On function spaces with fractional Fourier transform in weighted

Toksoy and Sandıkçı Journal of Inequalities and Applications (2015) 2015:87 DOI 10.1186/s13660-015-0609-4

RESEARCH

Open Access

On function spaces with fractional Fourier transform in weighted Lebesgue spaces Erdem Toksoy and Ay¸se Sandıkçı* Dedicated to Professor Ravi P Agarwal. *

Correspondence: [email protected] Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayıs University, Samsun, Turkey

Abstract ω d Let w and ω be weight functions on Rd . In this work, we define Aw, α ,p (R ) to be the 1 d vector space of f ∈ Lw (R ) such that the fractional Fourier transform Fα f belongs to p ω = f 1,w + Lω (Rd ) for 1 ≤ p < ∞. We endow this space with the sum norm f Aw, α ,p ω (Rd ) becomes a Banach space and invariant under Fα f p,ω and show that Aw, α ,p time-frequency shifts. Further we show that the mapping y → Ty f is continuous from ω d d w,ω d Rd into Aw, α ,p (R ), the mapping z → Mz f is continuous from R into Aα ,p (R ) and ω d 1 d Aw, α ,p (R ) is a Banach module over Lw (R ) with  convolution operation. At the end of this work, we discuss inclusion properties of these spaces. Keywords: fractional Fourier transform; convolution; Banach module

1 Introduction In this work, for any function f : Rd → C, the translation and modulation operator are defined as Tx f (t) = f (t – x) and Mw f (t) = eiwt f (t) for all y, w ∈ Rd , respectively. Also we write the Lebesgue space (Lp (Rd ),  · p ), for  ≤ p < ∞. Let w be a weight function on Rd , that is, a measurable and locally bounded function w satisfying w(x) ≥  and w(x + y) ≤ w(x)w(y) for all x, y ∈ Rd . We define, for  ≤ p < ∞,      Lpw Rd = f |fw ∈ Lp Rd . p

It is well known that Lw (Rd ) is a Banach space under the norm f p,w = fwp . Let w and w are two weight functions. We say that w ≺ w if there exists c > , such that w (x) ≤ cw (x) for all x ∈ Rd [, ]. The Fourier transform fˆ (or F f ) of f ∈ L (R) is given by  fˆ (w) = √ π



+∞

f (t)e–iwt dt. –∞

The fractional Fourier transform is a generalization of the Fourier transform with a parameter α and can be interpreted as a rotation by an angle α in the time-frequency plane. The fractional Fourier transform with angle α of a function f is defined by 

Fα f (u) =

+∞

Kα (u, t)f (t) dt, –∞

© 2015 Toksoy and Sandıkçı; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Toksoy and Sandıkçı Journal of Inequalities and Applications (2015) 2015:87

Page 2 of 10

where ⎧   –i cot α i( u +t ) cot α–iut cosec α ⎪ ⎪ e , if α is not multiple of π, ⎨ π Kα (u, t) = δ(t – u), if α = kπ, k ∈ Z, ⎪ ⎪ ⎩ δ(t + u), if α = (k + )π, k ∈ Z, and δ is a Dirac delta function. The fractional Fourier transform with α = π corresponds to the Fourier transform [–]. The fractional Fourier transform can be extended to higher dimensions as []: (F α ,...,αn f )(u , . . . , un )  +∞  +∞ ··· Kα ,...,αn (u , . . . , un ; t , . . . , tn )f (t , . . . , tn ) dt · · · dtn , = –∞

–∞

or shortly 

Fα f (u) =

+∞ –∞

 ···

+∞

Kα (u, t)f (t) dt, –∞

where Kα (u, t) = Kα ,...,αn (u , . . . , un ; t , . . . , tn ) = Kα (u , t )Kα (u , t ) · · · Kαn (un , tn ). In this work we define the function spaces with fractional Fourier transform in weighted Lebesgue spaces and discuss some properties of these spaces.

