Topics in Generalized Differentiation

Transcription

1 Topics in Generalized Differentiation J. Marsall As Abstract Te course will be built around tree topics: ) Prove te almost everywere equivalence of te L p n-t symmetric quantum derivative and te L p Peano derivative. 2) Develop a teory of non-linear generalized derivatives, for example of te form a n fx + b n )fx + c n ). 3) Classify wic generalized derivatives of te form a n fx + b n ) satisfy te mean value teorem. Lecture I will discuss tree types of difference quotients. Te first are te additive linear ones. Tese ave been around for a long time. One can see teir sadow already in te notation Leibnitz used for te dt derivative, d d dx d. For an example wit d = 2, let = dx, and consider te Scwarz generalized second derivative d 2 f x) f x + ) 2f x) + f x ) lim 2 = lim 2. ) Te difference quotient associated wit te generalized additive linear derivative as te form D uw f x) = lim D uw f x, ) uw f x, ) = lim = lim d d+e i= w if x + u i ) d. 2) 57

2 58 Seminar of Matematical Analysis Here f will be a real-valued function of a real variable. Te w i s are te weigts and te u i s are te base points, u > u > > u d+e. Tis is a generalized dt derivative wen w and u satisfy d+e j =,,..., d w i u j i =. 3) i= d! j = d Te scale of suc a derivative will be te closest tat two base points come wen is set equal to. Tus te scale of te Scwarz generalized second derivative is min { x + ) x, x + ) x ), x x ) } =. Tat is wy we will systematically write, for example, te tird symmetric Riemann derivative as f x lim ) 3f x + 2 ) + 3f x 2 ) f x 3 2 ) 3 4) instead of te completely equivalent f x + 3) 3f x + ) + 3f x ) f x 3) lim 8 3. In oter words, tere are an infinite number of ways to write te difference quotient for a single generalized derivative, but only at most 2 if te scale is fixed at. Note tat te substitution does not cange ), it does fx+) fx) cange te standard first difference quotient, Hencefort te scale will always be taken to be. We will systematically abuse notation, referring to any of te difference quotient, or its numerator, or its limit, as te derivative. Tus, te second symmetric Riemann derivative also called te Scwarz derivative) will refer eiter to te quotient fx ) fx).) or to its numerator f x + ) 2f x) + f x ) 2, f x + ) 2f x) + f x ), or to its limit as given in equation ) above.

3 J. M. As 59 Te tird symmetric Riemann derivative, given by te quotient 4) is an instance of te kt symmetric Riemann derivative. Tere is one of tese as well as a forward Riemann derivative for every natural number k. Te first tree forward Riemann derivatives are given by and f x + ) f x), f x + 2) 2f x + ) + f x) 2, f x + 3) 3f x + 2) + 3f x + ) f x) 3. Zygmund and Marcinkiewicz in te middle 93s did some of te best early work on generalized derivatives. Here is an important family of generalized derivatives tat tey introduced. Tey began wit a family of differences. = f x + ) 2 f x), 2 = [f x + 2) f x)] 2 [f x + ) f x)] = f x + 2) 2f x + ) + f x), 3 = [f x + 4) 2f x + 2) + f x)] 2 2 [f x + 2) 2f x + ) + f x)] 5) = f x + 4) 6f x + 2) + 8f x + ) 3f x). Te next one is formed by taking te tird difference at 2 and subtracting 2 3 times te tird difference from it; and tus as base points 8, 4, 2,, and. It sould be clear ow to inductively create te entire family of tese differences. Te dt difference corresponds to a dt derivative, after multiplication by a constant. Oter generalized derivatives sow up in numerical analysis. One measure for evaluating ow good a derivative is comes from considering Taylor expansions. If f is sufficiently smoot, we write f x + ) = f x) + f x) + f x) f x) 3 3! +...

