Here, it is observe that ΛBV(p)(σ,B), the class of functions of boundedp−Λ−variation from a non-empty compact subsetσofRinto a com- mutative unital Banach algebraB, is a commutative unital Banach algebra

Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 4 Issue 1 (2012), Pages 181-189.

ON BANACH ALGEBRA VALUED FUNCTIONS OF BOUNDED GENERALIZED VARIATION OF ONE AND SEVERAL

VARIABLES

(COMMUNICATED BY VIJAY GUPTA)

R. G. VYAS AND K. N. DARJI

Abstract. Here, it is observe that ΛBV^(p)(σ,B), the class of functions of boundedp−Λ−variation from a non-empty compact subsetσofRinto a com- mutative unital Banach algebraB, is a commutative unital Banach algebra.

Moreover, (Λ¹, ...,Λ^N)^∗BV^(p)(Π^N_i=1σi,B), the class ofN−variables functions of boundedp−(Λ¹, ...,Λ^N)^∗−variation from Π^N_i=1σiintoB, is a Banach space, whereσiare non-empty compact subsets ofR, for alli= 1to N.

2010 MSC: 47B40, 26B30, 26A45, 46A04.

Key words and phrases: Banach space, commutative unital Banach alge- bra, functions of several variables, functions of boundedp−Λ−variation over a compact set and isometry.

1. Introduction.

Berkson and Gillespie [3] proved thatBVH([a, b]×[c, d],C), the class of functions of bounded variation (in the sense of Hardy) over [a, b]×[c, d], is a Banach al- gebra with respect to the pointwise operations and the variation norm (also see [1], [6] and [7]). Here, first we prove that ΛBV^(p)(σ,B), the class of functions of p−Λ− bounded variation from a non-empty compact subset σ of R into a commutative unital Banach algebra B, is a commutative unital Banach algebra with respect to the pointwise operations and the Λp-variation norm. Finally, we show that (Λ¹,Λ², ...,Λ^N)^∗BV^(p)(∏N

i=1σi,B), the class ofN-variables functions of bounded p−(Λ¹,Λ², ...,Λ^N)^∗−variation over∏N

i=1σi is a Banach space with re- spect to the pointwise linear operations and (Λ¹,Λ², ...,Λ^N)_p-variation norm, where σ₁, σ₂, ..., σ_N are non-empty compact subsets ofR.

2. The classΛBV^(p)(σ,B).

Letσbe any non-empty compact subset ofRandI= [a, b] be the smallest closed interval containing σ. Let ⨿(σ) be a class of all partitions of σ. That is ⨿(σ) = {t : t={ti}ⁿi=0 is an increasing finite sequence inσ}.

⃝c2012 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.

Submitted October 7, 2011. Published February 10, 2012.

181

Definition 2.1. For a given non-empty compact subsetσ ofR, a Banach algebra B, a non-decreasing sequence of positive numbers Λ ={λi}^∞i=1 such that ∑_∞

i=1 1 λi

diverges andp≥1. A function f :σ→B is said to be of bounded p−Λvariation overσ(that is,f ∈ΛBV^(p)(σ,B))if

V_Λp(f, σ,B) = sup

t∈⨿(σ)

V_Λp(f, σ,B, t)<∞, where

V_Λp(f, σ,B, t) = (

∑n

i=1

||△f(t_i)||^p_B λi

)^1/p, in which△f(t_i) =f(t_i)−f(t_i−1).

In the above definition, forσ= [a, b] one gets the class ΛBV^(p)([a, b],B); forp= 1 andλn= 1, for alln, one gets the class BV(σ,B) [1] and forλn= 1, for alln, one gets the class BV^(p)(σ,B). ForB=C, we omit writingC, the class ΛBV^(p)(σ,B) reduces to the class ΛBV^(p)(σ).

Definition 2.2. For a given function f :σ→B, whereσ is a non-empty compact subset ofR. Define the functionEf :I→Bby Ef(x) =f(α(x)), where

α(x) =

{ x, if x∈σ, sup{t : [x, t]⊂I\σ}, otherwise.

