MATH48E FOURIER ANALYSIS AND ITS APPLICATIONS 7.. Cesàro summability. 7. Summability methods Arithmetic meas. The followig idea is due to the Italia geometer Eresto Cesàro (859-96). He shows that eve if the Fourier series of a give cotiuous fuctio does ot coverge, the ifiite sum produced by his average summatio method always coverge. Let us cosider the ifiite series u +u +u +u 3 +..., ad let s = u +u + +u. Defiitio. We say the above series is Cesàro summable to the limit σ (or summable by the method of arithmetic meas to σ) if lim σ s +s + +s = lim We ote that a series is Cesàro summable does ot ecessary imply that it is summable i the ordiary sese. For example, the very simple diverget series give by + +... = σ. gives Thus, s =, s =, s =,... s +s + +s = /, as. Hece although the diverget series is Cesàro summable. Coversely, we have Theorem 7.. If a series is coverget ad has a limit σ, the it is Cesàro summable to the same limit σ. Proof. Suppose we are give that lim s = σ. For ay give ɛ >, there is m > such that s σ < ɛ,
FOURIER ANALYSIS AND APPLICATIONS for all m. Cosider the differece σ σ = (s σ)+(s σ)+ +(s σ) = m (s j σ)+ (s j σ), j= j=m = (s j σ) j= where we have assumed that > m. Sice m is fixed, so we may choose so large such that Thus, σ σ m s j σ < ɛ. j= m j= s j σ + < ɛ + m j=m s j σ ɛ < ɛ + ɛ = ɛ, for > m, that is, whe is sufficietly large. This proves the theorem. Cesàro summable of Fourier series. If we defie f(x) a + a k coskx+b k sikx s (x) = a + a k coskx+b k sikx, σ (x) = s (x)+s (x)+...s (x) = a + ( k = s j (x) j= ) (a k coskx+b k sikx).
Thus, FOURIER ANALYSIS AND APPLICATIONS 3 σ (x) = s (x)+ s (x)+ + s (x) = π. f(x+u) si(+ )u si( u ) du + π f(x+u) si(+ )u si( u ) du + f(x+u) si( + )u π si( u ) = ( f(x+u) π si u si ( k + ) ) u k= du du. But So k= We obtai the ew represetatio that si ( u) ( ) si k+ u = cosku cos(k +)u. si ( k + ) ( ) u = cosku cos(k +)u si( u ) σ (x) = π = si( u )( cosu) = si ( u si( u ) = si ( u ) si( u ). f(x+u) si ( u ) si u du. ) We ote i the above cosideratio whe f(x) = ad hece σ (x) = give (7.) = π for each =,, 3,... si ( u ) si ( u ) du = π si u si u du Defiitio. Let f be defied o [a, b] with at most a fiite umber of discotiuities. The f is said to be absolutely itegrable if f(x) is absolutely itegrable over [a, b]. That is, exists. b a f(x) dx
4 FOURIER ANALYSIS AND APPLICATIONS Theorem 7.. The Fourier series of a absolutely itegrable fuctio f(x) of period π is Cesàro summable to the limit f(x) at every poit of cotiuity ad to the limit at every poit of jump discotiuity. f(x+)+f(x ) Proof. It will be sufficiet to prove (7.) lim π ad (7.3) lim sice they imply f(x+u) f(x+u) π si u si u si u si u du = f(x+) du = f(x ) lim σ (x) = f(x+)+f(x ) which gives lim σ (x) = f(x) at a cotiuity poit a x. We will oly prove the limit (7.) above. This is equivalet to the statemet lim π ( ) si u f(x+u) f(x+) π si u du =. Give ɛ >, there is δ > such that for < u δ, f(x+u) f(x+) < ɛ. Writig the above itegral i the form of the sum of two itegrals δ ad δ which we deote by I ad I respectively. Thus, Also, I = π ɛ π < ɛ π I = π π δ δ δ ( f(x+u) f(x+) ) si u si u si u si u si u si u du du = ɛ. du ( )si u f(x+u) f(x+) si u du π f(x+u) f(x+) du, si δ δ
FOURIER ANALYSIS AND APPLICATIONS 5 sice si δ si u for u [δ,π]. It follows that I < ɛ whe we choose to be sufficietly large (f is absolutely itegrable). This proves that I + I ca be made arbitrary small whe teds to ifiity. This establishes (7.). The remaiig itegral from to follows similarly. This completes the proof of the Theorem. Remark. We ote that the estimatio of I depeds o x. So the covergece of σ (x) is poitwise. Defiitio. The series σ (x) = u (x)+u (x)+... is said to be uiformly (Cesàro) summable o [a, b] to f, if give ɛ >, there is N such that σ (x) f(x) < ɛ for > N ad for all x [a, b]. k= We ca prove Theorem 7.3. The Fourier series of a absolutely itegrable fuctio f of period π is uiformly Cesàro summable to f o every [α, β] [a, b] where f is cotiuous. Proof. Let x [α, β]. We choose δ > so small such that x+u lyig withi [α, β]. We apply the (7.) ad to write σ (x) f(x) = ( ) si u f(x+u) f(x) π si u du = J +J. where J ad J stad for the itegrals ad respectively. We split the J ito the itegrals as the sum of δ ad δ as it the proof of the last theorem. Sice f is cotiuous over [α, β], so give ɛ >, we choose δ > such that f(x+u) f(x) < ɛ/. holds for all x [α, β] provided u < δ. Moreover, we ca defie M = max α x β f(x). It is easy to see that I < ɛ, x [α, β], O the other had, I = π πsi δ/ πsi δ/ M πsi δ/ δ ( )si u f(x+u) f(x) si u du π f(x+u) f(x) du δ f(x+u) du+mπ
