FUNCTIONAL SEQUENTIAL AND TRIGONOMETRIC SUMMABILITY OF REAL AND COMPLEX FUNCTIONS

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1 International Journal of Analysis Applications ISSN Volume 15, Number (17), -8 DOI: 1.894/ FUNCTIONAL SEQUENTIAL AND TRIGONOMETRIC SUMMABILITY OF REAL AND COMPLEX FUNCTIONS M.H. HOOSHMAND Abstract. Limit summability of functions was introduced as a new approach to extensions of the summation of real complex functions, also evaluating antidifferences. Also, limit summ functions generalize the (logarithm of) Gamma-type functions satisfying the functional equation F (x + 1) = f(x)f (x). Recently, another approach to the topic entitled analytic summability of functions, has been introduced studied by the author. Since some functions are neither limit nor analytic summable, several types of summabilities are needed for improving the problem. Here, I introduce study functional sequential summability of real complex functions for obtaining multiple approaches to them. We not only show that the analytic summability is a type of functional sequential summability but also obtain trigonometric summability (for functions with a fourier series) as another its type. Hence, we arrive at a class of real complex function spaces with various properties. Thereafter, we prove several properties of functional sequential, also many criteria for trigonometric summability. Finally, we state many problems future directions for the researches. 1. Introduction In the theory of indefinite sum, antidifference finite calculus, obtaining some special solutions of the difference functional equation F (x) := F (x) F (x 1) = f(x) ; x E, (1.1) is very important, where E is the domain of a real or complex function f or a subset of C (analogously for F (x) := F (x + 1) F (x) = f(x), where is the forward difference operator, e.g., see [1]). But the author discovered an approach to the solution of the equation (on 1) when he tried to generalize the Bohr-Mollerup theorem Gamma-type functions (see [,3,5]). The following are its summary. Let f be a real or complex function with domain D f N = {1,, 3, }. Put then for any x Σ f n N set Σ f = {x x + N D f }, R n (f, x) = R n (x) := f(n) f(x + n), n f σn (x) = f σl,n (x) := xf(n) + R k (x). The function f is called limit summable at x Σ f if the functional sequence {f σn (x)} is convergent at x = x. The function f is called limit summable on the set S Σ f if it is limit summable at all the points of S. Now, put f σ (x) = f σl (x) = lim f σ n (x), R(x) = R(f, x) = lim R n(f, x). n n Therefore D fσ = {x Σ f f is limit summable at x}, f σl = f σ is the same limit function of f σn with domain D fσ. The function f is called limit summable if it is summable on Σ f, R(1) = D f D f 1. In this k=1 Received 17 th September, 17; accepted 1 st October, 17; published 1 st November, Mathematics Subject Classification. 4A3; 39A1. Key words phrases. limit summability; analytic summability; antidifferences. c 17 Authors retain the copyrights of their papers, all open access articles are distributed under the terms of the Creative Commons Attribution License.

2 FUNCTIONAL SEQUENTIAL AND TRIGONOMETRIC SUMMABILITY OF FUNCTIONS 3 case the function f σ is referred to as the limit summ function of f. If f is limit summable, then D fσ = D f 1 f σ (x) = f(x) + f σ (x 1) ; x D f. Therefore, if f is limit summable then its limit summ function f σ satisfies (1.1). The paper [] states a framework for studying the limit summable functions including basic conditions, many criteria for limit summability, uniqueness conditions for its summ function so on. Example 1.1. If a < 1, then the complex function f(z) = a z is absolutely limit summable f σ (z) = a a 1 (az 1). Also, if < b 1 < a < 1, then the real function g(x) = ca x + log b x is limit summable g σ (x) = ca a 1 (ax 1) + log b Γ(x + 1). One can see the topic of limit summability in [,3]. Since some of the important special functions are not limit summable, recently, another type of summability entitled analytic summability has been introduced in [4]. We obtain it again in this paper (as a type of functional sequential summability).. Functional sequential summability Let δ = {δ n (x)} n= be a linearly independent functional sequence of complex (or real) functions on E C. Put S E (δ) := Span(δ) = Span(δ (x), δ 1 (x), δ (x), ). Similarly, if δ = {δ n (x)} n= β = {β n (x)} n= are two functional sequences on E such that δ β is linearly independent, then we may set S E (δ, β) := Span(δ β), (S E (δ 1,, δ k ) can be defined analogously). Now, let δ = {δ n (x)} n= be as mentioned = { n (x)} n= a functional sequence satisfying n (x) n (x 1) = δ n (x) ; x E, n N = {, 1,, }. (.1) Since there are infinitely many such functional sequence (because (1.1) has the general solution F (x) = F p (x) + ϕ(x) where F p is an its solution ϕ any 1-periodic function), we fix one of them. The equation implies each n (x) is defined on E (E 1). Also, since is linearly independent (see proof of Theorem.), we may put S σ E(δ; ) = S σ E(δ) := Span( ) = Span( (x), 1 (x), (x), ), so SE σ (δ) = S E( ). Note that we use the notation SE σ (δ) when is well-known in the topic there is no any risk of confusion. Example.1. If δ n (z) = z n, then δ n (z) n (z) = B n+1(z + 1) b n+1 n + 1 satisfies the conditions, where B n (z) is the Bernoulli polynomial b n = B n (1) (see [4]). Here S σ E (δ) is equal to the space of all polynomials with zero constant. Note. In continuation, we consider δ, E,, S E (δ) SE σ (δ) with the mentioned conditions. Lemma.1. We have f S E (δ) if only if there exists a unique F SE σ (δ) satisfying the difference functional equation (1.1). (Notation. Hence we shall denote F by f σδ, or simply f σ when there is no any risk of confusion.) Proof. If f S E (δ) then there exist δ n1,, δ nr δ coefficients c n1,, c nr such that f = r 1 c n k δ nk. Putting F = r 1 c n k nk we have F SE σ (δ) F satisfies the equation. Also, if G SE σ (δ) satisfies (.1) then G = t 1 d n j nj, for some coefficients d m1,, d mt, r t c nk δ nk = F (x) F (x 1) = f(x) = G(x) G(x 1) = d mj δ nj (x) ; x E. k=1 j=1