Obviously,Ef is an extension off and is constant on the gaps inσ.

We prove the following theorems.

Theorem 2.1. If f, g ∈ ΛBV^(p)(σ,B), where B is a commutative unital Banach algebra and σis a non empty compact subset of R. Then the following hold:

(i) f and g are bounded.

(ii) VΛ_p(f +g, σ,B)≤VΛ_p(f, σ,B) +VΛ_p(g, σ,B).

(iii)VΛp(αf, σ,B) =|α|VΛp(f, σ,B), for anyα∈C. (iv)V_Λp(f, σ^′,B)≤V_Λp(f, σ,B), if a compact set σ^′⊂σ.

(v)V_Λp(f g, σ,B)≤ ||f||∞V_Λp(g, σ,B) +||g||∞V_Λp(f, σ,B), where

||f||_∞= sup

x∈σ||f(x)||_B < ∞.

(vi) Ifσ=σ₁∪σ₂, whereσ₁ andσ₂ are non empty compact subsets ofRsuch that σ₁⊂[a, c], σ₂⊂[c, b]and σ₁∩σ₂={c}, then

VΛ_p(f, σ,B)≤VΛ_p(f, σ1,B) +VΛ_p(f, σ2,B).

Corollary 2.2. Letσ1andσ2be non empty compact subsets ofRsuch thatσ1⊂σ2. Iff ∈ΛBV^(p)(σ2,B)thenf|σ1∈ΛBV^(p)(σ1,B)and||f|σ1||Λp(σ1,B)≤ ||f||Λp(σ2,B), where

||f||Λp(σ,B)=||f||_∞+VΛ_p(f, σ,B).

Theorem 2.3. Letσbe a non empty compact subset ofRandBbe a commutative unital Banach algebra. If f ∈ ΛBV^(p)(σ,B) then VΛ_p(f, σ,B) = VΛ_p(Ef, I,B), whereI= [a, b]is the smallest closed interval containingσ.

Corollary 2.4. For a given functionf :σ →B. f ∈ΛBV^(p)(σ,B)if and only if E_f ∈ΛBV^(p)(I,B).

Corollary 2.5. The map F : ΛBV^(p)(σ,B)→ ΛBV^(p)(I,B), defined as F(f) = E_f f or all f ∈ΛBV^(p)(σ,B), is a linear isometry.

Theorem 2.6. (ΛBV^(p)(σ,B),∥.∥Λ_p(σ,B))is a commutative unital Banach algebra with respect to the pointwise operations, where B is a commutative unital Banach algebra and σis a non empty compact subset of R.

Corollary 2.7. (ΛBV^(p)([a, b]),∥.∥Λ_p([a,b]))is a commutative unital Banach algebra with respect to the pointwise operations.

Proof of Theorem 2.1. For anyx∈σ,

||f(x)||_B ≤ ||f(a)||_B + (λ1)^(1/p) VΛ_p(f, σ,B). (1) Thus, Theorem 2.1(i) follows.

Proof of the remaining part of the Theorem is obvious.

Proof of Theorem 2.3. I = [a, b] is the smallest closed interval containing σ andEf|σ=f impliesVΛ_p(f, σ,B)≤VΛ_p(Ef, I,B).

For anyt∈ ⨿(I) define s=t∩σ∈ ⨿(σ). For the simplicity of the proof, suppose that there is only one tk ∈ t\s and (tk−1, tk)∩σ = (tk, tk+1)∩σ = ϕ. Then, Ef(tk) =f(α(tk)) and

∑

t

||△Ef(ti)||^p_B

λ_i =∑k−2 i=1

||△Ef(ti)||^p_B

λ_i +^{||△Ef(tk−1)||}

pB

λ_k−1 +^{||△Ef(tk)||}

pB

λ_k +∑

i≥k+1

||△Ef(ti)||^p_B λ_i . Fors^∗=s∪ {α(tk)}we get

∑

t

||△Ef(ti)||^p_B

λ_i ≤∑

s∗

||△f(si)||^p_B λ_i . Hence, the result follows fromVΛ_p(Ef, I,B)≤VΛ_p(f, σ,B).