4 6 Seminar of Matematical Analysis Substitute tis into, say, te tird difference 5) to get f x + 4) 3 = 6f x + 2) +8f x + ) 3f x) = f x) + 4f x) f x) f x) 3 3! f x) 6 2f x) f x) f x) 3 3! f x) +8 f x) +8 2 f x) f x) 3 3! f x). Observe tat te rigt and side simplifies to 4f x) so we get a generalized tird derivative ere wen we divide by 4 3. In oter words, 4 3 corresponds to a generalized tird derivative.) Tis calculation gives an explanation for te conditions 3). To make tis more explicit we will consider a very basic generalized derivative, te dt Peano derivative. Say tat f as an dt Peano derivative at x, denoted f d x) if for x fixed tere are numbers f x),f x),..., f d x) suc tat f x + ) = f x) + f x) + + f d x) d d )! + f d x) d d! + od ) 6) as. Tis notion, te notion of approximation near x to dt order, is very basic indeed. In fact a serious argument can be made tat tis is a better definition of iger order differentiation tan te standard one wic defines te dt derivative iteratively wit te dt derivative f d) x) being defined as te ordinary first derivative of f d ) x). I will ave more to say about tis later on. Note tat f x) = f x) if and only if f is continuous at x, and tat f x) and f x) are identical by definition. Also, if f d) x) exists, ten f d x) exists and equals f d x). 7) Tis is te content of Peano s powerful, but little known version of Taylor s Teorem.) Tis implication is irreversible for every d 2. I will give a simple example of tis in my next lecture. Here is a connection between te two generalizations considered so far. If f d x) exists, ten every D uw f x) wit w and u satisfying 3) exists and equals f d x). 8)

5 J. M. As 6 Tis fact follows immediately from substituting ) into 3) and intercanging te order of summation. Tis calculation is te basis for te conditions 3). Te implication 8) is also irreversible; for example, D 2, 2), ) x = at x =, but x is not differentiable tere. Tis D uw is te symmetric first derivative.) Neverteless, te implication 8) is more reversible tan is te implication 7). Muc of wat I ave to say in tese lectures will be about te possibilities of partial converses to implication 8), to generalizations of implication 8), and to partial converses of tose generalizations. From a certain numerical analysis point of view, te ordinary first derivative is bad, wile te symmetric first derivative is good. More specifically, for te ordinary first derivative we ave te expansion were f x + ) f x) = f x) + f x) + f 2 x) o 2) f x) = f x) + E, E = f 2 x) 2 + o ) ; wile for te symmetric first derivative of te same scale we ave f x + ) 2) f x 2 = { ) } f x + 2 f x ) 2 { = f x) +f x) 2 +f 2 x) 2 8 +f 3 x) o 3) f x) +f x) 2 f 2 x) 2 8 +f 3 x) o 3) = f x) + F, were F = f 3 x) o 2). Comparing te error terms E and F gives a measure of te goodness of approximation of te first derivative at x by tese two difference quotients for fixed small decidedly favoring te symmetric first derivative. Tere are many generalized first derivatives. One way to make new ones is to slide, tat is to add a constant multiple of to every base point. Tus, sliding by 2, {x 2, x + 2 } {x 2 + 2, x } = {x, x + }, }

6 62 Seminar of Matematical Analysis canges te first symmetric derivative into te ordinary first derivative. So sliding does not preserve te numerical version of good tat we ave just mentioned. Furtermore, te function x does ave a first symmetric derivative at x =, but it fails to be differentiable tere, so sliding does not preserve differentiability eiter. Sliding does preserve te property of being a derivative. To see tis, note tat if c is any real number, ten te equation system d+e j =,,..., d w i u i + c) j =. i= d! j = d is equivalent to te equation system 3), so tat d+e i= w if x + u i ) is a generalized dt derivative if and only if d+e i= w if x + u i + c)) is. Tere is a probably apocrypal story from te middle of te nineteent century tat wen H. A. Scwarz was examining a P.D. candidate wo wrote te formula for a general quadratic in two variables as ax 2 + bxy + cy 2 + dx + ey + f, were e is not te natural logaritm, e insisted tat te student be more precise by inserting necessarily after not. Te symbol e ere stands for excess and is a nonnegative integer. Te minimum number of points for a dt derivative is first derivative 2 points second derivative 3 points dt derivative d + points and tese correspond to te cases of zero excess, e =. Wen e =, te system of equations 3) may be tougt of as a system of d + equations in te d + unknowns w, w,..., w d, wic could be written in te matrix notation Aw = d!. Ten te matrix A is a Vandermonde matrix and it is easy to see tat tere is a unique solution for w [, p. 82]. But if te excess e >, ten tere

