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

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On approximation processes defined by the cosine

perator 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)

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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 2π-periodic continuous functions. In notations: for f ∈ C2π the Fourier partial sums Sn(f, x) = a0 2 +

n

k=1

ak cos kx + bk sin kx define the Rogosinski means by Rn(f, x) := 1 2

Sn(f, x +

π 2(n + 1)) + Sn(f, x − π 2(n + 1))

.
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Introduction

X - be an arbitrary (real or complex) Banach space. [X] - be the Banach algebra of all bounded linear operators U : X → X X ⊃ Aσ - be a dense family of subsets, meaning that for every f ∈ X there exists a family {gσ}σ>0, gσ ∈ Aσ such that limσ→∞ f − gσ = 0. S : Aσ → Aσ - be a linear projection operator Definition A cosine operator function Th ∈ [X] (h ≥ 0) is defined by the properties:

(i)

T0 = I(identity operator),

(ii)

Th1 · Th2 = 1

2(Th1+h2 + T|h1−h2|),

(iii)

Thf ≤ Tf, 0 < T− not depending on h > 0.

A. Kivinukk, A. Saksa (Tallinn University)

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Introduction

Let τh ∈ [X], h ∈ R, be a translation operator, defined by the properties

(i)

τ0 = I,

(ii)

τh1 · τh2 = τh1+h2,

(iii)

τhf ≤ Tf, 0 < T− not depending on h ∈ R. Then Th := 1

2(τh + 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, T C

h (f, x) := 1

2

f(x cos h +
1 − x2 sin h) + f(x cos h −
1 − x2 sin h)
.
A. Kivinukk, A. Saksa (Tallinn University)

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Introduction

But for some spaces we cannot define the translation operator τh ∈ [X], h ∈ R, nevertheless the cosine operator function does exist. An example: X = C−

2π - space of π-symmetric and 4π-periodic

continuous functions, i.e. f(π − x) = f(π + x) and f(4π + x) = f(x) for all x ∈ R. For example, the functions y = sin(k − 1

2)x, k ∈ N are in space C− 2π.

Here τh

sin

1

2◦

, x
= sin 1

2(x + h) /

∈ C−

2π for some h ∈ R, but

Thf ∈ C−

2π, where Th := 1 2(τh + τ−h) and τh is the ordinary translation

perator.
A. Kivinukk, A. Saksa (Tallinn University)

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Introduction

Example = Trigonometric approximation

X = C2π, the space of 2π-periodic continuous functions, with dense subspaces An ⊂ C2π consisting of trigonometric polynomials of degree not exceeding n; Fourier partial sum operators Sn : An → An are here the linear projection operators. In this setting the trigonometric Rogosinski means have the shape Rn(f, x) = T

π 2(n+1)Sn(f, x).

Important: The projection operators, hence the Rogosinski means, are defined on the whole space C2π.

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Introduction

Example = Shannon sampling operators

X = C(R), the space of uniformly continuous and bounded functions

n 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)| ≤ eσ|y|fC (z = x + iy ∈ C). Linear projection operator in this case is the classical Whittaker-Kotel’nikov-Shannon operator, for g ∈ B∞

σ , σ < πw defined

by (Ssinc

w g)(t) := ∞

k=−∞

g( k w ) sinc(wt − k), where the kernel function sinc(t) := sin πt

πt .

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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 < πw, then

(Ssinc

w g)(t) = g(t).

Important: The projection operators Ssinc

w

: B∞

σ → B∞ σ are defined only

n dense subspaces B∞

σ ⊂ C(R).

Theorem Extension Theorem ([Kantorovich-Akilov], Ch.V, Sect. 8.2, 8.3) Let Aσ ⊂ X be a family of dense subsets of a Banach space X and

B : Aσ → 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 = limσ→∞ Bgσ, where {gσ}σ>0 ⊂ Aσ is an arbitrary family with f = limσ→∞ gσ.

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Rogosinski-type operators

Definition Let Aσ ⊂ X be a dense family of subsets and Sσ : Aσ → Aσ be a linear projection operator, moreover, Th ∈ [X] be a cosine operator function. The Rogosinski-type operator Rσ,h,a : X → X is defined as an extension of Rσ,h,a : Aσ → X, which is defined by

Rσ,h,ag := aTh(Sσg) + (1 − a) T3h(Sσg) (h ≥ 0, a ∈ R).
Example. Let An ⊂ C2π be the set of trigonometric polynomials and

Sn : An → An 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 = C2π, 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.

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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 EA(f) := inf

g∈A f − g.

Remark We often may suppose that there exists an element g∗ ∈ A of best approximation, i.e. EA(f) = f − g ∗ . Definition The modulus of continuity of order k is defined via the cosine

perator function by

ωk(f, δ) := sup

0≤hδ

(Th − I)kf, k ∈ N.