Proof of Theorem 2.6. Let{fk} be any Cauchy sequence in the given normed linear space. Therefore it converges uniformly to some function say f. For any t∈ ⨿(σ),we get

VΛ_p(fk, σ,B, t)≤VΛ_p(fk−fl, σ,B, t) +VΛ_p(fl, σ,B, t)

≤V_Λp(f_k−f_l, σ,B) +V_Λp(f_l, σ,B).

This implies,

VΛ_p(fk, σ,B)≤VΛ_p(fk−fl, σ,B) +VΛ_p(fl, σ,B) and

|V_Λp(f_k, σ,B)−V_Λp(f_l, σ,B)| ≤V_Λp(f_k−f_l, σ,B)→0as k, l→ ∞.

Hence, {VΛp(fk, σ,B)}^∞k=1 is a Cauchy sequence in R and it is bounded by some constant sayM >0. Therefore

VΛ_p(f, σ,B, t) = lim

k→∞VΛ_p(fk, σ,B, t)

≤ lim

k→∞V_Λp(f_k, σ,B)≤M <∞.

Thus,f ∈ΛBV^(p)(σ,B).

Since{fk} is a Cauchy sequence, for anyϵ >0 there existsn0∈Nsuch that VΛ_p(fk−fl, σ,B, t)< ϵ, f or all k, l≥n0.

Lettingl→ ∞and taking supremum on the both sides of the above inequality we getVΛ_p(fk−f, σ,B)≤ϵ, f or all k≥n0.

Thus,||f_k−f||Λ_p(σ,B)→0 as k→ ∞.

Hence, (ΛBV^(p)(σ,B),∥.∥Λ_p(σ,B)) is a Banach space.

For anyg, h∈ΛBV^(p)(σ,B),

||g.h||Λ_p(σ,B) = ||g.h||∞ + V_Λp(g.h, σ,B)

≤ ||g||_∞ ||h||_∞+||g||_∞ VΛ_p(h, σ,B) +||h||_∞VΛ_p(g, σ,B)

≤(||g||_∞+V_Λp(g, σ,B))(||h||_∞+V_Λp(h, σ,B))

=||g||Λ_p(σ,B)||h||Λ_p(σ,B). This completes the proof.

3. Generalizations to several variables.

For the sake of simplicity, first we prove these results for functions of two variables and then we extend the results for functions of several variables.

Letσ1andσ2be two non empty compact subsets ofRand letR= [a1, b1]×[a2, b2]⊂ R²be the smallest closed rectangle containing σ=σ1×σ2.

Definition 3.1. Let Lbe the class of all non-decreasing sequences Λ^′ ={λ^′_n}(n= 1,2, ...)of positive numbers such that∑

n 1

λ^′_n diverges. For given a Banach algebra B, p≥1, Λ = (Λ¹,Λ²) andσ =σ₁×σ₂; where Λ¹ ={λ¹_n}, Λ² ={λ²_n} ∈L and σ1,σ2 are non empty compact subsets ofR. A function f :σ→Bis said to be of boundedp−Λ−variation(that is, f ∈ΛBV^(p)(σ,B)), if

VΛ_p(f, σ,B) =sup

D VΛ_p(f, σ,B, D)<∞, where

VΛ_p(f, σ,B, D) = (

∑n

i=1

∑m

j=1

||△f(s_i, t_j)||^p_B λ¹_i λ²_j )^1/p in which

△f(si, tj) =f(si, tj)−f(si, tj−1)−f(si−1, tj) +f(si−1, tj−1), and D=s×tis a rectangular grid onσobtained from any two partitionss={s_i}ⁿi=0∈

⨿(σ₁)andt={t_j}^mj=0∈ ⨿(σ₂).