7 J. M. As 63 are many w s for a given u. Some examples of first derivative wit positive excess e = are 2a)f x + a + )) + 4afx + a) + 2a)f x + a )) lim 2 9) for any constant a. Te case a = / 3 is special. 2 Lecture 2 Tere are tree best generalized first derivatives based on tree points. Te first is te derivative defined by relation 9) wit a = 3. Performing Taylor expansions ere yields [ 2 ) ) )f x fx + )) ) )] )f x + 3 f x) +.6f 4 x) 3 + o 3). 3 Te second, given ere togeter wit its approximating relation is f z + / 3 ) + ω 2 f z + ω/ 3 ) + ωf z + ω 2 / 3 ) 3 f z) +.8f 4 z) 3 + o 3), were ω is a complex cube root of so tat ω 2 + ω + =, and te tird may be written scaled as 5fx + 2) + 32f x + ) 27fx 2 3 ) f x) +.56f 4 x) 3 + o 3). How are tese best? For any generalized first derivative te dominant error term in approximating f by te difference quotient will ave te form Cf n+ n, were n 2 is an integer. We will say tat a difference quotient wit dominant error term Cf n+ n is better tan one wit dominant error term Df m+ m if n > m, or if n = m and C < D. All tree ave an error term tat is one power of iger tan migt be expected from linear algebra considerations of te system 3). Te first minimizes truncation error if we restrict ourselves to te general setting of tese talks, tat of real-valued functions of a real variable. Te second only makes sense in te setting of

8 64 Seminar of Matematical Analysis analytic functions of a complex variable. It is peraps out of place ere, but arises naturally from te algebra involved wit attempts to minimize. Te tird example, wic can be written in equivalent but unscaled form as 5fx + 6) + 32f x + 3) 27fx 2), 3 is almost as good as te oter two. It is te best you can do if you are limited to functions defined as tabulated values. Tere are oter notions of goodness tat migt be considered. Tese are discussed in reference [6]. In reference [8], best additive linear generalized first and second derivatives for eac excess e are found. Let us return to te ordinary dt derivative f d), te Peano dt derivative f d, and te generalized linear) dt derivatives D d uwf. Explicitly, f d x) is defined as f x + ) f x) f x) f 2 x) f d x) = lim d d! 2 2!... f d x) As we mentioned above, for eac fixed x we ave te implications and d d!). f d) x) = f d x) ) f d x) = D d uwf x). ) Te first implication is reversible wen d = since te definitions of f x) = f ) x) and f x) coincide. Te second implication is reversible for te same reason wen d = and uw is eiter c, )/c, /c) or, c)/c, /c), for some positive c. Except for tese trivial cases, no oter implication is reversible. Here is an example wic sows wy te first implication is x 3 sin/x) x irreversible. Let f x) =. Ten for nonzero, x = f + ) = sin/)) were te last term is o 2) since sin/) as. In particular, f 2 ) =.

9 J. M. As Te function x 3 sin/x) x cos However, by direct calculation, f x ) + 3x2 sin x ) x x) = x = so tat f cos/) sin/) ) = lim = lim cos/) does not exist. An instance were te second implication is irreversible given above was te function x wic as first symmetric derivative at x =, but wic does not ave a first Peano derivative tere. Anoter interesting example sowing te second implication to be irreversible is te if x > function sgnx = if x =. Tis as a Scwarz generalized second if x < derivative equal to at x = see definition )), but certainly is not well approximated by a quadratic polynomial near x =, so tat sgn does not ave a second Peano derivative at x =. My tesis advisor, Antoni Zygmund, used to say tat te purpose of counterexamples is to lead us to teorems, so since te exact converses of implications ) and ) are false, let us consider te following partial converses. For all x E, f d x) = for almost every x E, f d) x) 2) For all x E, D d uwf x) = for almost every x E, f d x). 3)