3 4 M.H. HOOSHMAND Hence, independence of δ implies t = r {c n1,, c nr } = {d m1,, d mr }. Therefore, G = r c nk nk = F. 1 The converse is obvious (by (.1)). Definition.1. Let f(x) = c nδ n (x) be convergent on E. We say the function f is (δ, )- summable (or simply δ-summable) if the series c n n (x) is convergent on E (the terms uniformly absolutely δ-summable are defined analogously). Moreover, if it is the case, then we may put f σδ (x) = f σ (x) := c n n (x), call it δ-summ function of f on E (this symbol agrees with the above notation, i.e., if f S E (δ) then the function f σ that is gotten from Lemma. this definition are the same). Now, we set S E (δ) := {f : E C f is δ-summable} S σ E(δ) := {f σδ f S E (δ)} It is obvious that S E (δ) S E (δ) the space of all functions with a Fourier series on E, S σ E (δ) S σ E(δ). Theorem.1. The sets S E (δ) S σ E(δ) are functions spaces the map σ = σ δ : S E (δ) S σ E(δ), defined by σ(f) := f σ, is a surjective linear map with SE σ (δ) σ(s E(δ)). Also, we have (a) f σ satisfies the difference functional equation (1.1). (b) The surjective linear map σ is bijective ( so S E (δ) = S σ E(δ)) if only if δ = {δ n (x)} n= is infinitely linearly independent (i.e., c nδ n (x) = on E implies c n =, for all n N, e.g., δ n (x) = x n or δ n (x) = e inx ). Therefore, if this is the case then SE σ (δ) = σ(s E(δ)). Proof. It is obvious that S E (δ) S σ E(δ) are vector spaces σ is a surjective linear map from S E (δ) to S σ E(δ). Then, Lemma. implies S σ E (δ) σ(s E(δ)). Now, note that linearly independence (resp. infinitely linearly independence) of δ on E implies linearly independence (resp. infinitely linearly independence) of on E (E 1). Because if c nj nj (t) = for all t E (E 1) then cnj nj (x) =, cnj nj (x 1) = ; x E. Therefore cnj ( nj (x) nj (x 1)) = c nj δ nj (x) = ; x E, so all c nj are zero. Now, if f, g S E (δ) then f(x) = c nδ n (x), g(x) = d nδ n (x), for all x E, f σ (x) = c n n (x), g σ (x) = d n n (x), for all x E (E 1). So f σ (x) f σ (x 1) = c n ( n (x) n (x 1)) = c n δ n (x) = f(x) ; x E. Also, if σ(f) = σ(g), then (c n d n ) n (x) = on E (E 1) so c n d n = (because δ is infinitely linearly independent), which means σ is injective. Hence the proof is complete.