If f ∈ΛBV^(p)(σ,B)is such that the marginal functions f(a1, .)∈Λ²BV^(p)(σ2,B) andf(., a2)∈Λ¹BV^(p)(σ1,B)then f is said to be of boundedp−Λ^∗−variation over σ (that is, f ∈Λ^∗BV^(p)(σ,B)). If f ∈ Λ^∗BV^(p)(σ,B) then each of the marginal functionsf(., t)∈Λ¹BV^(p)(σ1,B) andf(s, .)∈Λ²BV^(p)(σ2,B), where t∈σ2 and s∈σ₁ are fixed.

For B = C, we omit writing C, the class Λ^∗BV^(p)(σ,B) reduces to the class Λ^∗BV^(p)(σ). Forσ=Rthe class Λ^∗BV^(p)(σ,B) reduces to the class Λ^∗BV^(p)(R,B).

For p= 1, we omit writing p, the classes ΛBV^(p)(R) and Λ^∗BV^(p)(R) reduce to classes ΛBV(R) and Λ^∗BV(R) respectively.

Definition 3.2. For a given function f : σ→B, where σ=σ1×σ2. Define the function Ef :R→B byEf(x1, x2) =f(α(x1), α(x2))where

α(xi) =

{ x_i, if x_i∈σ_i, sup{t : [x_i, t]⊂[a_i, b_i]\σ_i}, otherwise, fori= 1,2.

Theorem 3.1. If f, g∈Λ^∗BV^(p)(σ,B), where B is a commutative unital Banach algebra and σ=σ₁×σ₂, in whichσ₁ andσ₂ are non empty compact subsets ofR. Then the following hold:

(i) f andg are bounded.

(ii) VΛ_p(f+g, σ,B)≤VΛ_p(f, σ,B) +VΛ_p(g, σ,B).

(iii)VΛ_p(αf, σ,B) =|α|VΛ_p(f, σ,B), for anyα∈C.

(iv)VΛ_p(f, σ^′,B)≤VΛ_p(f, σ,B), ifσ^′ =σ₁^′ ×σ₂^′ ⊂σ, in which σ₁^′ and σ₂^′ are non empty compact subsets ofR.

Corollary 3.2. Let σ1, σ2,τ1 andτ2be non empty compact subsets ofRsuch that σ=σ1×σ2 ⊂τ =τ1×τ2. If f ∈Λ^∗BV^(p)(τ,B) then f|σ∈Λ^∗BV^(p)(σ,B)and

||f|σ||Λp(σ,B)≤ ||f||Λp(τ,B), where

∥f∥Λp(σ,B)

=∥f∥∞+V_Λp(f, σ,B) +V_(Λ1)_p(f(., a₂), σ₁,B) +V_(Λ2)_p(f(a₁, .), σ₂,B).

Theorem 3.3. Let σ1 and σ2 be non empty compact subsets of R such that σ= σ1×σ2 andBbe a commutative unital Banach algebra. Iff ∈Λ^∗BV^(p)(σ,B)then VΛ_p(f, σ,B) = VΛ_p(Ef,R,B), where R is the smallest closed rectangle containing σ.

Corollary 3.4. For a given functionf :σ→B. f ∈Λ^∗BV^(p)(σ,B) if and only if E_f ∈Λ^∗BV^(p)(R,B).

Corollary 3.5. The mapF : Λ^∗BV^(p)(σ,B)→Λ^∗BV^(p)(R,B), defined asF(f) = E_f for allf ∈Λ^∗BV^(p)(σ,B), is a linear isometry.

Theorem 3.6. (Λ^∗BV^(p)(σ,B),∥.∥Λp(σ,B)) is a Banach space with respect to the pointwise operations, where B is a commutative unital Banach algebra and σ = σ1×σ2 in whichσ1 andσ2 are non empty compact subsets ofR.

Corollary 3.7. (Λ^∗BV^(p)([a₁, b₁]×[a₂, b₂]),∥.∥Λ_p([a₁,b₁]×[a₂,b₂]))is a Banach space with respect to the pointwise operations.

