CHARACTERIZATION OF THE FOURIER SERIES OF A DISTRIBUTION HAVING A VALUE AT A POINT

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 4, April 1996 CHARACTERIZATION OF THE FOURIER SERIES OF A DISTRIBUTION HAVING A VALUE AT A POINT RICARDO ESTRADA (Communicated by J. Marshall Ash) Abstract. Let f be a periodic distribution of period 2π. Let aneinθ be its Fourier series. We show that the distributional point value f(θ )exists and equals γ if and only if the partial sums x n ax aneinθ converge to γ in the Cesàro sense as x for each a>. 1. Introduction The study of the relationship between the local behavior of a periodic function at a point and the convergence and summability of the corresponding Fourier series has been one of the main concerns of classical harmonic analysis, a subject with many years of history. In particular, the characterization of the Fourier series having a value at a point is a very interesting problem, whose solution depends on the notion of value used. Among the various useful notions of value we could mention the value of a continuous function at a point, the measure-theoretical approximate value of an integrable function, or the distributional point value of a generalized function. In this article we shall be concerned with the characterization of the Fourier series of generalized functions having a distributional point value. The notion of distributional point value was introduced by Lojasiewicz [8] and corresponds, roughly, to the existence of value on the average. If f D (R) is a distribution and x R, we say that f has the distributional value γ at x = x,andwritef(x )=γin D if for each φ Dwehave lim ε f(x +εx),φ(x) =γ φ(x) dx (if f is given by an integral and φ(x) dx = 1 this means that the average f(x + εx)φ(x) dx tends to γ). It can be shown that f(x )=γin D if and only if there exists a primitive of order k of f, i.e., F (k) = f, which is continuous in a neighborhood of x = x and satisfies lim x x k!(x x ) k F(x)=γ. This corresponds to the notion of generalized derivatives (see [13] and the references therein) used in the theory of trigonometric series well before the definition of Lojasiewicz and even before the introduction of the notion of distributions by Schwartz [1]. It turns out that the existence of distributional point values is equivalent to the existence in the Cesàro sense of the limits of certain partial sums of the corresponding Fourier series. Convergence in the Cesàro sense is also a kind of convergence Received by the editors May 2, 1994 and, in revised form, October 18, Mathematics Subject Classification. Primary 46F1, 42A24. c 1996 American Mathematical Society 125

2 126 RICARDO ESTRADA on the average and is defined as follows [6]: if f is an integrable function defined for x wesaythatf(x) tends to L in the (C, 1) sense as x +, 1 x and write lim x + F (x) =L(C, 1) if lim x + x F (t) dt = L. Convergence in the (C, k) sense, k =2,3,..., is defined recursively by lim x F (x) =L(C, k) 1 x if lim x + x F (t) dt = L (C, k 1). We say that F (x) convergestolin the Cesàro sense and write lim x + F (x) =L(C) if lim x + F (x) =L(C, k) for some k. The Cesàro convergence of a sequence {x n } can be defined similarly and it is equivalent to the Cesàro convergence of the function F defined by F (t) =x [[t]].in particular a series n=1 a n is Cesàro summable to the sum S, written as n=1 a n = n S (C), if lim n j=1 a j = S (C). A series a n is (C) summable to S if both series n= a n and n=1 a n are (C) summable to S 1 and S 2, respectively, with S = S 1 + S 2. A series a n is principal value Cesàro summable to S, written as p. v. a n n = S (C), if lim n j= n a j = S (C). Now let f S be a periodic distribution of period 2π and let a ne inθ be the associated Fourier series. Let θ R. Then the following two results are known [3, 12]: (1) If a ne inθ = γ (C), i.e., if the partial sums M n= N a ne inθ tend to γ in the Cesàro sense as N and M tend to infinity independently, then f(θ )=γ in D. The converse does not hold (f(θ) = n=2 einθ (2) If f(θ )=γin D,thenp.v. a ne inθ lim N N n= N The converse does not hold (f(θ) = Here we prove the following a n e inθ = γ (C). n=1 n log n,atθ=,isanexample). = γ (C), i.e., n sin nθ, atθ=,isanexample). Theorem. Let f S be a periodic distribution of period 2π and let a ne inθ be its Fourier series. Let θ R. Then (1.1) f(θ )=γ, in D, if and only if there exists k such that (1.2) for each a>. lim x + x n ax a n e inθ = γ, (C, k) Our approach is based on the theory of distributional asymptotic expansions [4, 5, 11] and is inspired by the work of Ramanujan [9] who was one of the first to study a sequence {a n } by studying the asymptotic behavior of series of the type n=1 a nφ(nε), as ε +,forsmoothφ. See also [1, 2]. Here we study the Fourier series a ne inθ by analysing the behavior as ε oftheseries a ne inθ φ(nε) forφ S. Notice that we use the standard notation concerning spaces of distributions and test functions [7, 1].