2 On function spaces with fractional Fourier transform in weighted Lebesgue spaces w,ω Definition  Let w and ω be weight functions on Rd and  ≤ p < ∞. The space Aα,p (Rd ) p w,ω consist of all f ∈ Lw (Rd ) such that F α f ∈ Lω (Rd ). The norm on the vector space Aα,p (Rd ) is = f ,w + F α f p,ω . f Aw,ω α,p w,ω (Rd ),  · Aw,ω ) is a Banach space for  ≤ p < ∞. Theorem  (Aα,p α,p w,ω (Rd ). Thus (fn )n∈N and (F α fn )n∈N are Cauchy Proof Let (fn )n∈N is a Cauchy sequence in Aα,p p p sequences in Lw (Rd ) and Lω (Rd ), respectively. Since Lw (Rd ) and Lω (Rd ) are Banach spaces, p there exist f ∈ Lw (Rd ) and g ∈ Lω (Rd ) such that fn – f ,w → , F α fn – gp,ω →  and hence fn – f  →  and F α fn – gp → . Then (F α fn )n∈N has a subsequence (F α fnk )nk ∈N that converges pointwise to g almost everywhere. Also it is easy to see that fnk – f  → . Then we have

Fα f (u) – g(u) ≤ Fα (fn – f )(u) + Fα fn (u) – g(u) k k d 

 – i cot αj ≤ π j=

Toksoy and Sandıkçı Journal of Inequalities and Applications (2015) 2015:87

 ×

Rd

Page 3 of 10

d i   (fn – f )(t) e j= (  (uj +tj ) cot αj –iuj tj cosec αj ) dt k

+ F α fnk (u) – g(u) d 

 – i cot αj fn – f  + F α fn (u) – g(u) . = k k π j= →  and From this inequality, we obtain Fα f = g almost everywhere. Thus fn – f Aw,ω α,p w,ω d w,ω d f ∈ Aα,p (R ). Hence (Aα,p (R ),  · Aw,ω ) is a Banach space.  α,p The following proposition is generalization of the one-dimensional and two-dimensional versions. The proof of this proposition is very similar to the proofs of onedimensional and two-dimensional versions in [, , , ], and we omit the details. Proposition  Let α = (α , α , . . . , αd ), where αi = kπ for each index i with  ≤ i ≤ d and k ∈ Z. Then d

i  j= (  yj sin αj cos αj –iuj yj sin αj )

() Fα (Ty f )(u) = e

F α f (u – y cos α , . . . , ud – yd cos αd ) ()

for all f ∈ L (Rd ) and y ∈ Rd ; d

() Fα (Mv f )(u) = e

i  j= (–  vj sin αj cos αj +iuj vj cos αj )

F α f (u – v sin α , . . . , ud – vd sin αd )

for all f ∈ L (Rd ) and v ∈ Rd . Theorem  Let α = (α , α , . . . , αd ), where αi = kπ for each index i with  ≤ i ≤ d and k ∈ Z. w,ω (Rd ) is () Let  ≤ p < ∞, w and ω be weight functions on Rd . Then the space Aα,p translation invariant. () Let ω be a bounded weight function on Rd . Then the mapping y → Ty f of Rd into w,ω (Rd ) is continuous. Aα,p p

w,ω Proof () Let f ∈ Aα,p (Rd ). Then f ∈ Lw (Rd ) and F α f ∈ Lω (Rd ). It is well known that the space Lw (Rd ) is translation invariant and holds Ty f ,w ≤ w(y)f ,w for all y ∈ Rd []. Let b = (y cos α , . . . , yd cos αd ). By using the equality (), we get

  F α (Ty f ) = p,ω

 Rd

 =

Rd

 p F α (Ty f )(u) p ωp (u) du F α f (u – y cos α , . . . , ud – yd cos αd ) p

 p d i  p × e j= (  yj sin αj cos αj –iuj yj sin αj ) ωp (u) du  ≤

Rd

 p F α f (u – b) p ωp (u – b)ωp (b) du

= ω(b)F α f p,ω

Toksoy and Sandıkçı Journal of Inequalities and Applications (2015) 2015:87

Page 4 of 10

for all y ∈ Rd . Hence, we have ≤ w(y)f ,w + ω(b)F α f p,ω < ∞. Ty f Aw,ω α,p w,ω (Rd ) is translation invariant. This means that Aα,p w,ω () Let f ∈ Aα,p (Rd ). We will show that if limn→∞ yn =  for any sequence (yn )n∈N ⊂ Rd , then limn→∞ Tyn f = f , which will complete the proof. It is well known that the mapping y → Ty f is continuous from Rd into Lw (Rd ) (see []). Thus, we have