Proof of Theorem 3.1. For any (x, y)∈σ,

||f(x, y)||B

≤ ||f(a1, a2)||B+||f(x, y)−f(x, a2)−f(a1, y)+f(a1, a2)||B+||f(x, a2)||B+||f(a1, y)||B

≤3||f(a1, a2)||_B+ (λ¹₁ λ²₁)^1/p VΛ_p(f, σ,B) + (λ¹₁)^1/p V_(Λ1)_p(f(., a2), σ1,B) +(λ²₁)^1/p V_(Λ2)_p(f(a₁, .), σ₂,B).

Thus, Theorem 3.1(i) follows.

Proof of remaining part of the Theorem is obvious.

Proof of Theorem 3.3. Ris the smallest closed rectangle containingσ=σ₁×σ₂ andE_f|σ=f impliesV_Λp(f, σ,B)≤V_Λp(E_f,R,B).

Let R = s×t be any rectangle grid of [a₁, b₁]×[a₂, b₂], where s = {s_i}^mi=1 ∈

⨿([a1, b1]) and t = {tj}ⁿj=1 ∈ ⨿([a2, b2]). Consider u = s∩σ1 ∈ ⨿(σ1) and v=t∩σ2∈ ⨿(σ2).ThenD=u×vis a rectangle grid ofσ=σ1×σ2. For the sim- plicity of the proof, suppose there is only one (sk, tl)∈Rsuch that (sk, tl)∈R\D, where (sk−1, sk)∩σ1= (sk, sk+1)∩σ1=ϕand (tl−1, tl)∩σ2= (tl, tl+1)∩σ2=ϕ.

Then

∑

s×t

||△E_f(s_i,t_j)||^p_B

λ¹_iλ²_j =∑k−2 i=1

∑l−2 j=1

||△E_f(s_i,t_j)||^p_B

λ¹_iλ²_j +^{||△Ef(sk−1,tl−1)||}

p B

λ¹_k−1λ²_l−1

+||△E_f(s_k−1, t_l)||^p_B

λ¹_k−1λ²_l +||△E_f(s_k, t_l−1)||^p_B

λ¹_kλ²_l−1 +||△E_f(s_k, t_l)||^p_B λ¹_kλ²_l

+

∑m

i≥k+1

∑n

j≥l+1

||△Ef(si, tj)||^p_B λ¹_iλ²_j .

SinceEf(sk, tl) =f(α(sk, tl)), for (u×v)^∗= (u×v)∪ {α(sk, tl)}, we get

∑

s×t

||△Ef(si, tj)||^p_B

λ¹_iλ²_j ≤ ∑

(u×v)^∗

||△f(ui, vj)||^p_B λ¹_iλ²_j .

Hence, the result follows fromVΛp(Ef,R,B)≤VΛp(f, σ,B).

Proof of Theorem 3.6. Let {f_k}^∞k=1 be a Cauchy sequence in Λ^∗BV^(p)(σ,B).

Therefore it converges uniformly to some function sayf onσ. From Theorem 2.6, we get

lim

k→∞V_(Λ2)_p((f_k(a₁, .)−f(a₁, .)), σ₂,B) = 0 (2) and

lim

k→∞V_(Λ1)_p((f_k(., a₂)−f(., a₂)), σ₁,B) = 0. (3) Now, for any rectangular grid D ofσ

VΛ_p(fk, σ,B, D)≤VΛ_p(fk−fl, σ,B, D) +VΛ_p(fl, σ,B, D)

≤VΛ_p(fk−fl, σ,B) +VΛ_p(fl, σ,B).

This implies,

VΛ_p(fk, σ,B)≤VΛ_p(fk−fl, σ,B) +VΛ_p(fl, σ,B) and

|V_Λp(f_k, σ,B)−V_Λp(f_l, σ,B)|

≤V_Λp(f_k−f_l, σ,B)→0 as k, l→ ∞.