4 FUNCTIONAL SEQUENTIAL AND TRIGONOMETRIC SUMMABILITY OF FUNCTIONS 5.1. Analytic summability. Let δ n (z) = z n (z C) consider all terms of Example.1 on an open domain D = E C. Then S D (δ) is a subspace of all analytic functions on D. Also, f(z) = c nz n S D (δ) is δ-summable if only if the series B n+1 (z + 1) b n+1 c n n + 1 is convergent on D, this is the same Analytic summability of f from [4]. Therefore (according to [4]) we denote it by f σa instead of f σδ call this f Analytic summable, also denote the space by S D (A). Theorem.4 implies σ A is a surjective linear map from S D (A) to S σ D(A), so S D (A) = S σ D(A) that is a new result about analytic summability. One can see many criteria, results open problems about this type of functional sequential summability in it. Example.. The exp function is analytic summable in C 1 N n+1 1 n! exp σ (z) = n! σ(zn ) = lim N n! k!(n + 1 k)! b n+1 kz k So exp σa (z) = n= N+1 = lim N N k=n 1 = e e 1 n= k=1 1 b k+1 n n! (k + 1 n)! zn = e e 1 (ez 1) (details can bee seen in [4]). z n n! = e e 1 (ez 1). 3. Trigonometric summability 1 n! ( j= b j j! )zn Now, we can introduce another important type of functional sequential summability in two ways: (a) Consider γ = {e inx } n= (x R, n Z). Here, the indices set N is replaced by Z. Therefore, in the definition of functional sequential summability, f has the Fourier series form f(x) = c ne inx. Putting Γ n (x) = ein e in 1 (einx 1) for n Γ (x) = c x, one see that Γ n satisfies the conditions (for all real numbers x). Hence, f is γ-summable on E if only if the following series is convergent c n Γ n (x) ; x E. (3.1) (b) Put δ = {cos(nx)} n= β = {sin(nx)} n= (x R). The sequence δ β is infinitely linearly independent. For every positive integer n, put n (x) = cos(nx) + cos(n) cos(nx + n) 1, (1 cos(n)) sin(nx) + sin(n) sin(nx + n) B n (x) =, (1 cos(n)) also (x) = 1, B (x) =. It is easy to see that they satisfy the conditions. It is obvious that f is δ, β-summable on E (i.e., f S E (δ, β)) if only if it has the Fourier series of the form f(x) = a + a n cos(nx) + b n sin(nx) the following series is convergent a n n (x) + b n B n (x). (3.) Now, let f(x) = n= c n e inx = a + a n cos(nx) + b n sin(nx). It is easy to see that the series (3.1) is convergent if only if (3.) is convergent, so we arrive at the following definition.

5 6 M.H. HOOSHMAND Definition 3.1. Let E, f, γ δ, β be as the above. We call f trigonometric summable (on E) if the series (3.) (or equivalently (3.1)) is convergent. Also, we put f σtrg (x) = f σ (x) := a x + a n n (x) + b n B n (x) = c x + c n Γ n (x) ; x E, call it trigonometric summ of f (on E). Now, we prove many equivalent conditions for trigonometric summability also several criteria for trigonometric summability of functions with a Fourier series. Theorem 3.1. (Some criteria for trigonometric summability of real complex functions) Let f(x) = a + a n cos(nx) + b n sin(nx) = c n e inx is defined on E R. For n put A n = a n cot( n )b n, B n = b n + cot( n )a n, C n = c n i cot( n )c n. Also, set A = a, B = C =. Then (a) The following statements are equivalent: - f is trigonometric summable on E; - The series A n cos(nx) + B n sin(nx) A n is convergent (to f σ (x) a x) on E ; - The series C ne inx C n is convergent (to f σ (x) c x) on E; - The series csc( n nx ) sin( )(a n cos(n x+1 ) + b n sin(n x+1 )) is convergent (to f σ(x) 1 a x). (b) If the Fourier series B n cos(n x ) A n sin(n x ) (i.e., the series is gotten from the Fourier series of f, replacing a n, b n x by B n, A n x respectively) is absolutely convergent, then f is absolutely trigonometric summable f σ (x) 1 a x B n cos(n x ) A n sin(n x ). (c) If the Fourier series a n x+1 sin( n ) cos(n ) + bn a Fourier series of f, replacing a n, b n x by n sin( n sin( n ), x+1 ) sin(n ) (i.e., the series is gotten from the b n x+1 sin( n ) respectively) is absolutely convergent, then f is absolutely trigonometric summable f σ (x) 1 a a n x sin( n ) cos(nx + 1 ) + b n sin( n ) sin(nx + 1 ). Proof. By using the identity sin(n) 1 cos(n) = cot( n ), we obtain (for all n 1) n (x) = cos(nx) 1 + cot( n ) sin(nx) = sin(nx ){cot(n ) cos(nx ) sin(nx )} = csc( n ) sin(nx + n ) 1 = csc(n ) sin(nx ) cos(nx + n ), B n (x) = sin(nx) + cot( n )(1 cos(nx)) = sin(nx ){cos(nx ) + cot(n ) sin(nx )} = cot( n ) csc(n ) cos(nx + n ) = csc(n ) sin(nx ) sin(nx + n ), Therefore, Γ n (x) = (1 i cot( n ))(einx 1). a n n (x) + b n B n (x) = A n cos(nx) + B n sin(nx) A n = C n (e inx 1) =,n c n Γ n (x),