Tyn f – f ,w → 

()

as n → ∞. Also,     F α (Ty f – f ) = Fα (Ty f ) – F α f  n n p,ω p,ω  d ( i (yj ) sin αj cos αj –iuj yjn sin αj )  = e j=  n T(yn cos α ,...,ydn cos αd ) (F α f ) – F α f p,ω   ≤  T(yn cos α ,...,ydn cos αd ) (F α f ) – F α f p,ω j  d i j    +  e j= (  (yn ) sin αj cos αj –iuj yn sin αj ) –  F α f p,ω .

p

p

Since F α f ∈ Lω (Rd ), the mapping y → Ty (F α f ) is continuous from Rd into Lω (Rd ) for all y ∈ Rd []. Then we obtain T(yn cos α ,...,ydn cos αd ) (F α f ) – F α f p,ω →  as n → ∞. Now d

i

j 

j

let hyn (u) = |e j= (  (yn ) sin αj cos αj –iuj yn sin αj ) – ||F α f (u)|. Since limn→∞ yn =  and ω is a p bounded weight function on Rd , we see that limn→∞ hyn (u)ωp (u) =  for all u ∈ Rd . Also, since j d i j  hyn (u) = e j= (  (yn ) sin αj cos αj –iuj yn sin αj ) –  F α f (u) ≤  F α f (u)

p

p

and Fα f ∈ Lω (Rd ), we can write hyn (u)ωp (u) ≤ p |F α f (u)|p ωp (u). Thus, by the Lebesgue dominated convergence theorem,   d ( i (yjn ) sin αj cos αj –iuj yjn sin αj )   e j=  –  F αf 

p,ω

→

as limn→∞ yn = . Hence, → Tyn f – f Aw,ω α,p

()

as n → ∞. Combining () and (),   = Tyn f – f ,w + F α (Tyn f – f )p,ω →  Tyn f – f Aw,ω α,p as n → ∞. This is the desired result.



Theorem  Let α = (α , α , . . . , αd ), where αi = kπ for each index i with  ≤ i ≤ d and k ∈ Z.

Toksoy and Sandıkçı Journal of Inequalities and Applications (2015) 2015:87

Page 5 of 10

w,ω () Let  ≤ p < ∞, w and ω be weight functions on Rd . Then Aα,p (Rd ) is invariant under modulations. () Let ω be a bounded weight function on Rd . Then the mapping z → Mz f is continuous w,ω (Rd ). from Rd into Aα,p p

w,ω Proof () Let f ∈ Aα,p (Rd ). Then f ∈ Lw (Rd ) and F α f ∈ Lω (Rd ). It is easy to see that Mη f ,w = f ,w and Mη f ∈ Lw (Rd ). Let c = (η sin α , . . . , ηd sin αd ) ∈ Rd . Thus by Proposition , we have

  F α (Mη f ) = p,ω

 Rd

 =

Rd

 p F α (Mη f )(u) p ωp (u) du p F α f (u – η sin α , . . . , ud – ηd sin αd )

d (– i η sin αj cos αj +iuj ηj cos αj ) p p ω (u) du × e j=  j  ≤

Rd

F α f (u – c) p ωp (u – c)ωp (c) du



p

 p

= ω(c)F α f p,ω for all η ∈ Rd . Hence, we get Mη f Aw,ω ≤ f ,w + ω(c)F α f p,ω < ∞. α,p () The proof technique of this part is the same as that of Theorem (). So, for the sake of brevity, we will not prove it.  The following definition is an extension of the convolution in [, ] of two functions to n dimensions. Definition  Let α = (α , α , . . . , αd ), where αi = kπ for each index i with  ≤ i ≤ d and k ∈ Z. Then the convolution of two functions f , g ∈ L (Rd ) is the function f g defined by  (f g)(x) =

d

Rd

f (y)g(x – y)e

j= iyj (yj –xj ) cot αj

dy.