Hence, {V_Λp(f_k, σ,B)}^∞k=1 is a Cauchy sequence in R and it is bounded by some constant sayM >0. Therefore

VΛ_p(f, σ,B, D) = (∑m i=1

∑n j=1

||△f(si,tj)||^p_B λ¹_iλ²_j )^1/p

= lim

k→∞(

∑m

i=1

∑n

j=1

||△f_k(s_i, t_j)||^p_B

λ¹_iλ²_j )^1/p≤ lim

k→∞V_Λp(f_k, σ,B)≤M <∞. This together with (2) and (3) impliesf ∈Λ^∗BV^(p)(σ,B). Moreover, VΛp(fk−f, σ,B, D) = (∑m

i=1

∑n j=1

||△(f_k−f)(s_i,t_j)||^p_B λ¹_iλ²_j )^1/p

= lim

l→∞(

∑m

i=1

∑n

j=1

||△(f_k−f_l)(s_i, t_j)||^p_B λ¹_iλ²_j )^1/p

≤ lim

l→∞VΛp(fk−fl, σ,B)→0as k→ ∞. Thus the result follows form (2) and (3).

Finally, we extends these results for functions of several variables as follow.

Let f(x) = f(x₁, ..., x_N) be a function from R^N into a Banach algebra B. Given x= (x₁, ..., x_N)∈R^N andh= (h₁, ..., h_N)∈R^N, define (see [5])

△f(x;h) =T_hf(x)−f(x) =△f(x₁, ..., x_N;h₁, ..., h_N)

=

∑1

η₁=0

...

∑1

η_N=0

(−1)^η1+...+ηNf(x1+η1h1, ..., xN+ηNhN).

Letσ1, σ2, ..., σN be non-empty compact subsets ofRandR=∏N

i=1[ai, bi]⊂R^N be the smallest closed parallelepiped containingσ=∏N

i=1σi.

Definition 3.3. Let L, p and B be as in the definition 3.1. For given Λ = (Λ¹,Λ², ...,Λ^N); whereΛⁱ={λⁱ_n} ∈L, ∀i= 1to N; andσ=∏N

i=1σi. A function f :σ→Bis said to be of boundedp−Λ−variation(that is,f ∈ΛBV^(p)(σ,B))if V_Λp(f, σ,B)

= sup P (

s1

∑

r₁=1

...

sN

∑

r_N=1

||△f(x^r₁¹⁻¹, ..., x^r_N^N−1;h^r₁¹, ..., h^r_N^N)||^p_B

λ¹_r1...λ^N_rN )^1/p<∞,

where the supremum is extended over all partitionsP =P1×P2×...×PN of the σ, Pj = {aj = x⁰_j < x¹_j < ... < x^s_j^j = bj} and sj ≥ 1; rj = 1,2, ..., sj; h^r_j^j = x^r_j^j −x^r_j^j−1; j = 1,2, ..., N.

Moreover, a function f ∈ΛBV^(p)(σ,B)is said to be of bounded p−Λ^∗−variation (that is, f ∈Λ^∗BV^(p)(σ,B)), if for each of its marginal functions

f(x1, ..., xi−1, ai, xi+1, ..., xN)∈(Λ¹, ...,Λⁱ⁻¹,Λⁱ⁺¹, ...,Λ^N)^∗BV^(p)(σ(ai),B),

∀i= 1,2, ..., N,whereσ(a_i) ={(x₁, ..., x_i−1, x_i+1, ..., x_N)∈∏N

j=1j̸=iσ_j}.

Definition 3.4. For a given function f :σ→ B, where σ=∏N

i=1σi. Define the function Ef :R→B byEf(x) =f(α(x1), α(x2), ..., α(xN)), x= (x1, x2, ...xN), where

α(xi) =

{ xi, if xi∈σi, sup{t : [xi, t]⊂[ai, bi]\σi}, otherwise, for alli= 1,2, ..., N.

Theorem 3.8. If f, g∈Λ^∗BV^(p)(σ,B), where B is a commutative unital Banach algebra and σ=∏N

i=1σ_i in which σ_i are non empty compact subsets of R, for all i= 1 toN . Then the following hold:

(i) f andg are bounded.