10 66 Seminar of Matematical Analysis Except for te trivial d = case, converse 2) is still false. H. William Oliver proved tat for any interval I and any open, dense subset O of I, tere is a function f wit te second Peano derivative f 2 x) existing for every x I, te ordinary second derivative f x) existing for all x O, and f x) existing at no point of I \ O.Tis example is especially daunting since te measure of I \ O can be close to te measure of I. In fact, if ǫ > is given, let {r n } be te rational numbers of I. Ten te open set r ǫ 4, r + ǫ ) 4 r2 ǫ 8, r 2 + ǫ 8 and its intersection wit I is open and dense in I. But Oliver was Zygmund s student, so e found and proved tis partial converse. ) as Lebesgue measure ǫ 2 + ǫ 4 + = ǫ Teorem Oliver). If te dt Peano derivative f d x) exists at every point of an interval I and if f d x) is bounded above, ten te ordinary dt derivative f d) x) also exists at all points of I. So te oped for converse 2) is false. But te second converse is true! Almost all of te deep results tat I will discuss involve tis converse or analogues of it. Te first big paper ere is te 936 work of Marcinkiewicz and Zygmund tat I ave already mentioned. It is On te differentiability of functions and summability of trigonometric series. [9] In tis paper, along wit many oter tings, is proved te validity of te second converse 3) for te special cases wen D uw is te dt symmetric Riemann derivative, D d f x) = lim d ) ) n d f x + d/2 n)). n n= Later on, in my tesis, te second converse 3) is proved for every D uw.[] Let d 2. Define an equivalence relation on te set of all te dt derivatives tat we ave defined ere by saying tat two derivatives are equivalent if for any measurable function, te set were one of te derivatives exists and te oter one does not as Lebesgue measure zero. Ten tere are only two equivalence classes. One is te singleton consisting of only te ordinary dt derivative. Te oter contains everyting else, te dt Peano derivative and every D uw satisfying te conditions 3). Earlier I said tat a case could be made tat te standard dt derivative sould be te dt Peano derivative, polynomial approximation to order d, rater tan te ordinary dt derivative. Te fact tat ordinary differentiation stands alone from tis almost everywere point of view is one of te reasons for tis.

11 J. M. As 67 3 Lecture 3 In 939 Marcinkiewicz and Zygmund proved tat for all d, te existence of te dt symmetric Riemann derivative, D d f x) = lim d d ) ) n d f n n= ) ) d x + 2 n = lim d d 4) for every x in a set E implies te existence of te Peano derivative f d x) at almost every x E.[9] One of te pieces of teir proof is a sliding lemma. Lemma 2. If d = d ) ) n d f n n= ) ) d x + 2 n = O d) 5) for all x E, ten any of its slides is also O d) at almost every point of E. For example, from for all x E, follows 2 = f x + ) 2f x) + f x ) = O 2) 6) f x + 2) 2f x + ) + f x) = O 2) 7) for a. e. x E. We cannot drop te a. e. from te lemma s conclusion. For >, sgn + ) 2 sgn) + sgn ) = = O 2), but sgn + 2) 2 sgn + ) + sgn) = O 2) as. If we start from te existence of te second symmetric derivative on a set, so tat f x + ) 2f x) + f x ) = D d f x) 2 + o 2) tere, ten pass to te weaker condition 6), and ten pass to condition 7), we lose information at bot steps. At te first step we lose te limit itself and at te second step we lose a set of measure. Te second loss obviously does not matter, since we are only aiming for an almost everywere result. Te first loss will not matter eiter, since we may lower our sigts

12 68 Seminar of Matematical Analysis from te goal of acieving te existence of f d x) a. e. to tat of acieving merely f x + ) = f x) + f x) + + f d x) d d )! + Od ) 8) a. e. Tis is because Marcinkiewicz and Zygmund prove a lemma stating tat if condition 8) olds on a set, ten we also ave te existence of te Peano derivative f d x) itself after discarding a subset of zero measure. Suc a lemma is a generalization of te fact tat Lipscitz functions are differentiable almost everywere. Te second big idea is te introduction of te derivatives based on te d + points d D d f x) = c d lim d, x + 2 d, x + 2 d 2,...,x + 2, x +, x. I mentioned ow tese differences d were formed in te first lecture. See formula 5) for 3.) Te constants c d are required to normalize te last equation of conditions 3), for example, we sowed in lecture tat c 3 = 4. It turns out tat it is muc easier to prove teir next lemma, d = O d) on E = formula 8) olds a. e. on E, 9) tan to prove te same ting starting from d = O d) on E. Already I ave mentioned enoug results to produce te d = 2 case of te teorem. Suppose D 2 f x) exists for all x E. By te sliding lemma we know tat condition 7) olds a. e. on E. But 2 and 2 coincide, so by implication 9), formula 8) olds a. e. on E and tis is enoug to get f 2 x) a. e. on E. For d 3, tere is only one more step needed. To completes te logical flow of teir argument, Marcinkiewicz and Zygmund sow tat d is a linear combination of slides of d. An example of tis last lemma is te equation f x + 4) 6f x + 2) + 8f x + ) 3f x) = f x + 4) 3f x + 3) + 3f x + 2) f x + ) + 3 {f x + 3) 3f x + 2) + 3f x + ) f x)}

