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arXiv:1808.07751v4 [math.GM] 25 Dec 2019 Abstract: [email protected] University, E-mail: Bayar Celal Manisa , of Department Yavuz Enes generalize via generalized numbers and fuzzy Dirichlet of series of resummation the On http: through Press IOS at available is publication final The eishv lydipratrlsi nltcnme theor number analytic in roles important real played in have primarily series Introduced mathematics. of fields series various factorial and Dirichlet invention, their Following Introduction 1 le of defin summation the we concerning Besides, results obtain and convergence. functions its implies number fuzzy series of series factorial a of summation which under conditions ehdaeas eue scorollaries. Res as deduced . also Fourie original are level the method that to prove series and Dirichlet functions generalized valued ordin fuzzy its of imply series series summat factorial which generalized under and introdu conditions Dirichlet now Tauberian we obtain literature and the numbers in Taub various achievements given the and space Following number fuzzy in converge to fail summ introduced authors intro Furthermore, are numbers [17–19]. m vestigated fuzzy classical of in series concepts and many particular, and developed satisfa also achieved have and concept introduc problems its real-world Since modeling sets. for classical tool using by handle to hard is regula in utilized also are series both paper, the solutio series of factorial context concerning studies giv also and convergence are rapid There more of series series factorial of to series transformation in used and the numb properties stu On analytical analytic [3–6]. to in recently theorists equations usage differential number the of for Beside solutions tool 2]. [1, a identities as related also serve function zeta htteeaen eo ftezt ucino h line the on function zeta the of zeros nu no prime are with relation there deep that has and primes over product Euler ahmtc ujc Classification: Subject Mathematics e od n phrases: and words Key setnino lsia es uz esaeitoue by introduced are sets fuzzy sets, classical of extension As eitouesmcniuu umto ehd o eiso series for methods summation semicontinuous introduce We eiso uz ubr,Tueinterm,fzyFourier fuzzy theorems, Tauberian numbers, fuzzy of series 37,4A5 40E05 40A05, 03E72, Re aefsiae ayrsacesadfudapiain in applications found and researchers many fascinated have aia Turkey. Manisa, so ifrneeutos[11] ncneto ihthe with connection In [11–13]. equations difference of ns 1 r u.Bsds edfietecneto ee level of concept the define we Besides, sum. ary te ad atra eisaesuidwt eadt thei to regard with studied are series factorial hand, other n nepnin ffntos uhr ovre power converted Authors functions. of expansions in and iaino iegn eis[,1,1,1,15]. 14, 12, 10, [8, series divergent of rization lsfrsm atclrcsso h eea summation general the of cases particular some for ults in eerhr sdfzyst sasatmathematical smart a as sets fuzzy used researchers tion, nfcoilsre xasosfrsm ucin [7–10]. functions some for expansions series factorial en ( ue n orsodn ovrec rprisaein- are properties convergence corresponding and duced blt ehd ohnl eiso uz ubr which numbers fuzzy of series handle to methods ability //dx.doi.org/10.3233/JIFS-190632 s rter,te rs sepnin ffntosadas and functions of expansions as arise they theory, er tr eut.I h enie hoeia ai fthe of theoretical meantime, the In results. ctory e ore series. Fourier vel aibe oase obntra usin,Dirichlet questions, combinatorial answer to variables h ocp flvlFuirsre ffzyvalued fuzzy of series Fourier level of concept the e .TeReanzt function zeta Riemann The y. 1 = ) eiso uz audfntosaesmal via summable are functions valued fuzzy of series r eagnrlsmainmto o eiso fuzzy of series for method summation general a ce ra odtost civ h ovrec [20–27]. convergence the achieve to conditions erian brtermwihi,i at qiaett h fact the to equivalent fact, in is, which theorem mber i eeaie iihe eisadvageneralized via and series Dirichlet generalized via s ybhvoso rtmtcfntosadt derive to and functions arithmetic of behaviors dy ae 1]t ersn mrcs nweg which knowledge imprecise represent to [16] Zadeh teaisaeetne ofzymteais In mathematics. fuzzy to extended are athematics o fasre ffzynmesvageneralized via numbers fuzzy of series a of ion iihe eisadi atclrteRiemann the particular in and series Dirichlet . series uz ubr n ieTauberian give and numbers fuzzy f ζ ( s ) srpeetdby represented is d r On the resummation of series of fuzzy numbers via generalized Dirichlet and generalized factorial series