6 FOURIER ANALYSIS AND APPLICATIONS for some costat M sice f is absolutely itegrable. Hece we ca choose N > large eough so that I < ɛ, x [α, β]. It follows that J < ɛ. Similarly, we ca show J < ɛ, so that J + J < ɛ for all x [α, β]. Previous cosideratio about covergece of Fourier series requires the f to be piecewise cotiuous ad absolutely itegrable. If, however, f is merely cotiuous, the Fourier series may diverge at certai poits. But we obtai Theorem 7.4. The Fourier series of a cotiuous fuctio f(x) of period π is uiformly Cesàro summable to f(x). Thus, Cesàro summable is both superior ad surprisig. It follows from our study of Cesàro summability that Theorem 7.5. If the Fourier series of a absolutely itegrable fuctio f coverges at a poit x of cotiuity (respectively a jump discotiuity), the the Fourier series must coverge to f(x) (respectively ( f(x+)+f(x ) ) ). We recall from a earlier theorem that a square itegrable fuctio f is completely defied by its trigoometric Fourier series. Theorem 7. implies that Theorem 7.6. Ay absolutely itegrable fuctio is completely determied (except for its values at a fiite umber of poits) by its trigoometric Fourier series, whether or ot the series coverges.
7.. Abel summability ad Poisso kerel. FOURIER ANALYSIS AND APPLICATIONS 7 Abel s summability. (N. H. Abel: Norwegia mathematicia (8-89)) Cosider the series (7.4) u +u + +u +..., ad (7.5) σ(r) = u +u r + +u r +... We assume that the series (7.5) coverges for < r < (which will always be the case if the terms of the series (7.4) are bouded) ad that the limit lim σ(r) = σ r exists. If this is the case, the we say that the series (7.4) is summable by Abel s method to the value σ. Abel s method ca be used to sum certai diverget series. For example, the series + +... already ecoutered i the begiig of this chapter is summable by by the method of arithmetic meas ad by Abel s method to the value σ =. I fact, the latter is give by case σ(r) = r+r r 3 + = +r ad therefore which agrees with the Cesàro sum. lim r σ(r) =, A importat questio is if the series (7.4) coverges, will Abel s method of summatio gives the sum of the series (7.5) that agrees with (7.4) as r. Theorem 7.7. If the series (7.4) coverges ad its sum equals σ, the the series is summable by Abel s method to the same umber σ. To prove this result, first we eed the followig lemma. Lemma 7.8. Let (7.4) ad (7.5) be the series defied above. If the series (7.4) is a coverget series (with real or complex terms), the (7.5) coverges for r, ad its sum σ(r) is cotiuous o the iterval [, ].
8 FOURIER ANALYSIS AND APPLICATIONS Proof. If σ = u +u + +u +..., the for every ɛ >, there exists a iteger N such that if N σ σ ɛ, where σ is the partial sum of the series (7.4). Cosider the remaider of the series (7.4) (7.6) R = σ σ = u + +u + + +u +, Sice σ σ ɛ/, we have u + +u + + +u +m = R R +m R + R +m ɛ for m =,,... Thus every partial sum of the series (7.6), just like its etire sum, is less tha ɛ i absolute value. Sice the umbers r +, r +,... decreasemootoically tozeroas forayr ( r < ), Abel slemma(lemma6.) isapplicable to the series R (r) = u + r + +u + r + +... It follows that this series, ad hece the series (7.5) coverges. Moreover, R (r) ɛr + ɛ for r <. Now let σ (r) deote the partial sum of the series (7.5) coverges. The we have (7.7) σ(r) σ (r) = R (r) ɛ for r <. Notig that σ() = σ ad σ () = σ, ad because σ σ ɛ/, we ca assert that the iequality (7.7) holds everywhere o the iterval r. This proves that the series (7.5) coverges uiformly o r ad hece implies that the fuctio σ(r) is cotiuous o r. We ow prove the theorem. Proof. If the series (7.4) coverges, the by the Lemma 7.8, the series (7.5) coverges ad its sum σ(r) is cotiuous o the iterval r. This meas that limσ(r) = σ() = σ, r which proves the theorem.