13 J. M. As 69 or 3 = 3 slid up by ) Te first generalization of te work of Marcinkiewicz and Zygmund is tis. Teorem 3. If a generalized dt derivative D uw f exists on a set E, ten te dt Peano derivative of f exists for a. e. x E. To prove tis, one more big idea is needed. I will restrict my discussion to applying tat idea to one very simple case not covered by Marcinkiewicz and Zygmund. Assume tat te derivative Df x) = lim t o f x + 3t) + f x + 2t) 2f x + t) 3t exists for all x E. We will sow tat f x) exists for a. e. x E. Note tat D is indeed a first derivative wit excess e =, since ) 3 = and ) 2 3 =. Taking into account wat we know about Marcinkiewicz and Zygmund s approac, we start wit on E. Te plan is to sow tat f x + 3t) + f x + 2t) 2f x + t) = O t) 2) satisfies a. e. on E, F x) = x a f t) dt F x + 2) 2F x + ) + F x) = O 2). 2) We will suppress various tecnical details, suc as sowing tat our ypotesis 2) allows us to assume tat f is actually locally integrable in a neigborood of almost every point of E. Ten by te Marcinkiewicz and Zygmund result, F will ave two Peano derivatives a. e. on E. It is plausible and actually not ard to prove from tis tat F = f a. e. and tat f as one less Peano derivative tan F does almost everywere. So our goal is reduced to proving relation 2). Te idea is to exploit te noncommutativity of te operations of integrating and sliding. Te substitution u = x+bt, du = bdt allows f x + bt)dt =

14 7 Seminar of Matematical Analysis x+b x f u) du b = b [F x + b) F x)]. So integrating our assumption 2), taking Ot)dt = o 2) into account gives or 3 [F x + 3) F x)] + [F x + 2) F x)] 2 [F x + ) F x)] 2 = O 2) 2F x + 3) + 3F x + 2) 2F x + ) + 7F x) = O 2) 22) a. e. on E. So integrating and ten sliding by gives 2F x + 4) + 3F x + 3) 2F x + 2) + 7F x + ) = O 2) 23) for a. e. x E. Now start over from assumption 2). Tis time slide by t, and ten integrate to get or f x + 4t) + f x + 3t) 2f x + 2t) = O t) 4 [F x + 4) F x)] + 3 [F x + 3) F x)] 2 [F x + 2) F x)] 2 = O 2) 3F x + 4) + 4F x + 3) 2F x + 2) + 5F x) = O 2) 24) for a. e. x E. Te crucial point is tat te left and sides of equations 23) and 24) are different. Te rest is aritmetic. Tink of te tree expressions on te left sides of relations 22), 23), and 24) as tree vectors in te span of five basis elements v i = F x + ih),i =,,..., 4. We can eliminate v 3 and v 4 by taking te following linear combination of tese tree vectors. 2 2 v {2v 4 + 3v 3 2v 2 + 7v } 4 {3v 4 + 4v 3 2v 2 + 5v } {2v 3 + 3v 2 2v + 7v } v = +42 v 2 + v v