2 Preliminaries

A fuzzy number is a fuzzy set on the real axis, i.e. u is normal, fuzzy convex, upper semi-continuous and supp u = t R : u(t) > 0 is compact [16]. RF denotes the space of fuzzy numbers. α-level set [u] is defined by { ∈ } α t R : u(t) α if 0 < α 1 [u]α := { ∈ ≥ } ≤ ( t R : u(t) > α if α = 0. { ∈ } r R may be seen as a fuzzy number r defined by ∈ 1 if t = r r(t) := 0 if t = r.  6 − + Theorem 2.1 (Representation Theorem). [28] Let [u]α = [u (α), u (α)] for u RF and for each α [0, 1]. Then, the following statements hold: ∈ ∈ (i) u−(α) is a bounded and non-decreasing left on (0, 1]. (ii) u+(α) is a bounded and non-increasing left continuous function on (0, 1]. (iii) The functions u−(α) and u+(α) are right continuous at the point α = 0. (iv) u−(1) u+(1). ≤ Conversely, if the pair of functions γ and β satisfies the conditions (i)-(iv), then there exists a unique u RF such ∈ that [u] := [γ(α), β(α)] for each α [0, 1]. The fuzzy number u corresponding to the pair of functions γ and β α ∈ is defined by u : R [0, 1], u(t) := sup α : γ(α) t β(α) . → { ≤ ≤ } Let u, v RF and k R. The addition and scalar multiplication are defined by ∈ ∈ [ku−, ku+] if k 0 (i) [u + v] = [u− + v−, u+ + v+] (ii) [ku] = α α (2.1) α α α α α α + − ≥ ([kuα , kuα ] if k< 0 where [u] = [u−, u+], for all α [0, 1]. α α α ∈ Fuzzy number 0 is identity element in (RF , +) and none of u = r has inverse in (RF , +). For any k , k R 6 1 2 ∈ with k k 0, distribution property (k + k )u = k u + k u holds but for general k , k R it fails to hold. On 1 2 ≥ 1 2 1 2 1 2 ∈ the other hand properties k(u + v) = ku + kv and k (k u) = (k k )u holds for any k, k , k R. It should be 1 2 1 2 1 2 ∈ noted that RF with addition and scalar multiplication defined above is not a linear space over R. Partial ordering relation on RF is defined as follows: u v [u] [v] u− v− and u+ v+ for all α [0, 1].  ⇐⇒ α  α ⇐⇒ α ≤ α α ≤ α ∈ The metric D on RF is defined as − − + + D(u, v):= sup max uα vα , uα vα , α∈[0,1] {| − | | − |} and it has the following properties D(ku, kv)= k D(u, v), D(u + v, w + z) D(u, w)+ D(v, z) | | ≤ where u, v, w, z RF and k R. ∈ ∈ Definition 2.2. [29] Let (u ) be a of fuzzy numbers. Denote s = n u for all n N, if the k n k=0 k ∈ sequence (s ) converges to a fuzzy number u then we say that the series u of fuzzy numbers converges to u and n k P write u = u which implies that k P n n P u−(α) u−(α) and u+(α) u+(α) (n ) k → k → →∞ Xk=0 Xk=0 uniformly in α [0, 1]. Conversely, for the sequence (u ) of fuzzy numbers if u−(α)= γ(α) and u+(α)= ∈ k k k k k β(α) converge uniformly in α, then (γ(α), β(α)) : α [0, 1] defines a fuzzy number u represented by [u] = { ∈ } P P α [γ(α), β(α)] and uk = u. P 2 On the resummation of series of fuzzy numbers via generalized Dirichlet and generalized factorial series