A. Kivinukk, A. Saksa (Tallinn University)

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Order of approximation by Rogosinski-type operators

Remark

Let τh ∈ [X], h ∈ R, be a translation operator, defined by the properties

(i)

τ0 = I,

(ii)

τh1 · τh2 = τh1+h2,

(iii)

τhf ≤ Tf, 0 < T− not depending on h ∈ R. Since Th := 1

2(τh + τ−h), h ≥ 0 is a cosine operator and

Th − I = 1

2

τh/2 − τ−h/2

2 , then the modulus of continuity, defined above, can be represented by ωk(f, δ) = 1 2k ω2k(f, δ), where

ωk(f, δ) := sup

0≤h≤δ

τh/2 − τ−h/2

k f

, k ∈ N.
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Order of approximation by Rogosinski-type operators

Theorem For every f ∈ X, a ∈ R for the Rogosinski-type operators Rσ,h,a : X → X there holds Rσ,h,af − f ≤

Rσ,h,a[X] + |a|T + |1 − a|T
EAσ(f)

+ |a|ω(f, h) + |1 − a|ω(f, 3h). 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

f approximation. Attention: The next one is not a corollary of the

previous one. Theorem Denote Rσ,h = Rσ,h,9/8. Then we have Rσ,hf − f ≤

Rσ,h[X] + 5

4T

EAσ(f) + 3

2ω2(f, h) + 1 2ω3(f, h).

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Blackman-Harris-type operators

Definition Let Aσ ⊂ X be a dense family of subsets and Sσ : Aσ → Aσ be a linear projection operator, moreover, Th ∈ [X] be a cosine operator function. The Blackman-Harris-type operator Cσ,h,c : X → X is defined as an extension of Cσ,h,c : Aσ → X, which is defined by

Cσ,h,cg :=

m

k=0

ckTkh(Sσg), h ≥ 0, c = (c0, ..., cm) ∈ Rm+1 with

m

k=0

ck = 1.

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Blackman-Harris-type operators

The Blackman-Harris-type operators in Trigonometric Approximation reduce to the ϕ-means or ϕ-summation methods generated by a single function ϕc(t) =

m

k=0

ck cos kt with the corresponding operators Cn,h,c(f, x) := a0 2 +

n

k=1

ϕc (kh) (ak cos kx + bk sin kx) . The choice of the increment h depends on the concrete operator, e.g., for the trigonometric Rogosinski operator h = π/(2(n + 1)).

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Blackman-Harris-type operators

Theorem For every f ∈ X, c = (c0, ..., cm) ∈ Rm+1 for the Blackman-Harris-type

perator Cσ,h,c : X → X there holds

Cσ,h,cf − f ≤

Cσ,h,c[X] + |c0| + T

m

k=1

|ck|

EAσ(f)

+

m

k=1

|ck|ω(f, kh). Remark The higher order of approximation via ωl(f, h) depends, in fact, due to an isomorphism between cosines and the cosine operator functions, on a trigonometric equation in form

m

k=0

ck cos kx = (cos x − 1)l

m−l

k=0

dk cos kx, m ≥ l, dk ∈ R (k = 0, ..., m−l)

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Blackman-Harris-type operators

A similar abstract framework can be applied for the integral operators Kσ,h,cf := 1 Cσ,uh,cfdu, where the integral here is the Riemann integral in a Banach space (see [L. Schwartz, Cours d’analyse (Vol. 1) Analyse Mathématique Cours, 1967 ]) Example In Shannon sampling theory the Lanczos operators Lw : C(R) → C(R) can be defined by Lw(f, t) = 1 Tu/(2w)(Rwf, t)du, where the Rogosinski-type sampling operator is defined by Rw(f, t) :=

∞

k=−∞

f( k w )r(wt − k) with the kernel function r ∈ L1(R), r(t) =

cos πt 2π(1/4−t2).

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Concluding remarks

Positive moments Formally the abstract Blackman-Harris – type operators possess good approximation properties, since the modulus of continuity and operators both are defined by a cosine operator function; these have also practical applications, e.g. in signal and imaging processes. The given abstract setting has many concrete applications: Shannon sampling series, singular integrals of Fourier transform, trigonometric Fourier approximation, Fourier-Chebyshev series, Fourier-Walsh series (keeping in mind the dyadic translation ) etc. A negative moment In abstract setting we cannot say very much about the uniform boundedness of this operators family, but in concrete function spaces we have computed exact operator norms, also important in applications.

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Concluding remarks

About boundedness: As the Rogosinski-type operators

Rσ,h,ag := aTh(Sσg) + (1 − a) T3h(Sσg)

possess the representation Ua = aV + W, V, W ∈ [X], then for every a, a0, a1 ∈ R we get (a1 − a0)Ua = (a1 − a)Ua0 + (a − a0)Ua1. This equality shows that for boundedness of Ua[X] for arbitrary a ∈ R it is enough to know the boundedness of Ua0[X] and Ua1[X] for two specific values a0, a1 ∈ R.

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References

Butzer, P . L., Gessinger, A. Ergodic theorems for semigroups and cosine operator functions at zero and infinity with rates; applications to partial differential equations. A survey. In: Mathematical Analysis, Wavelets, and Signal Processing., Contemporary Mathematics, 1995, 190, 67-94. Kivinukk, A., Tamberg, G., On Blackman-Harris windows for Shannon sampling series. Sampling Theory in Signal and Image Processing, 2007, 6, 87–108. Kivinukk, A., Saksa, A., Zeltser, M. On approximation processes defined by a cosine operator function. Proc. of the Estonian Acad.

f Sci., 2017, 66, Issue 2, 214 - 224.

Rogosinski, W. W., Reihensummierung durch

Abschnittskoppelungen. Math. Z., 1926, 25, 132-149.
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References

KÖSZÖNÖM SZÉPEN ! THANK YOU !

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