It is easy to see that f g belongs to L (Rd ) by Fubini’s theorem. Theorem  Let α = (α , α , . . . , αd ), where αi = kπ for each index i with  ≤ i ≤ d and k ∈ Z, and f , g ∈ L (Rd ). Then 

Fα (f g)(u) =

d

j=



 d π – i u cot αj e j=  j Fα f (u)Fα g(u),  – i cot αj

where Fα f and Fα g are the fractional Fourier transforms of functions f and g, respectively.

Toksoy and Sandıkçı Journal of Inequalities and Applications (2015) 2015:87

Page 6 of 10

Proof Let α = (α , α , . . . , αd ), where αi = kπ for each index i with  ≤ i ≤ d and k ∈ Z, and f , g ∈ L (Rd ). We can write from the definition of the fractional Fourier transform 

Fα (f g)(u) =

d



j=

 =

d



j=



 – i cot αj π

d

Rd



 – i cot αj π



Rd

dt

d

Rd

f (y)g(t – y)e

d

i   j= (  (uj +tj ) cot αj –iuj tj cosec αj )

×e

i   j= (  (uj +tj ) cot αj –iuj tj cosec αj )

(f g)(t)e

j= iyj (yj –tj ) cot αj

dt dy.

We make the substitution t – y = k and obtain 

Fα (f g)(u) =

d



j=



 – i cot αj π



Rd

d

=

× g(k)e  d 







×

i   j= (  (uj +yj ) cot αj –iuj yj cosec αj )

 j

j

j=

d

f (y)e

=

Rd

dk   d 

 – i cot αj – i u cot α

d



j=



×  =

π

i   j= (  (uj +yj ) cot αj –iuj yj cosec αj )

Rd d i   ( (k +u ) cot αj –iuj kj cosec αj ) dk × g(k)e j=  j j



 dy

  d   d i u cot α  – i cot α π j – j e j=  j  – i cot αj π j= d

Rd

d

F α f (u)g(k)e



j=

 dy

i  j= (  kj cot αj –iuj kj cosec αj )

d π e j=  – i cot αj

j=

Rd

d

f (y)e

i   j= (  (kj +uj ) cot αj –iuj kj cosec αj )

dk

 d π – i u cot αj e j=  j Fα f (u)F α g(u).  – i cot αj



Theorem  Let α = (α , α , . . . , αd ), where αi = kπ for each index i with  ≤ i ≤ d and k ∈ Z. Lw (Rd ) is a Banach algebra under  convolution. Proof It is well known that Lw (Rd ) is a Banach space []. Let f , g ∈ Lw (Rd ), then we have  |f g|w(x) dy f g,w = Rd

 d iyj (yj –xj ) cot αj j= = dy w(x) dx d f (y)g(x – y)e d R R    g(x – y) w(x – y) dx f (y) w(y) dy ≤ 

Rd

= g,w

Rd



Rd

f (y) w(y) dy

= g,w f ,w . It is easy to show that the other conditions of the Banach algebra are satisfied.

() 

Toksoy and Sandıkçı Journal of Inequalities and Applications (2015) 2015:87

Page 7 of 10

Theorem  Let α = (α , α , . . . , αd ), where αi = kπ for each index i with  ≤ i ≤ d and k ∈ Z. w,ω (Rd ) is a Banach -convolution module over Lw (Rd ). Aα,p w,ω w,ω Proof It is well known that Aα,p (Rd ) is a Banach space by Theorem . Let f ∈ Aα,p (Rd )  d and g ∈ Lw (R ). By using the inequality (), we get

 d      d   π  – i uj cot αj j= F α (f g) =  e F f (u) F g(u)   α α p,ω    – i cot αj j=

 d π =  – i cot α j=

 d π =  – i cot α j=

 × ≤

Rd

g(t)e

j



Rd

p d  p  – i cot α j F α f (u) d π R j=

i   j= (  (uj +tj ) cot αj –iuj tj cosec αj )

F α f (u) p

 = g

j

 p F α f (u) p F α g(u) p ωp (u) du

d

Rd



p,ω



Rd

 Rd

g(t) dt

 p p p dt ω (u) du

p p

 p

ω (u) du

 p F α f (u) p ωp (u) du

≤ g,w F α f p,ω .

()

Combining () and (), we obtain   = f g,w + F α (f g)p,ω f gAw,ω α,p ≤ g,w f ,w + g,w F α f p,ω = f Aw,ω g,w . α,p This is the desired result. It is easy to see that the other conditions of the module are satisfied. 