Besides, we say that series uk is bounded if the sequence (sn) is bounded. The set of bounded series of fuzzy numbers is denoted by bs(F ). P Theorem 2.3. [30] If u and v converge, then D ( u , v ) D(u , v ). k k k k ≤ k k A fuzzy valued functionPf : I RP RF has the parametricP representationP P ⊂ → − + [f(x)]α = fα (x),fα (x) , for each x I and α [0, 1].  ∈ ∈ A function f : R RF is 2π-periodic if f(x) = f(x + 2π) for all x R. The space of all 2π-periodic → R (F) R ∈ and continuous fuzzy valued functions on is denoted by C2π ( ) where the continuity is meant with respect to metric D. Besides the space of all 2π-periodic and real valued continuous functions on R is denoted by C2π(R) F and equipped with the supremum norm . Here we note that if f C( )(R) then f ∓ C (R) in view of k · k ∈ 2π α ∈ 2π metric D. Let f, g : I RF be fuzzy valued functions. The distance between f and g is defined by → ∗ − − + + D (f, g) = sup D(f(x), g(x)) = sup sup max fα (x) gα (x) , fα (x) gα (x) . x∈I x∈I α∈[0,1] {| − | | − |}

Let L : CF (R) CF (R) be an where CF (R) denotes the space of all continuous fuzzy valued functions → on R. Then, we call L a fuzzy linear operator iff

L(c1f1 + c2f2)= c1L(f1)+ c2L(f2), for any c , c R,f ,f CF (R). Also, operator L is called fuzzy positive linear operator if it is fuzzy linear 1 2 ∈ 1 2 ∈ and the condition L(f; x) L(g; x) is satisfied for any f, g CF (R) with f(x) g(x) and for all x R .  ∈  ∈ (F) Theorem 2.4. [31] Let L N be a sequence of fuzzy positive linear operators from C (R) into itself. Assume { n}n∈ 2π that there exists a corresponding sequence Ln of positive linear operators defined on C2π(R) with the n∈N property n o e L (f; x) ∓ = L f ∓; x , (2.2) { n }α n α F for all x R, α [0, 1],n N and f C( )(R). Assume further that ∈ ∈ e ∈ ∈ 2π lim Ln(fi) fi = 0, (2.3) n→∞ k − k for i = 0, 1, 2ewith f (x) = 1,f (x) = cos x,f (x) = sin x. Then, for all f C(F)(R) we have 0 1 2 ∈ 2π ∗ lim D (Ln(f),f) = 0. n→∞

3 Main results

We now introduce semicontinuous summation methods of series of fuzzy numbers, defined by means of nonneg- ative and uniformly bounded sequences of continuous real valued functions on [0, ). Introduced method acts ∞ on the terms of the series of fuzzy numbers directly and does not require computation of the partial sums which may be challenging even in the case of series of real numbers. The method also includes many known summation methods. Definition 3.1. Let u be a series of fuzzy numbers and φ (s) be a nonnegative uniformly bounded sequence n { n } of continuous real valued functions defined on [0, ) such that φ (s) φ (s) for all s [0, ) and φ (0) = 1. P ∞ n+1 ≤ n ∈ ∞ n Then, series un of fuzzy numbers is said to be (φ) summable to fuzzy number µ if unφn(s) exists for s > 0 and ∞P P lim unφn(s)= µ. s→0+ n=0 X 3 On the resummation of series of fuzzy numbers via generalized Dirichlet and generalized factorial series

Theorem 3.2. If series un of fuzzy numbers converges to fuzzy number µ, then it is (φ) summable to µ.