6 FUNCTIONAL SEQUENTIAL AND TRIGONOMETRIC SUMMABILITY OF FUNCTIONS 7 A n (cos(nx) 1) + B n sin(nx) = sin(n x ){B n cos(n x ) A n sin(n x )} = Now one can obtain the results, directly. sin( nx )( a n sin( n ) cos(nx + 1 ) + b n sin( n ) sin(nx + 1 )). The above theorem together with the identity A n + Bn = (a n + b n) csc ( n ) imply the following corollary. Corollary 3.1. Let a n b n are two sequences of real or complex numbers. a n +b n (a) If one of the series a n + b n sin( n ), sin( n ) is convergent, then the function f(x) := a + a n cos(nx) + b n sin(nx) (is defined in R) is absolutely uniformly trigonometric summable on whole R f σ (x) = 1 a x + 1 A n cos(nx) + B n sin(nx) A n = 1 a x + csc( n nx ) sin( )(a n cos(n x+1 ) + b n sin(n x+1 )) = 1 a x + = c x + sin(n x )(B n cos(n x ) A n sin(n x )) C ne inx C n f σ(x) = Hence, f σ(x) has a Fourier series in R. nb n cos(nx) na n sin(nx). Note. If a n cot( n )b n (= A n) is convergent then trigonometric summability of the function f(x) = a + a n cos(nx) + b n sin(nx) is equivalent to convergence of the Fourier series A n cos(nx) + B n sin(nx), so f σ (x) 1 a x (if exists) has a Fourier series. Example 3.1. Consider the real function f(x) := 1 + sin( n ) n cos(nx) + sin( n ) n sin(nx) It is defined on whole R by using the above results, f is absolutely uniformly trigonometric summable f σ (x) = x π ( csc( n ) cos( n ) n ) cos(nx) + ( cos( n ) + csc( n ) n ) sin(nx) Also f σ(x) = 1 + = x + csc ( n ) n sin( nx )(cos(nx + 1 = x + C n e inx C n. ) + sin(n x + 1 )) ( cos( n ) + csc( n ) ) cos(nx) + ( cos( n ) csc( n ) ) sin(nx) ; x R, n n f σ (x) x n = π 3 f σ(x) π + x ; x R. 3

7 8 M.H. HOOSHMAND At last, there are some important questions for future studies of trigonometric, analytic limit summability of functions (similar to the open problems of [4]). Open problem 1. Let f be an analytic function defined on open domain D = E. If f is both analytic trigonometric summable, then is it true that f σtrg = f σa on D? Open problem. Let f be a function with a Fourier series on E with the property N E Σ f. If f is both limit trigonometric summable, then is it true that f σl = f σtrg on E? Open problem 3. If f is trigonometric summable on E = D f, then under what conditions is it a unique solution of the equation (1.1) with the initial property f σ () =? (compare to the uniqueness Theorem 3.1, Corollary 3.4 of [] Theorem A, Corollary 3.4 of [3]). Finally, as another future direction for the researches, one may study discover other functional sequential summability (by taking some appropriate functional sequences δ = {δ n (x)} n= ), also intersection of the spaces of limit, trigonometric analytic summable functions. Note that every constant function f(x) = c lies in the intersection σ l (c) = σ A (c) = σ rg (c) = cx. References [1] P.M. Batchelder, Introduction to Linear Difference Equations, Dover Publications Inc., [] M.H. Hooshm, Limit Summability of Real Functions, Real Anal. Exch. 7(1), [3] M.H. Hooshm, Another Look at the Limit Summability of Real Functions, J. Math. Ext. 4(9), [4] M.H. Hooshm, Analytic Summability of Real complex Functions, J. Contemp. Math. Anal., 5(16), [5] R. J. Webster, Log-Convex Solutions to the Functional Equation f(x + 1) = g(x)f(x): Γ -Type Functions, J. Math. Anal. Appl., 9 (1997), Young Researchers Elite Club, Shiraz Branch, Islamic Azad University, Shiraz, Iran Corresponding author: hadi.hooshm@gmail.com