3 THE FOURIER SERIES OF A DISTRIBUTION Asymptotically homogeneous functions In this section we define and give the basic properties of the asymptotically homogeneous functions of degree. These functions play a central role in our analysis. Lemma 1. Let τ be a real-valued continuous function defined in an interval of the form [A, ) for some A R. Suppose (2.1) τ(ax) =a µ τ(x)+o(1), as x +, for each a>.ifµ<,then (2.2) τ(x)=o(1), as x +. Proof. Let ε>. Let x > be such that τ(2x) 2 µ τ(x) ε for x>x. Let M =max{ τ(x) : x x 2x }. An inductive argument shows that if x [2 n x, 2 n+1 x ], n =,1,2,...,then τ(x) 2 nµ M + n 1 j=1 2jµ ε. Therefore, lim x + τ(x) 2µ 1 2 εand (2.2) follows. µ The lemma does not hold if µ =. Indeed, there are functions like ln(ln x), ln x α, α<1, or cos ln x that satisfy τ(ax) =τ(x)+o(1), as x +, foreach a>, but which do not tend to zero at infinity. Definition. Let τ be a continuous function defined in an interval of the form [A, ) for some A R. We say that τ is asymptotically homogeneous of degree if for each a>wehave (2.3) τ(ax) =τ(x)+o(1), as x +. The asymptotically homogeneous functions of degree are related to the slowly oscillating functions [5], also known as regularly varying functions of order. These are positive functions that satisfy ρ(ax) =ρ(x)+o(ρ(x)), as x,foreach a >. However, the two concepts are different: ln x is slowly oscillating but not asymptotically homogeneous of degree while cos ln x is asymptotically homogeneous of degree but not slowly oscillating. Observe that the argument of the proof of Lemma 1 shows that if τ is an asymptotically homogeneous function of order, then τ(x) =o(ln x), as x +. Notice also that we did not ask for any uniform behavior with respect to a in (2.3). The fact that (2.3) holds uniformly on a [A, B] if[a, B] (, ) follows from the definition, as we shall see. Lemma 2. Let τ be an asymptotically homogeneous function of degree, continuous in [, ). LetHbe the Heaviside function. Then (2.4) τ(λx)h(x) =τ(λ)h(x)+o(1), as λ, in S. If [A, B] (, ), then(2.3) holds uniformly for a [A, B]. Proof. Suppose first that τ is bounded in [, ). Then if φ S we can apply the Lebesgue dominated convergence theorem to obtain lim λ (τ(λx) τ(λ))φ(x) dx =. This is (2.4). The uniform convergence follows from the distributional formula (2.4). Indeed, weak convergence in S implies strong convergence [1]. Thus if K is a

4 128 RICARDO ESTRADA compact subset of S, then τ(λx)h(x) τ(λ)h(x),φ(x) = o(1) uniformly for φ K. Let φ S be a function that satisfies φ (x) dx = 1. Then if [A, B] (, ) thesetk={a 1 φ (a 1 x):a [A, B]} is a compact set of S. Then τ(λx)h(x),φ (x) =τ(λ)+o(1) as λ,andsoτ(λa) = τ(λax),φ (x) + o(1) = τ(λx),a 1 φ (a 1 x) +o(1) = τ(λ)+o(1) uniformly for a [A, B]. Let us now return to the general case when τ is not necessarily bounded. We shall first show that if [A, B] (, ), then τ(ax) =τ(x)+o(1), as x +, uniformly on a [A, B]. Observe first that the functions cos τ(x) andsinτ(x) are bounded asymptotically homogeneous functions. By what we have already proven it follows that if [A, B] (, ), then e iτ (ax) = e iτ (x) + o(1), as x, uniformly on a [A, B]. Let ε>. Suppose ε<πand A<1<B. There exists x > such that e iτ (ax) e iτ (x) 1 e iε for x x and a [A, B]. For each x x the set {τ(ax) :A