa Tallinn University 6th Workshop on Fourier Analysis and Related Fields August 24-31, 2017, Pécs, Hungary A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 1 / 20

Introduction Introduction The aim of this presentation is to introduce an abstract framework of certain approximation processes using a cosine operator functions concept. Historical roots of these processes go back to W.W. Rogosinski, 1926, who proved that the arithmetical mean of shifted Fourier partial sums converges uniformly to a given 2 π -periodic continuous functions. In notations: for f ∈ C 2 π the Fourier partial sums n � S n ( f , x ) = a 0 2 + a k cos kx + b k sin kx k = 1 define the Rogosinski means by � � R n ( f , x ) := 1 π π S n ( f , x + 2 ( n + 1 )) + S n ( f , x − 2 ( n + 1 )) . 2 A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 2 / 20

Introduction X - be an arbitrary (real or complex) Banach space. [ X ] - be the Banach algebra of all bounded linear operators U : X → X X ⊃ A σ - be a dense family of subsets, meaning that for every f ∈ X there exists a family { g σ } σ> 0 , g σ ∈ A σ such that lim σ →∞ � f − g σ � = 0 . S : A σ → A σ - be a linear projection operator Definition A cosine operator function T h ∈ [ X ] ( h ≥ 0) is defined by the properties: T 0 = I (identity operator), (i) T h 1 · T h 2 = 1 (ii) 2 ( T h 1 + h 2 + T | h 1 − h 2 | ) , � T h f � ≤ T � f � , 0 < T − not depending on h > 0 . (iii) A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 3 / 20

Introduction Let τ h ∈ [ X ] , h ∈ R , be a translation operator , defined by the properties (i) τ 0 = I , (ii) τ h 1 · τ h 2 = τ h 1 + h 2 , � τ h f � ≤ T � f � , 0 < T − not depending on h ∈ R . (iii) Then T h := 1 2 ( τ h + τ − h ) , h ≥ 0 , is a cosine operator function. It means that if we can define a translation operator, then we have also the cosine operator function. A non-trivial cosine operator function related with the Fourier-Chebyshev series: x ∈ [ − 1 , 1 ] , 0 ≤ h ≤ π, � � � � h ( f , x ) := 1 1 − x 2 sin h ) + f ( x cos h − 1 − x 2 sin h ) T C f ( x cos h + . 2 A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 4 / 20

Introduction But for some spaces we cannot define the translation operator τ h ∈ [ X ] , h ∈ R , nevertheless the cosine operator function does exist. An example : X = C − 2 π - space of π -symmetric and 4 π -periodic continuous functions, i.e. f ( π − x ) = f ( π + x ) and f ( 4 π + x ) = f ( x ) for all x ∈ R . For example, the functions y = sin( k − 1 2 ) x , k ∈ N are in space C − 2 π . � � 1 � � = sin 1 Here τ h ∈ C − 2 π for some h ∈ R , but sin 2 ◦ , x 2 ( x + h ) / 2 π , where T h := 1 T h f ∈ C − 2 ( τ h + τ − h ) and τ h is the ordinary translation operator. A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 5 / 20

Introduction Example = Trigonometric approximation X = C 2 π , the space of 2 π -periodic continuous functions, with dense subspaces A n ⊂ C 2 π consisting of trigonometric polynomials of degree not exceeding n ; Fourier partial sum operators S n : A n → A n are here the linear projection operators. In this setting the trigonometric Rogosinski means have the shape R n ( f , x ) = T 2 ( n + 1 ) S n ( f , x ) . π Important: The projection operators, hence the Rogosinski means, are defined on the whole space C 2 π . A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 6 / 20

Introduction Example = Shannon sampling operators X = C ( R ) , the space of uniformly continuous and bounded functions on R with dense subspaces B ∞ σ ⊂ C ( R ) consisting of bounded functions on R , which are entire functions f ( z ) ( z ∈ C ) of exponential type σ , i.e. | f ( z ) | ≤ e σ | y | � f � C ( z = x + iy ∈ C ). Linear projection operator in this case is the classical Whittaker-Kotel’nikov-Shannon operator, for g ∈ B ∞ σ , σ < π w defined by ∞ � g ( k ( S sinc w g )( t ) := w ) sinc( wt − k ) , k = −∞ where the kernel function sinc( t ) := sin π t π t . A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 7 / 20