Proof. Let un be a seriesP of fuzzy numbers and un = µ. It is sufficient to show that series unφn(s) exists for s> 0 and P∞ P P lim unφn(s)= µ. s→0+ nX=0 From Definition 2.2, series u φ (s) exists for s> 0 if series u∓(α)φ (s) converge uniformly in α [0, 1]. n n n n ∈ By convergence of the series u we know that u∓(α) converge uniformly in α [0, 1]. Besides, φ (s) is P n n P ∈ { n } uniformly bounded and φ (s) φ (s). So by Abel’s test, series u∓(α)φ (s) converge n+1 P ≤ n P n n uniformly in α. Then, series u φ (s) converges in fuzzy number space for s> 0. Now we aim to show that n n P ∞ P lim unφn(s)= µ. s→0+ nX=0 n Define the sequence of continuous fuzzy valued functions fn(s) on [0, 1] such that fn(s) = k=0 ukφk(s). ∞ { } Then, limn→∞ fn(s) = k=0 ukφk(s) = f(s). From Theorem 3.3 and Theorem 3.5 in [32], f is continuous ∓ ∓ P on [0, 1] if and only if sequence [fn(s)] (α) uniformly converge to [f(s)] (α) on [0, 1] [0, 1]. Series ∓ P × u (α)φ (s) converge uniformly on [0, 1] [0, 1] by Abel’s uniform convergence test in view of the facts that k k ×   P u∓(α) converge uniformly on [0, 1] [0, 1], • k × Pφ (s) is uniformly bounded on [0, 1] [0, 1] and φ (s) φ (s) for (α, s) [0, 1] [0, 1]. • { k } × k+1 ≤ k ∈ × So, f(s)= ukφk(s) is continuous on [0, 1] which implies ∞ ∞ P lim ukφk(s)= uk = µ. s→0+ Xk=0 Xk=0 This completes the proof.

In view of the result above we see that convergence of a series of fuzzy numbers implies its (φ) summability in fuzzy number space. However (φ) summability of a series of fuzzy numbers may not imply its convergence. Moti- vated by this fact now we aim to investigate conditions under which (φ) summability of a series of fuzzy numbers −λns λ0λ1...λn imply convergence in fuzzy number space in particular cases φn(s)= e and φn(s)= (s+λ0)(s+λ1)...(s+λn) · 3.1 Resummation of series of fuzzy numbers via generalized Definition 3.3. Given a sequence 0 λ < λ < , a series u of fuzzy numbers is said to be (A, λ) ≤ 0 1 ···→∞ n summable to a fuzzy number µ if generalized Dirichlet series u e−λns converges for s> 0 and n P ∞ P −λns lim une = µ. s→0+ n=0 X We note that a of fuzzy numbers is (A, λ) summable by Theorem 3.2 but converse statement is not necessarily to hold which can be seen by series un of fuzzy numbers whose general term is defined by

(n + 1)2(t + ( 1)n+1), P( 1)n t ( 1)n + (n + 1)−2 2 − n+1 − n ≤ ≤ − −2 n −2 un(t)= 2 (n + 1) t + ( 1) , ( 1) + (n + 1) t ( 1) + 2(n + 1)  − − − ≤ ≤ −  0,  (otherwise). 

4 On the resummation of series of fuzzy numbers via generalized Dirichlet and generalized factorial series

∞ Series n=0 un of fuzzy numbers is (A, ln(n + 1)) summable to fuzzy number

P 6(t−1/2) π2 π2 , 1/2 t 1/2+ 6 6(t−(1/2)) ≤ π2≤ π2 µ(t)= 2 2 , 1/2+ t 1/2+  − π 6 ≤ ≤ 3 0, (otherwise), but it is not convergent. 

Theorem 3.4. If series u of fuzzy numbers is (A, λ) summable to fuzzy number µ and λn D(u , 0)¯ = n λn−λn−1 n o(1), then u = µ. n P λn Proof. LetP series un of fuzzy numbers be (A, λ) summable to µ and D(un, 0)¯ = o(1) as n . λn−λn−1 → ∞ ∞ −λ s ∞ −λ /λ Then, since lim + u e k = µ we have lim →∞ u e k n = µ in view of the fact that 1/λ 0 s→P0 k=0 k n k=0 k n ↓ as n . Hence, we get →∞ P P n n ∞ ∞ D u ,µ D u , u e−λk/λn + D u e−λk/λn ,µ k ≤ k k k k=0 ! k=0 k=0 ! k=0 ! X n X X ∞ X ∞ −λk/λn −λk/λn −λk/λn 1 e D(uk, 0)+¯ D(uk, 0)¯ e + D uke ,µ ≤ − ! Xk=0   k=Xn+1 Xk=0 n ∞ ¯ ∞ 1 λkD(uk, 0) λk λk−1 −λk/λn −λk/λn λkD(uk, 0)+¯ − e + D uke ,µ ≤ λn λk λk−1 λk ! Xk=0 k=Xn+1 − Xk=0 n ∞ 1 λk λk−1 −λk/λn = o(1) (λk λk−1)+ − e (λn − λk ) Xk=0 k=Xn+1 ∞ ∞ λ λ λ − o(1) k = o(1) k − k 1 e−λk/λn = e−λk/λn dx λk λn+1 k=n+1 k=n+1 Zλk−1 X∞ X o(1) λk o(1) ∞ λ = e−x/λn dx = e−x/λn dx = o(1) n λn+1 λk−1 λn+1 λn eλn+1 k=Xn+1 Z Z   = o(1), with agreements n 1 and λ− = 0, which completes the proof. ≥ 1 Replacing o(1) with O(1) in the proof of Theorem 3.4, we can get following theorem.