a B}is a connected set contained in [τ(x) ε +2nπ, τ(x) +ε+2nπ] and it follows that it is contained in the component that contains τ(x); that is, τ(ax) τ(x) ε,fora [A, B]. To prove (2.4) it would be enough to prove that there are constants A,A 1 such that τ(λx) τ(λ) A ln x + A 1 for x>andλ>λ for then the dominated convergence theorem could be invoked again. Let λ > be such that τ(λx) τ(λ) 1ifλ λ and x [1/2, 2]. Then it follows by induction that τ(λx) τ(λ) n+1ifx [2 n, 2 n+1 ],λ λ, n = ln x,1,2... Therefore τ(λx) τ(λ) ln 2 +1 if x 1,λ λ. Proceeding similarly, ln x ln if x [ 2, 1 n+1 2 ],λ 2 n λ n, n=,1,2,...,then τ(λx) τ(λ) n+1 Recall now that τ(x) =o(ln x), as x. Then we can find constants M,M 1 such 1 that τ(x) M ln x + M 1,x>. Therefore, if x [ 2, 1 n+1 2 ] but λ<2 n λ n,then λ<λ /x and so τ(λx) τ(λ) 2(M ln λ +M 1 ) 2(M ln x +M ln λ +M 1 ). Summarizing, if A =max{2m,1/ln 2},A 1 =max{2(m ln λ + M 1 ), 1}, then τ(λx) τ(λ) A ln x + A 1 for x>andλ λ. The asymptotically homogeneous functions of degree do not have to be smooth, i.e., C. However, we have Lemma 3. Let τ be an asymptotically homogeneous function of degree, continuous in [, ). Then there exists a function σ, asymptotically homogeneous of degree, smoothin[, ), such that (2.5) τ(x) =σ(x)+o(1), as x. Proof. Define σ(λ) = τ(λx)φ (x) dx, λ 1, where φ Ssatisfies φ (x) dx = 1, and extend to [, + ) in any smooth way. Then (2.4) gives (2.5). 3. Primitives of a distributionally null sequence If {f n } is a sequence of continuous functions that converge uniformly to zero on an interval of the form [ A, A], then the sequence of primitives { x f n(t) dt} also converges uniformly to zero while, in general, the sequence of derivatives {f n}, even if defined, does not. On the other hand, the situation with the distributional convergence is the opposite: if f n ins,thenalsof n ins but the sequence of primitives { x f n(t) dt}, even if defined, might be divergent. We now turn our attention to the study of this last problem. Let F S. If F is integrable near x =, then we can define the primitive F (t) dt. If F is more singular near x =, then there is no canonical way to x

5 THE FOURIER SERIES OF A DISTRIBUTION 129 define x F (t) dt, that is, there is no canonical way to single out a primitive of F that vanishes at x =. However, if F is locally even near x =,ithasa unique locally odd primitive, which we denote as x F (t) dt. Observe that with this notation x δ(t) dt = 1 2 sgn x, wheresgnx=x/ x,x, is the sign function. We shall use the notation x F (t) dt so long as F = F + F 1,whereF is locally integrable near x =andwheref 1 is locally even. Let {F n } n=1 be a sequence of distributions of S. Suppose that lim n F n = in S. Then [1] there are primitives of the F n that also tend to zero. In particular, if x F n(t) dt is defined for each n, there are constants c n such that x F n(t) dt = c n + o(1), as n in S. The example F n (x) = 2nxe x2 (1 e x2 ) n 1, x F n(t) dt = 1 (1 e x2 ) n =1+o(1) shows that the c n s cannot be replaced by, in general. Observe also that if the F λ depend smoothly on the parameter λ and F λ = o(1), as λ,then x F λ(t)dt = σ(λ)+o(1), where σ is smooth. Lemma 4. Let F S be a Radon measure such that x F (t) dt is defined. Define the distributions F n, n 1, recursively