15 J. M. As 7 = 27 F x + 2) 2F x + ) + F x)) = O 2) for a. e. x E. We ave reaced te desired goal of relation 2). Te next step in te program is to move to L p norms, p <. Here is a restatement of te Marcinkiewicz and Zygmund Teorem. Teorem 4. If d ) ) n d f n n= ) ) d x + 2 n D d f x) d = o d) for every x E, ten for a. e. x E, ) f x + ) f x) + f x) + + f d x) d = o d). d! And ere is te L p version. Teorem 5. Let p [, ). If d ) ) n d f n n= for every x E, ten for a. e. x E, ) ) d p /p x + 2 n t D dt) pd f x)td = o d), f x + t) f p x) + fp x)t + + fp d ) td p /p x) dt) = o d). d! Tis is true. Even a more general version, te obvious analogue of Teorem 3, wic we will call Teorem 3 p, is true. Wen I went to Zygmund in 963 for a tesis problem, I wanted to try te ten unsolved question of te almost everywere convergence of Fourier series. My plan was to try to find a counterexample, an L 2 function wit almost everywere divergent Fourier series. But Zygmund wanted me to try to prove Teorem 4. Luckily, I worked on Teorem 4 and Lennart Carleson worked on and proved) te almost everywere convergence of Fourier series of L 2 functions. Oterwise, I still migt not ave a P.D. You cannot prove Teorem 4 by just canging norms. A lot of te lemmas go troug easily, but te sliding lemma does not. At least I ave never been able to find a direct proof of it. In fact, ere is an important open question; peraps te best one I will offer in tese lectures.

16 72 Seminar of Matematical Analysis Problem 6. It is true tat if f x + t) f x t) dt = O 2) for all x E, ten f x + 2t) f x) dt = O 2) for almost every x E. Prove tis by a direct metod. Te use of te word direct is a little vague and will be explicated in a minute. Te metod tat I used to prove Teorem 4 immediately required Teorem 3. Furtermore, once I ad proved Teorem 3, Teorem 3 p was no arder to prove tan was Teorem 4. Let me sow you were Teorem 3 comes into play. Te d = case is easy, but already te d = 2 case is not. Let p =, so our assumption implies f x + t) + f x t) 2f x) dt = O 2). From tis and te triangle inequality follows or, equivalently, {f x + t) + f x t) 2f x)} dt = O 2), F x + ) F x ) 2f x) = O 3) 25) were F x + ) = x+ x f u) du. If we can remove te 2f x) term, ten we be in te realm of generalized derivatives of te form D uw F. Te obvious ting to do is to write formula 25) wit replaced by 2 and ten subtract twice formula 25) from tat. We get. F x + 2) F x 2) 2f x) 2 2 F x + ) F x ) 2f x)) = F x + 2) 2F x + ) + 2F x ) F x 2) = O 3). Note tat 2+2 =, ) 2) =, ) 2 2) 2 =, and ) 3 2) 3 = 2 = 2 3!, so tat te left side corresponds to te tird derivative D 2,,, 2)/2,,, /2) F. Tus arose te necessity of extending Marcinkiewicz and Zygmund s teorem to Teorem 3. Now it sould be a little more clear wat I mean by a direct proof in te problem given above: do not prove it by passing to te indefinite integral F, ten proving tat F as two Peano derivatives, etcetera.

17 J. M. As 73 4 Lecture 4 Stefan Catoiu, Ricardo Ríos, and I recently proved an analogue of te teorem of Marcinkiewicz and Zygmund. Te work will soon appear in te Journal of te London Matematical Society.[4] Assume trougout our discussion tat x. Making te substitution qx = x + and noting tat if and only if q eiter as f x + ) f x) lim f qx) f x) or as lim. q qx x Already someting new appens if we replace te additively symmetric points x +, x used to define te symmetric first derivative wit te multiplicatively symmetric points qx, q x and consider f qx) f q x ) D,q f x) = lim q qx q. x Similarly, f qx) + q)f x) + qf q x ) D 2,q f x) = lim q q qx x) 2 is someting new. Notice tat in bot cases tere would be noting at all defined if x were. Tese are te first two instances of q-analogues of te generalized Riemann derivatives. Using te polynomial expansion of f about x allows us to write f qx) = f x + q )x) = f x) + f x) qx x) + f 2 x) qx x)2 2 + and tereby quickly prove tat if at eac point x, te first Peano derivative f x) exists, ten at tat point D q f does also and f x) = D q f x); similarly te existence of f 2 x) implies f 2 x) = D q 2f x). As you would expect, te pointwise converses to tese two implications are false. In our paper, we sow tat te converses do old on an almost everywere basis. We do tis by more or less establising te analogue of every step of te Marcinkiewicz and Zygmund proof.