λn Theorem 3.5. If series un of fuzzy numbers is (A, λ) summable and D(un, 0)¯ = O(1), then (un) λn−λn−1 ∈ bs(F ). P 3.2 Resummation of series of fuzzy numbers via generalized factorial series

1 Definition 3.6. Given a sequence 0 < λ0 < λ1 < where diverges, a series un of fuzzy ··· → ∞ λn numbers is said to be summable by generalized factorial series to a fuzzy number µ if generalized factorial series P P unλ0...λn converges for s> 0 and (s+λ0)...(s+λn) ∞ P u λ . . . λ lim n 0 n = µ. s→0+ (s + λ0) . . . (s + λn) nX=0

5 On the resummation of series of fuzzy numbers via generalized Dirichlet and generalized factorial series

We note that a convergent series of fuzzy numbers is summable via generalized factorial series by Theorem 3.2 but converse statement is not necessarily to hold which can be seen by series un of fuzzy numbers where

(n + 1)4(t + ( 1)n+1(n + 1)), ( 1)n(n + 1) t P( 1)n(n + 1) + (n + 1)−4 4 − n+1 − n ≤ −≤4 − n −4 un(t)= 2 (n + 1) t + ( 1) (n + 1) , ( 1) n + (n + 1) t ( 1) n + 2(n + 1)  − − − ≤ ≤ − 0,  (otherwise). ∞  Series n=0 un of fuzzy numbers is summable via ordinary factorial series(with choice λn = n + 1) to fuzzy number P 90(t−1/4) π4 π4 , 1/4 t 1/4+ 90 90(t−(1/4)) ≤ π4≤ π4 µ(t)= 2 4 , 1/4+ t 1/4+  − π 90 ≤ ≤ 45 0, (otherwise), but it is not convergent.  Theorem 3.7. If series u of fuzzy numbers is summable to fuzzy number µ by factorial series and λ n 1 D(u , 0)¯ = n n r=0 λr n , then . o(1) un = µ P P  Proof. LetP series u of fuzzy numbers be summable by factorial series to µ and λ n 1 D(u , 0)¯ = o(1) n n r=0 λr n as n . Then, we get   →∞ P P n n ∞ ∞ ukλ0 . . . λk ukλ0 . . . λk D uk,µ D uk, + D ,µ ≤ (s + λ0) . . . (s + λk) (s + λ0) . . . (s + λk) k=0 ! k=0 k=0 ! k=0 ! X n X X X∞ λ0 . . . λk D(uk, 0)¯ λ0 . . . λk 1 D(uk, 0)¯ + ≤ − (s + λ0) . . . (s + λk) (s + λ0) . . . (s + λk) Xk=0   k=Xn+1 ∞ u λ . . . λ +D k 0 k ,µ (s + λ0) . . . (s + λk) ! Xk=0 n k ∞ ∞ 1 − s Pk 1 ukλ0 . . . λk s D(u , 0)¯ + D(u , 0)¯ e 2 r=0 λr + D ,µ ≤ k λ k (s + λ ) . . . (s + λ ) r=0 r 0 k ! Xk=0 X k=Xn+1 Xk=0 in view of the inequalities (see [33, p. 199], [34, p. 23]) n λ0 . . . λn − s Pn 1 λ0 . . . λn 1 < e 2 r=0 λr , 1 < s (s + λ ) . . . (s + λ ) − (s + λ ) . . . (s + λ ) | | λ · 0 n 0 n r=0 r X Now letting γ = k 1 , taking s = 1 and proceeding as in the proof of Theorem 3.4 we get k r=0 λr γn n n ∞ P 1 γ γ γk 2γ k k−1 − 2 n D uk,µ = o(1) (γk γk−1)+ − e γn = o(1) = o(1), ! (γn − γk ) √eγn+1 Xk=0 Xk=0 k=Xn+1   which completes the proof.