by F n (x) = x F n 1(t)dt, n 1. Suppose (3.1) F (λx) =o(1/λ), as λ +, in S. Then there exists an asymptotically homogeneous function of degree,σ(λ),such that F n (λx) = λn 1 x n 1 σ(λ) (3.2) + o(λ n 1 ), as λ +, in S, (n 1)! for n 1. There exists n such that the convergence in (3.2) is uniform on compacts for n n. Conversely, if (3.2) holds for some n 1, thenf (λx) =o(1/λ) in S. Proof. Suppose F (λx) =o(1/λ) ins. Then there exists a smooth function σ(λ) such that F 1 (λx) =σ(λ)+o(1), as λ,ins. Replacing λx by λxa and grouping in two different ways, we obtain σ(aλ) =σ(λ)+o(1), as λ for each a>. Thus σ is asymptotically homogeneous of degree. Hence (3.2) holds for n = 1. Suppose now it holds for some n 1. Then integrating again we obtain F n+1 (λx) =λ n x n σ(λ)/n! +ρ(λ)+o(λ n ), as λ, for some function ρ. Evaluating at λax thus yields ρ(λa) =ρ(λ)+o(λ n ) and thus by Lemma 1, applied to λ n ρ(λ), it follows that ρ(λ) =o(λ n ) and (3.2) is obtained for n +1. That the convergence in (3.2) is uniform for x in compacts if n is large follows by the definition of the convergence of distributions. The converse is obtained by differentiating (3.2) n times with respect to x. Observe that if F is even, then F 1 is odd and thus σ(λ) =o(1), as λ,and (3.2) becomes F n (λx) =o(λ n 1 ), as λ. The same conclusion is obtained if supp F [, ). The sequence of functions {F n } n= is also related to Cesàro summability. The following lemma is an immediate corollary of the results proved in [6, p. 11]. Lemma 5. Let F be a function defined for x>that, suitably extended to R, defines an element of S for which x F (t) dt is defined. Then lim x + F (x) =γ (C, n) if and only if lim x n! x n F n (x) =γ. 4. The main result In this section we apply the results of the previous sections to characterize the Fourier series of the periodic distributions having a distributional point value.

6 121 RICARDO ESTRADA Lemma 6. Let f S be a periodic distribution of period 2π and let a ne inθ be its Fourier series. Let θ R. Then (4.1) if and only if f(θ )=γ, in D, (4.2) Proof. f(θ )=γ, in D a n e inθ δ(λx n) = γδ(x) λ + o(1/λ), as λ, in S. lim f(θ + εθ),φ(θ) = γ,φ(θ) φ D ε lim f(θ + εθ),φ(θ) = γ,φ(θ) φ S ε lim a n e inθ e inεθ,φ(θ) =γ φ(x) dx ε lim ε lim ε lim ε a n e inθ ˆφ(nε) =γˆφ() a n e inθ ψ(nε) =γψ() φ S ψ S a n e inθ δ(x nε) =γδ(x) in S a n e inθ δ(λx n) = γδ(x) λ φ S + o(1/λ), as λ in S. We are now ready to give our main result. Theorem. Let f S be a periodic distribution of period 2π and let a ne inθ be its Fourier series. Let θ R. Then (4.3) f(θ )=γ, in D, if and only if there exists k such that (4.4) lim a n e inθ = γ (C, k) x x n ax for each a>. Proof. Let F (x) = a ne inθ δ(x n) and define F 1,F 2,... recursively by F n+1 (x) = x F n(t)dt. Leta>, put G (x) =G (a, x) = x n ax a ne inθ and define G 1,G 2,... recursively by G n+1 (x) = x G n(t)dt. Observe that (4.4) means that lim x k! x k G k (x) =γ. Suppose first that f(θ )=γ.thenf (λx) =γδ(λx)+o(1/λ), as λ,ins. Using Lemma 4, there exists an asymptotically homogeneous function of degree such that if n 1, then F n (λx) = γsgn(λx)(λx)n 1 2(n 1)! + σ(λ)(λx)n 1 (n 1)! + o(λ n 1 ), as λ, in S