18 74 Seminar of Matematical Analysis Te reason tat we don t go on to prove te q-analogue of Teorem 3 is tat we cannot find an analogue of te non-commutivity of integrating and sliding idea. Tis seems to be a great impediment to furter progress. I will spend a lot of time in tis lecture on posing questions. So far, we can do noting in te L p setting. I will specialize to p =, because in all te classical work involving te additive generalized derivatives noting special apped to distinguis te p = cases from any corresponding cases wit oter finite values of p, and tings are a little easier to write ere. Problem 7. Prove: If at every x E, q q ten at a. e. x E, f px) f p x ) px p x ) D,qf x) dp = o qx x)), p f x + t) f x) dt = O ). Problem 8. Prove: If at every x E, q f px) + p) f x) + pf p x ) p px p x ) 2 D q 2,q f x) dp p = o qx q x ) 2 ), ten at a. e. x E, f x + t) f x) f p x)t dt = O 2). A q-version of an L p sliding lemma would immediately solve question 7. It would also give a major boost to solving lots of oter problems. Here is wat a general sliding lemma migt say. Again, I will keep p = for simplicity of statement and I really do not tink tat te value of p makes any difference. Problem 9. Prove: If at every x E, q w k p) f p n k x) dp q p = o q )α ), ten, for any fixed c, at a. e. x E, q wk p)f p nk c x ) dp q p = o q )α ).

19 J. M. As 75 Te very first instance of Problem 9 would immediately resolve Problem 7, so I will state it separately. Problem. Prove: If at every x E, q ten at a. e. x E, q q q f px) f p x ) dp = O q ), p f p 2 x ) f x) dt = O q ). Incidentally, te cange of variable p 2 p, dp equation is equivalent to q q p dp p f px) f x) dt = O q ), sows tat te last since q 2 is equivalent to q for q near. Te reason tat I tink tat even Problem is very ard is tat, as I mentioned before, even in te simplest additive case te L p sliding lemma is by no means a staigtforward extension of te usual L sliding lemma. In fact, Problem is truly an open question, te issue is not just to find a direct proof. Problem. Fix α >. Prove: If at every x E, ten at a. e. x E, f x + t) f x t) dt = O α ), f x + 2t) f x) dt = O α ). As I discussed in te tird lecture, tere is a very tricky indirect proof of tis wen α = using te metods of my tesis. If it could be solved for general α, way, te metod of te solution migt very well lead to a full L p additive sliding lemma. Problem 2. Fix α >.Prove: If at every x E, w k f x + u k t) dt = O α ), ten, for any fixed c, at a. e. x E, wk f x + u k c)t) dt = O α ).

20 76 Seminar of Matematical Analysis Note tat tere are no conditions on te w k s and u k s ere. If in 964 I could ave solved Problem 2, I would ave simply copied te Marcinkiewicz and Zygmund s proof, lemma by lemma into L p format witout any need for te generalized derivatives D uw. Let us return now to te question of using generalized derivatives for approximation. Tis time we will consider multilinear, in particular bilinear, generalized derivatives. Te idea is to approximate a derivative of order d by means of finite sums of te form wij f x + u i )f x + v i ). Proceeding formally from Taylor series expansions we get tis expression equal to i,j ij) w f 2 + i w iju i + ) j w ijv j ff i + w iju 2 i + j w ijvj) 2 ff 2 ) 2 + i,j w iju i v j f To see ow to make use of suc an expression, we will given an example involving analytic functions of a complex variable. Tis is te second, and last, time in tese lectures tat I depart from talking about real functions of a real variable. Let ω = + 3i ) /2 be a cube root of, so tat ω 2 = ω and ω 3 =. If f is analytic, Taylor expansion yields f z + )f z + ω) f z )f z ω) = 2 ω + )f z)f z) + 2ω )f z)f z) 3 or +f f ff 5) ) w 2) 5 /6 + O 7) f z + )f z + ω) f z )f z ω) 2 ω + )f z) = 26) f z) + 2ω )f z)f z) 2 ω + )f z) 2 + f f ff 5) ) w 2) 4 + O 6) 2 ω + )f z)