(ii) V_Λp(f+g, σ,B)≤V_Λp(f, σ,B) +V_Λp(g, σ,B).

(iii)VΛ_p(αf, σ,B) =|α|VΛ_p(f, σ,B), for anyα∈C. (iv) V_Λp(f, σ^′,B)≤V_Λp(f, σ,B), if σ^′ =∏N

i=1σ^′_i ⊂σ, in which σ_i^′ are non empty compact subsets of R, for all i= 1 toN.

Corollary 3.9. Let σi and τi be non empty compact subsets of R, for all i = 1 to N, such that σ = ∏N

i=1σi ⊂ τ = ∏N

i=1τi. If f ∈ Λ^∗BV^(p)(τ,B) then f|σ ∈ Λ^∗BV^(p)(σ,B)and||f|σ||Λ_p(σ,B)≤ ||f||Λ_p(τ,B), where

∥f∥Λp(σ,B)=∥f∥_∞+VΛ_p(f, σ,B) +

∑N

i=1

V_(Λ1,...Λⁱ⁻¹,Λⁱ⁺¹,...,Λ^N)_p(f(..., ai, ...),

∏N

j=1j̸=i

σj,B).

Theorem 3.10. Let σ_i be non empty compact subsets of R, for all i = 1 to N, such that σ = ∏N

i=1σi and B be a commutative unital Banach algebra. If f ∈ Λ^∗BV^(p)(σ,B) then VΛ_p(f, σ,B) = VΛ_p(Ef,R,B), where R is the smallest closed parallelepiped containing σ=∏N

i=1σ_i.

Corollary 3.11. For a given function f :σ→B. f ∈Λ^∗BV^(p)(σ,B) if and only if E_f∈Λ^∗BV^(p)(R,B).

Corollary 3.12. The mapF : Λ^∗BV^(p)(σ,B)→Λ^∗BV^(p)(R,B), defined asF(f) = Ef for allf ∈Λ^∗BV^(p)(σ,B), is a linear isometry.

Theorem 3.13. (Λ^∗BV^(p)(σ,B),∥.∥Λ_p(σ,B))is a Banach space with respect to the pointwise operations, where B is a commutative unital Banach algebra and σ =

∏N

i=1σ_i in whichσ_i are non empty compact subsets ofRfor, alli= 1 toN.

Corollary 3.14. (Λ^∗BV^(p)(∏N

i=1[a_i, b_i]),∥.∥Λp(∏_N

i=1[ai,bi]))is a Banach space with respect to the pointwise operations.

Proof of all these extended theorems are similarly follows by induction arguments onN.

References

[1] B. Ashton and I. Doust, Functions of bounded variation on compact subsets of the plane, Studia Mathematica, 169(2)2005, 163-188.

[2] A. N. Bakhvalov, Fourier coefficients of Functions from many-dimensional classes of bounded Λ-variation, Moscow Univ. Math. Bulletin, 66(2011), No.1, 8-16.

[3] E. Berkson and T. A. Gillespie, Absolute continuous functions of two variables and well- bounded Operators, J. London Math. Soc., 30(2)(1984), 305-321.

[4] M. I. Dyachenko and D. Waterman, Convergence of double Fourier series and W-classes, Trans. Amer. Math. Soc., 357 (2004), No.1, 397-407.

[5] Ferenc Moricz and Antal Verse, Absolute convergence of multiple Fourier series revisited, Analysis Mathematica, 34(2008), 145-162.

[6] R. G. Vyas, Properties of functions of generalized bounded variation, Mat. Vesnik, 58(2006), No.3-4, 91-96.

[7] R. G. Vyas, On variation topology, Mat. Vesnik, 61(2010), No.1, 47-50.

Author’s address:

R. G. Vyas

Department of Mathematics, Faculty of Science,

The Maharaja Sayajirao University of Baroda, Vadodara, Gujarat, India.

Email address: [email protected] Kiran N. Darji

Department of Science and Humanity,

Tatva Institute of Technological Studies, Modasa, Sabarkantha, Gujarat, India.

Email address: [email protected]