7 THE FOURIER SERIES OF A DISTRIBUTION 1211 and there exists n such that if n n this holds uniformly on x [ A, A] for A>. Thus, if n n, G n 1 (x) =a 1 n F n (ax) ( 1) 1 n F n ( x) = γxn 1 2(n 1)! + σ(x) (n 1)! + γ 2(n 1)! σ(x) (n 1)! + o(xn 1 ) = γxn 1 (n 1)! + o(xn 1 ) and (4.4) follows with k = n 1. Conversely, suppose (4.4) holds. Let n = k +1,sothat lim (n 1)! x x1 n G n 1 (x) =γ. Define τ(x) =(n 1)! x 1 n F n (x) γ/2. Then τ is asymptotically homogeneous of degree. Thus there exists a smooth asymptotically homogeneous function of degree such that τ(x) =σ(x)+o(1), as x. It follows that γsgn xxn 1 F n (x) = 2(n 1)! and therefore, by Lemma 2, + σ(x)xn 1 (n 1)! + o(x n 1 ), as x, F n (λx) = γsgn(λx)(λx)n 1 2(n 1)! + σ(λ)(λx)n 1 (n 1)! + o(λ n 1 ), as λ, in S. Differentiating n times we obtain F (λx) =γδ(λx)+o(1/λ), as λ, as required. It is interesting to observe that (4.4) holds uniformly on a [A, B] if[a, B] (, ) but that it is not necessary to assume such uniform behavior to obtain (4.3). Notice also that the theorem remains valid if (4.4) holds uniformly for a H where H is a set dense in some interval, H =[A, B], < A < B <. In particular, f(θ )=γif and only if (4.5) lim [[N/q]] N n= [[N/p]] a n e inθ = γ (C, k) for each p, q {1,2,3,...}, uniformly if A<p/q<Bfor [A, B] (, ). Observe that (4.5) considers the convergence, in the Cesàro sense, of sequences. The following consequences are worth recording. Corollary 1. If f is symmetric about θ = θ, i.e., f(θ θ )=f(θ θ),then f(θ )=γ in D if and only if a ne inθ = γ (C). Corollary 2. If f is antisymmetric about θ = θ, i.e., f(θ θ )= f(θ θ),then f(θ )=γin D if and only if there is a function σ, asymptotically homogeneous of degree, such that N n=1 a ne inθ = σ(n)+o(1) (C), asn. In this case γ =. Corollary 3. If the Fourier series of f is of the power series type, i.e., f(θ) = n= a ne inθ,thenf(θ )=γ in D if and only if n= a ne inθ = γ (C).

8 1212 RICARDO ESTRADA References 1. B. C. Berndt and R. J. Evans, Extensions of asymptotic expansions from Chapter 15 of Ramanujan s second notebook, J. Reine Angew. Math. 361 (1985), MR 87b: R. Estrada, The asymptotic expansion of certain series considered by Ramanujan, Appl. Anal. 43 (1992), MR 95i: , Valores puntuales distribucionales de series de Fourier, Memorias II Encuentro Centroamericano de Matemáticos, San Ramón, Costa Rica, R. Estrada and R. P. Kanwal, Asymptotic separation of variables, J. Math. Anal. Appl. 178 (1993), MR 94g: , Asymptotic analysi: A distributional approach, Birkhäuser, Boston, MR 95g: G. H. Hardy, Divergent series, Clarendon Press, Oxford, MR 11:25a 7. R. P. Kanwal, Generalized functions: Theory and technique, Academic Press, New York, MR 85f: S. Lojasiewicz, Sur la valeur et la limite d une distribution en un point, Studia Math. 16 (1957), MR 19:433d 9. S. Ramanujan, Notebooks, 2 vols., Bombay, MR 2: L. Schwartz, Théorie des distributions Hermann, Paris, MR 35: V. S. Vladimirov, Y. N. Drozhinov, and B. I. Zavyalov Multidimensional Tauberian theorems for generalized functions, Nauka, Moscow, MR 87m: G. Walter, Pointwise convergence of distributions, Studia Math. 26 (1966), MR 32: A. Zygmund, Trigonometric series, 2 vols., Cambridge Univ. Press, Cambridge and New York, MR 21:6498 Department of Mathematics, Texas A & M University, College Station, Texas Current address: P. O. Box 276, Tres Ríos, Costa Rica address: restrada@cariari.ucr.ac.cr