3 Inclusion properties of the space Aαw,,pω (Rd ) d Proposition  For every  = f ∈ Aw, α,p (R ) there exists c(f ) >  such that ≤ w(x)f Aw, . c(f )w(x) ≤ Tx f Aw, α,p α,p d Proof Let  = f ∈ Aw, α,p (R ). By [], there exists c(f ) >  such that

c(f )w(x) ≤ Tx f ,w ≤ w(x)f ,w . By using () and the equality F α (Tx f )p = F α f p , we obtain   c(f )w(x) ≤ Tx f ,w ≤ Tx f ,w + F α (Tx f )p ≤ w(x)f ,w + F α f p

()

Toksoy and Sandıkçı Journal of Inequalities and Applications (2015) 2015:87

Page 8 of 10

≤ w(x)f ,w + w(x)F α f p = w(x)f Aw, α,p d for all f ∈ Aw, α,p (R ).



Lemma  Let w , w , ω and ω be weight functions on Rd . If Awα,p ,ω (Rd ) ⊂ Awα,p ,ω (Rd ), then Awα,p ,ω (Rd ) is a Banach space under the norm |f | = f Awα,p ,ω + f Awα,p ,ω . Proof Let (fn )n∈N is a Cauchy sequence in (Awα,p ,ω (Rd ), | · |). Then (fn )n∈N is a Cauchy sequence in (Awα,p ,ω (Rd ),  · Awα,p ,ω ) and (Awα,p ,ω (Rd ),  · Awα,p ,ω ). As these spaces are Ba-

nach spaces, there exist f ∈ Awα,p ,ω (Rd ) and g ∈ Awα,p ,ω (Rd ) such that fn – f Awα,p ,ω → , fn – gAwα,p ,ω → . Using the inequalities  ·  ≤  · ,w ≤  · Awα,p ,ω and  ·  ≤  · ,w ≤  · Awα,p ,ω , we obtain fn – f  →  and fn – g → . Also f – g ≤ fn – f  +

fn – g , we have f = g. Hence |fn – f | →  and f ∈ Awα,p ,ω (Rd ). That means (Awα,p ,ω (Rd ), | · |) is a Banach space. 

w , w , (Rd ) ⊂ Aα,p (Rd ) if and Theorem  Let w and w be weight functions on Rd . Then Aα,p only if w ≺ w .

Proof Suppose that w ≺ w . Thus there exists c >  such that w (x) ≤ c w (x) for all x ∈ w , Rd . Also let f ∈ Aα,p (Rd ). Then we write f ,w ≤ c f ,w < ∞. Hence we have f Aw , = f ,w + F α f p ≤ c f ,w + c F α f p = c f Aw , . α,p

α,p

w , Therefore, Awα,p , (Rd ) ⊂ Aα,p (Rd ). w , w , Conversely, suppose that Aα,p (Rd ) ⊂ Aα,p (Rd ). For every f ∈ Awα,p , (Rd ), we have f ∈ w , d Aα,p (R ). By Proposition , there are constants c , c , c , c >  such that

c w (x) ≤ Tx f Aw , ≤ c w (x)

()

c w (x) ≤ Tx f Aw , ≤ c w (x)

()

α,p

and

α,p

for all x ∈ Rd . It is well known from Lemma  that the space Awα,p , (Rd ) is a Banach space under the norm |f |, f ∈ Awα,p , (Rd ). Then by the closed graph theorem the norms  · Aw , α,p

and  · Aw , are equivalent on Awα,p , (Rd ). So, there exists c >  such that f Aw , ≤ f Aw , α,p

α,p

α,p

w , for all f ∈ Awα,p , (Rd ). Moreover, as Tx f ∈ Aα,p (Rd ), we have

Tx f Aw , ≤ cTx f Aw , . α,p

α,p

()

Toksoy and Sandıkçı Journal of Inequalities and Applications (2015) 2015:87

Page 9 of 10

Then, combining (), (), and (), we obtain c w (x) ≤ Tx f Aw , ≤ cTx f Aw , ≤ cc w (x). α,p

Thus, w (x) ≤

cc w (x). c 

α,p

Let

cc c

= k. Then we find w (x) ≤ kw (x) for all x ∈ Rd .