Replacing o(1) with O(1) in the proof of Theorem 3.7, we may get following theorem.

Theorem 3.8. If series u of fuzzy numbers is summable by factorial series and λ n 1 D(u , 0)¯ = n n r=0 λr n O(1), then (un) bs(F ).   ∈ P P

6 On the resummation of series of fuzzy numbers via generalized Dirichlet and generalized factorial series

4 Resummation of level Fourier series of fuzzy valued functions

Bede et al. [35] defined fuzzy Fourier sum of continuous fuzzy valued function f : [ π,π] RF by − → m a 0 + a cos(nx)+ b sin(nx) 2 n n n=1 X 1 π 1 π 1 π where a0 = π −π f(x)dx, an = π −π f(x) cos(nx)dx, bn = π −π f(x) sin(nx)dx and the are in the fuzzy Riemann sense [28]. The α level set of a convergent fuzzy Fourier series of f is of the form R − R R ∞ ∞ a − a + { 0}α + a cos(nx) − + b sin(nx) − , { 0}α + a cos(nx) + + b sin(nx) + . 2 { n }α { n }α 2 { n }α { n }α " n=1 n=1 # X X By Theorem 2.1 and Definition 2.2 , α level set of a convergent fuzzy Fourier series represents a fuzzy number. − However, it is a bit challenging to express a cos(nx) ∓ and b sin(nx) ∓ in terms of f −(x) and f +(x) ex- { n }α { n }α α α plicitly due to the effect of changing signs of sin(nx), cos(nx) on interval [ π,π](see (ii) of (2.1)). To tackle this − problem, we now define a new type Fourier series for fuzzy valued functions which may be useful in calculations − + via fα (x) and fα (x). Definition 4.1. Let f be a 2π-periodic continuous fuzzy valued function. Then, level Fourier series of f is defined by the pair of series

∞ a∓(α) 0 + a∓(α) cos(nx)+ b∓(α) sin(nx) (4.1) 2 n n n=1 X ∓ 1 π ∓ ∓ 1 π ∓ ∓ 1 π ∓ where a0 (α) = π −π fα (x)dx, an (α) = π −π fα (x) cos(nx)dx and bn (α) = π −π fα (x) sin(nx)dx. In (4.1), "-" and "+" signed series are called left and right level Fourier series of f, respectively. R R R − + Left and right level Fourier series of a fuzzy valued function f are expressed in terms of fα (x) and fα (x), respec- tively. Hence, using level Fourier series is advantageous in calculations through the endpoints of α level set of − f. However, it is disadvantageous in that the pair in (4.1) may not represent a fuzzy valued function(see Theorem 2.1). Besides, as the common problem of convergence of Fourier series, level Fourier series may not converge to ∓ fα (x) even in the pointwise sense. At this point we may use summation methods to achieve both fuzzification and convergence of level Fourier series. Let consider the summation method (A, n). Since (4.1) is of the form

∞ 1 π 1 π f ∓(x) f ∓(t)dt + f ∓(t)2 cos(n(x t))dt α ∼ 2π α 2π α − Z−π n=1 Z−π X ∞ 1 π 1 π = f ∓(x t)dt + f ∓(x t)2 cos(nt)dt 2π α − 2π α − Z−π n=1 Z−π ∞ X ˙∓ = fn (α, x) n=0 X where

1 π ∓ fα (x t)dt, n = 0 ˙∓ 2π −π − fn (α, x)=  π 1 R f ∓(x t) cos(nt)dt, n = 0,  π −π α − 6 (A, n) mean of level FourierR series of f is

π ∞ π 1 ∓ 1 f˙∓e−ns = f (x t)dt + f ∓(x t)2 cos(nt)dt e−ns n 2π α − 2π α − −π n=1 −π X Z X Z  7 On the resummation of series of fuzzy numbers via generalized Dirichlet and generalized factorial series