Poisso Kerel. Cosider FOURIER ANALYSIS AND APPLICATIONS 9 + r k coskφ ( r < ). We ca cosider the above series as the real part of the series + z k = + r k (coskφ+isikφ). where z = re iφ = r(cosφ+isiφ). We have Comparig the real parts yield + z k = + z z = +z ( z) = +rcosφ+irsiφ ( rcosφ ircosφ) = r +irsiφ ( rcosφ+r ) + r k coskφ = r rcosφ+r, r <. which is called the Poisso kerel. We ote that the kerel is i fact positive for all φ ad r < sice rcosφ+r = ( r) +4rsi φ/ >. Relatioship with Fourier series. We assume f to be absolutely itegrable. Suppose We defie f(x) a + (a k cosk +b k sikx). f(x, r) = a + r (a k cosk+b k sikx), r <. Sice f is assumed to be absolutely itegrable, the a k, b k as k. So oe ca fid a M > such that a k M, b k M, k N. Hece r (a k cosk+b k sikx) Mr k <
FOURIER ANALYSIS AND APPLICATIONS because r <. Writig the Fourier coefficiets of f i itegral form allows us to rewrite f(x, r) = a + r (a k cosk+b k sikx) = π = π f(t) dt+ π dt+ f(t) π = f(t) dt+ π π = π = π ( f(t) + f(t)r (cosktcoskx+siktsikx) dt f(t)r cosk(t x) dt f(t)r cosk(t x) dt ) r cosk(t x) dt r f(t) dt, r <, rcosφ+r where we have iterchaged the summatio ad the itegratio sigs above because the correspodig series coverges uiformly. The itegral (7.8) f(x, r) = π r f(t) dt, r <, rcos(t x)+r is called the Poisso itegral of f. I particular, we have, whe f(x), the a / =, a k = = b+k for all k ad f(x, r). Hece (7.9) = r dt, r <. π rcos(t x)+r Theorem 7.9. Let f(x) be a absolutely itegrable fuctio of period π. The at every cotiuity poit, ad limf(x, r) = f(x), r f(x+)+f(x ) limf(x, r) = r at every poit of jump discotiuity. Proof. Let u = t x i (7.8). The, as i a very similar i a previous cosideratio that we have (7.) f(x, r) = π r f(x+u) du, r <, rcosu+r i view of the itegral i (7.) beig periodic of period π. We ote that at a cotiuity poit x,
FOURIER ANALYSIS AND APPLICATIONS. Hece it is sufficiet to cosider the ad π lim r π f(x) = f(x+)+f(x ). r f(x+) f(x+u) du =, rcosu+r r f(x ) lim f(x+u) du = r π rcosu+r separately. But it would be sufficiet to cosider oly the first equatio sice the sice the secod oe ca be dealt with similarly. We ote that the (7.9) ca be writte as (7.) = π r du, r <. rcosu+r sice the itegrad is a eve fuctio. But the it would be sufficiet to prove (7.) lim r π [f(x+u) f(x+)] Let ɛ > be give. The there is a δ > such that r du =. rcosu+r f(x+u) f(x+) ɛ, < u δ. Hece π r [f(x+u) f(x+)] rcosu+r du = π δ [f(x+u) f(x+)] + [f(x+u) f(x+)] π δ = I +I, r rcosu+r du r rcosu+r du say. It follows from (7.) that I π ɛ π δ f(x+u) f(x+) r r rcosu+r du = ɛ. The estimate of the secod itegral I is of o less difficult: rcosu+r du
FOURIER ANALYSIS AND APPLICATIONS I r f(x+u) f(x+) du, 4πsi δ/ δ as r. Hece we ca choose r sufficietly close to such that I ɛ. This proves (7.) ad hece the theorem. The followig discusses about uiformity of Able s summatio. Theorem 7.. The Fourier series of a absolutely itegrable fuctio f(x) of period π is uiformly summable by Abel s method to f(x) o every iterval [α, β] lyig etirely withi a iterval of cotiuity [a, b]. We omit its proof. I particular, we have Theorem 7.. If f is a cotiuous fuctio of period π, the f(x, r) f(x) as r, uiformly for all x. We recall from the Theorem 6.5 that for a cotiuous fuctio f(x) of period π to have f (x) absolutely itegrable, while f (x) is cotiuous, the the Fourier series of f coverges to f (x). Now we have Theorem 7.. If a absolutely itegrable fuctio f(x) of period π has derivative up to order m, that is, f (m) (x) exists at x, the the series obtaied by term-wise differetiatio of the Fourier series of f(x) is summable by Abel s method to the value f (m) (x). We omit its proof. However, we explore its cosequece. We agai cosider the well-kow example: x = ( ) k+sikx, < x < π. k The the above theorem asserts that term-wise differetiatio gives ( ) k+ coskx for which the coefficiets do ot ted to zero. A further differetiatio yields ( ) k ksikx, for which the coefficiets clear ted to ifiity.
FOURIER ANALYSIS AND APPLICATIONS 3 Exercise Q Fid the followig sums by Cesàro s method: (i) + coskx, (ii) + ++ +. Q Let f(x) be square itegrable ad if a k, ad b k are the Fourier coefficiets of f. Suppose further that σ (x) is the th-cesàro sum, prove that π [σ (x) f(x)] dx = k (a k +b k )+ (a k +b k ). Q3 Let u k be Cesàro summable, ad let t = ku k. Show that (i) u k < ; t /. (ii) If u, the u k <. k=