Proposition  Let w , w , ω and ω be weight functions on Rd . If w ≺ w and ω ≺ ω , then Awα,p ,ω (Rd ) ⊂ Awα,p ,ω (Rd ). Proof Assume that w ≺ w and ω ≺ ω . Then there exist c , c >  such that w (x) ≤ c w (x) and ω (x) ≤ c ω (x) for all x ∈ Rd . Let f ∈ Awα,p ,ω (Rd ). As f ∈ Lw (Rd ) and Fα f ∈ p Lω (Rd ), we have f ,w ≤ c f ,w < ∞ and F α f p,ω ≤ c F α f p,ω < ∞. Hence, we obtain f ∈ Awα,p ,ω (Rd ), and then Awα,p ,ω (Rd ) ⊂ Awα,p ,ω (Rd ). 

4 Duality p w,ω Let the mapping  : Aα,p (Rd ) → Lw (Rd ) × Lω (Rd ) be defined by (f ) = (f , F α f ) for  ≤ w,ω p < ∞ and let H = (Aα,p (Rd )). Then     (f ) = (f , F α f ) = f ,w + F α f p,ω w,ω is a norm on H for all f ∈ Aα,p (Rd ). Moreover, we define a set K as

  d   p  R × Lω– Rd , K = (ϕ, ψ) : (ϕ, ψ) ∈ L∞ w– 

 Rd

f (x)ϕ(x) dx +

Rd

 F α f (y)ψ(y) dy =  for all (f , F α f ) ∈ H ,

where p + p = . The following proposition is proved by the duality theorem, Theorem . in []. Proposition  Let  ≤ p < ∞, and w and ω be weight functions on Rd . The dual space of p w,ω Aα,p (Rd ) is isomorphic to L∞ (Rd ) × Lω– (Rd )/K where p + p = . w–

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Received: 13 October 2014 Accepted: 24 February 2015 References 1. Feichtinger, HG, Gürkanlı, AT: On a family of weighted convolution algebras. Int. J. Math. Sci. 13, 517-526 (1990) 2. Reiter, H: Classical Harmonic Analysis and Locally Compact Group. Oxford University Press, Oxford (1968) 3. Namias, V: The fractional order of Fourier transform and its application in quantum mechanics. IMA J. Appl. Math. 25, 241-265 (1980) 4. Ozaktas, HM, Kutay, MA, Zalevsky, Z: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, Chichester (2001) 5. Almeida, LB: The fractional Fourier transform and time-frequency representations. IEEE Trans. Signal Process. 42, 3084-3091 (1994) 6. Zayed, AI: On the relationship between the Fourier and fractional Fourier transforms. IEEE Signal Process. Lett. 3, 310-311 (1996)

Toksoy and Sandıkçı Journal of Inequalities and Applications (2015) 2015:87

Page 10 of 10

7. Almeida, LB: Product and convolution theorems for the fractional Fourier transform. IEEE Signal Process. Lett. 4(1), 15-17 (1997) 8. Zayed, AI: Fractional Fourier transform of generalized function. Integral Transforms Spec. Funct. 7, 299-312 (1998) 9. Bultheel, A, Martinez, H: A shattered survey of the fractional Fourier transform. Report TW337, Department of Computer Science, K.U. Leuven (2002) 10. Sahin, A, Ozaktas, HM, Mendlovic, D: Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters. Appl. Opt. 37(11), 2130-2141 (1998) 11. Sharma, VD: Operational calculus on generalized two-dimensional fractional Fourier transform. Int. J. Eng. Innov. Technol. 3(2), 253-256 (2013) 12. Fischer, RH, Gürkanlı, AT, Liu, TS: On a family of weighted spaces. Math. Slovaca 46(1), 71-82 (1996) 13. Singh, AK, Saxena, R: On convolution and product theorems for FRFT. Wirel. Pers. Commun. 65, 189-201 (2012) 14. Zayed, AI: A convolution and product theorem for the fractional Fourier transform. IEEE Signal Process. Lett. 5, 101-103 (1998) 15. Liu, T, Rooij, AV: Sums and intersections of normed linear spaces. Math. Nachr. 42, 29-42 (1969)