π ∞ 1 ∓ = f (x t) 1 + 2 cos(nt)e−ns dt 2π α − Z−π ( n=1 ) ∓ X 1 e−2s π f (x t)dt = − α − 2π 1 2e−s cos t + e−2s · Z−π − The pair f˙∓e−ns (α, x) satisfy the conditions of Theorem 2.1 for each x R,s > 0 and so represent the n ∈ fuzzy valued function nP o 1 e−2s π f(x t)dt − − 2π 1 2e−s cos t + e−2s · Z−π − Now we aim to show that 1 e−2s π f(x t)dt lim − − − − = f(x). s→0+ 2π 1 2e s cos t + e 2s Z−π − It will be convenient to utilize sequential criterion for limits in metric spaces and to use Theorem 2.4. Let (sn) be a sequence in R+ that converges to 0 and consider the sequence of fuzzy positive linear operators 1 e−2sn π f(x t)dt Ln(f; x)= − − 2π 1 2e−sn cos t + e−2sn · Z−π − Since −2sn π ∓ ∓ 1 e fα (x t)dt Ln f ; x = − − α 2π 1 2e−sn cos t + e−2sn Z−π −  we havee −sn −sn Ln(1; x) = 1, Ln(cos t; x)= e cos x, Ln(sin t; x)= e sin x, and followingly we get e e e L (f ) f = 0, L (f ) f = 1 e−sn 0, L (f ) f = 1 e−sn 0. n 0 − 0 n 1 − 1 − → n 2 − 2 − →

So by Theorem 2.4, we have L n(f) f. Since (sn) is arbitrary, we conclude e e → e 1 e−2s π f(x t)dt lim − − − − = f(x). (4.2) s→0+ 2π 1 2e s cos t + e 2s Z−π − Hence, fuzzification of level Fourier series (4.1) is recovered via (A, n) means and convergence is achieved via (4.2). On the other hand, by taking r = e−s above, Abel means of level Fourier series of f represents the fuzzy Abel-Poisson operator [36] 1 r2 π f(x t) P (f; x)= − − dt r 2π 1 2r cos t + r2 Z−π − and limr→1− Pr(f; x) = f(x) is achieved in view of (4.2). Fuzzy Abel-Poisson convolution operator satisfies conditions (4.4) and (4.5) of the fuzzy Dirichlet problem(in the polar coordinates) on the unit disc ∂2u 1 ∂u 1 ∂2u (r, x)+ (r, x)+ (r, x) = 0, x R, 0

8 On the resummation of series of fuzzy numbers via generalized Dirichlet and generalized factorial series

5 Conclusion

In current paper we have introduced a general summation method (φ) for series of fuzzy numbers via sequences of continuous functions and proved the regularity of (φ) method. We also obtained conditions which guarantee the convergence of a series of fuzzy numbers from its generalized Dirichlet and generalized factorial series. In (A, λ) −s summability method if we choose λn = n and use transformation x = e , results for Abel summability of series 1 of fuzzy numbers are obtained [23, Theorem 12, Theorem14]. Similarly, if we choose φn(s)= Γ(1+sn) we obtain regularity result for Mittag-Leffler summation method for series of fuzzy numbers by Theorem 3.2. Besides, in (A, λ) method if we choose λn = ln n and λn = n ln n(n 1) we get new results for summation of series of fuzzy numbers via ordinary Dirichlet series and via Lindelöf summ≥ ation method, respectively.

∞ Theorem 5.1. If series n=1 un of fuzzy numbers is summable to fuzzy number µ by ordinary Dirichlet series and n ln nD(u , 0)¯ = o(1), then u = µ. n P n ∞ ¯ Theorem 5.2. If series n=1Pun of fuzzy numbers is summable by ordinary Dirichlet series and n ln nD(un, 0) = O(1), then (u ) bs(F ). n ∈ P ∞ ¯ Theorem 5.3. If series n=1 un of fuzzy numbers is Lindelöf summable to fuzzy number µ and nD(un, 0) = o(1), then u = µ. n P TheoremP 5.4. If series ∞ u of fuzzy numbers is Lindelöf summable and nD(u , 0)¯ = O(1), then (u ) n=1 n n n ∈ bs(F ). P References

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