Introduction Example = Shannon sampling operators continued To be the projection operator is a statement of famous Whittaker-Kotel’nikov-Shannon theorem : if g ∈ B ∞ σ , σ < π w , then ( S sinc w g )( t ) = g ( t ) . Important: The projection operators S sinc : B ∞ σ → B ∞ σ are defined only w on dense subspaces B ∞ σ ⊂ C ( R ) . Theorem Extension Theorem ([Kantorovich-Akilov], Ch.V, Sect. 8.2, 8.3) Let A σ ⊂ X be a family of dense subsets of a Banach space X and B : A σ → X is a bounded linear operator with the operator norm � � � B � . Then � B has a bounded linear extension B : X → X with � B � = � � B � . For f ∈ X the operator B ∈ [ X ] is defined by Bf = lim σ →∞ � Bg σ , where { g σ } σ> 0 ⊂ A σ is an arbitrary family with f = lim σ →∞ g σ . A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 8 / 20

Rogosinski-type operators Rogosinski-type operators Definition Let A σ ⊂ X be a dense family of subsets and S σ : A σ → A σ be a linear projection operator, moreover, T h ∈ [ X ] be a cosine operator function. The Rogosinski-type operator R σ, h , a : X → X is defined as an extension of � R σ, h , a : A σ → X , which is defined by � R σ, h , a g := aT h ( S σ g ) + ( 1 − a ) T 3 h ( S σ g ) ( h ≥ 0 , a ∈ R ) . Example. Let A n ⊂ C 2 π be the set of trigonometric polynomials and S n : A n → A n be the Fourier partial sums operators. Then the given Definition for a = 1 leads to the historical Rogosinski means. Remark. The Fourier partial sums operators are defined on the whole space X = C 2 π , but this is not true for the Whittaker-Kotel’nikov-Shannon operator. This is the reason why we first define our approximation processes on a dense subspace. A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 9 / 20

Order of approximation by Rogosinski-type operators Order of approximation by Rogosinski-type operators To measure the order of approximation, two traditional quantities are known - the best approximation and the modulus of continuity . Definition The best approximation of f ∈ X by elements of A is defined by E A ( f ) := inf g ∈ A � f − g � . Remark We often may suppose that there exists an element g ∗ ∈ A of best approximation, i.e. E A ( f ) = � f − g ∗ � . Definition The modulus of continuity of order k is defined via the cosine operator function by � ( T h − I ) k f � , k ∈ N . ω k ( f , δ ) := sup 0 ≤ h � δ A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 10 / 20

Order of approximation by Rogosinski-type operators Remark Let τ h ∈ [ X ] , h ∈ R , be a translation operator, defined by the properties (i) τ 0 = I , (ii) τ h 1 · τ h 2 = τ h 1 + h 2 , � τ h f � ≤ T � f � , 0 < T − not depending on h ∈ R . (iii) Since T h := 1 2 ( τ h + τ − h ) , h ≥ 0 is a cosine operator and � � 2 , then the modulus of continuity, defined T h − I = 1 τ h / 2 − τ − h / 2 2 above, can be represented by ω k ( f , δ ) = 1 2 k � ω 2 k ( f , δ ) , where � � � � k f � � ω k ( f , δ ) := sup � τ h / 2 − τ − h / 2 � , k ∈ N . � 0 ≤ h ≤ δ A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 11 / 20

Order of approximation by Rogosinski-type operators Theorem For every f ∈ X, a ∈ R for the Rogosinski-type operators R σ, h , a : X → X there holds � � � R σ, h , a � [ X ] + | a | T + | 1 − a | T � R σ, h , a f − f � ≤ E A σ ( f ) | a | ω ( f , h ) + | 1 − a | ω ( f , 3 h ) . + The importance of parameters a ∈ R will be explained by the next theorem. It appears that the particular value a = 9 8 yields a better order of approximation. Attention: The next one is not a corollary of the previous one. Theorem Denote R σ, h = R σ, h , 9 / 8 . Then we have � � � R σ, h � [ X ] + 5 E A σ ( f ) + 3 2 ω 2 ( f , h ) + 1 � R σ, h f − f � ≤ 4 T 2 ω 3 ( f , h ) . A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 12 / 20

Blackman-Harris-type operators Blackman-Harris-type operators Definition Let A σ ⊂ X be a dense family of subsets and S σ : A σ → A σ be a linear projection operator, moreover, T h ∈ [ X ] be a cosine operator function. The Blackman-Harris-type operator C σ, h , c : X → X is defined as an extension of � C σ, h , c : A σ → X , which is defined by m � c k T kh ( S σ g ) , h ≥ 0 , c = ( c 0 , ..., c m ) ∈ R m + 1 � C σ, h , c g := k = 0 with m � c k = 1 . k = 0 A. Kivinukk, A. Saksa (Tallinn University) 6th Workshop on Fourier Analysis and ... 13 / 20