21 J. M. As 77 Note tat te main first) error term involves only derivatives of order 2. If one ad used a linear generalized derivative instead to generate a similar estimate, te main error term would necessarily involve f n) x) wit n 3. So it is possible tat tis kind of approximation could prove useful for functions wit small low order derivative and large ig order derivatives. Tis is te best non-linear example I ve found so far, but I don t tink it compares very favorably wit, say, tis excellent linear approximation for f x), f x ) + 27f x + 2 ) 27f x 2 ) + f x 3 2 ) 24 = f x) 3 64 f5) x) 4 + O 6), wic requires only 4 function evaluations.[8] So te problem ere is to improve substantially on te example I ave given. Problem 3. Find at least one bilinear or multilinear, or even more general) numerical difference quotient wic approximates te first derivative in a way tat compares favorably wit te known good linear quotients. Te last topic will be generalizations of te mean value teorem. Let +e Df x) = lim w n f x + u n ) n= be a generalized first derivative, so tat w n = and w n u n =. Recall tat u > u >. Let A and H > be fixed real numbers. Assume tat Df x) exists for every x [A, A + H]. Te question is, does tere necessarily exist a number c A, A + H) suc tat Df c) = +e n= w nf A + u n H). 27) H Wen uw =, ), ), te derivative D = D uw reduces to te ordinary derivative and formula 27) reduces to te usual mean value formula. Te standard ypotesis is tat f be continuous on [A, A + H] and differentiable on A, A + H). Te first non-standard derivative tat comes to my mind is te first symmetric derivative, uv = 2, 2), ). Here we quickly run into trouble. For example, te absolute value function is continuous on [, 2] and its

22 78 Seminar of Matematical Analysis symmetric derivative exists and is equal to sgnx) at eac point of, 2). Neverteless, wit A = and H = 3, we ave A + H A H = 2 3 = 3, wic is not equal to any of,,, te tree possible values of sgnx. To sarpen te question, let us make tree definitions. Definition 4. Say tat D uw as te weak mean value property if te existence of f on [A, A + H] implies tat tere is a c A, A + H) so tat te mean value formula 27) olds. Definition 5. Say tat D uw as te mean value property if te continuity of f on [A, A + H] and te existence of D uw on A, A + H) implies tat tere is a c A, A + H) so tat te mean value formula 27) olds. Definition 6. Say tat D uw as te strong mean value property if te existence of D uw on [A, A + H] implies tat tere is a c A, A + H) so tat te mean value formula 27) olds. Wen Roger Jones and I noticed te counterexample involving te symmetric derivative and te absolute value function, our response was to prove tat te symmetric derivative as te weak MVP. We went on to prove tat many oter generalized first derivatives also ave tis property. However, some first derivatives do not ave te MVP. In our paper [7], we give a sufficient condition, wic is also necessary wen tere are two or tree base points. A complete classification as not been acieved. If D uw as te MVP, ten it as te weak MVP. Ordinary differentiation as all tree MVPs. Even toug symmetric differentiation does ave te weak MVP, te last example sows tat it does not ave eiter te MVP, nor te strong MVP. Since te symmetric derivative is so close to ordinary differentiation, Jones and I conjectured tat tere migt very well be no first derivatives oter tan te ordinary derivative wit eiter te MVP or te strong MVP. Just as I was giving tese lectures Ricardo Ríos and I discovered tat tere are some derivatives wit te MVP. In particular, we ave te ironical result tat of all two base point generalized derivatives, te symmetric derivative stands out as te only one not aving te MVP. Explicitly, te generalized first derivative, lim fx + ) fx + a), a a as te mean value property for eac value of a except a =.

23 J. M. As 79 Problem 7. Wic generalized derivatives ave te weak mean value property, wic ave te mean value property, and wic ave te strong mean value property? References [] As, J. M., Generalizations of te Riemann derivative, Trans. Amer. Mat. Soc., 26967), [2], A caracterization of te Peano derivative, Trans. Amer. Mat. Soc., 4997), [3], Very generalized Riemann derivatives, generalized Riemann derivatives and associated summability metods, Real Analysis Excange, ), 29. [4] As, J. M., Catoiu, S., and Ríos, R., On te nt quantum derivative, J. London Mat. Soc., 6622), 4-3. [5] As, J. M., Erdös P., and Rubel, L. A., Very slowly varying functions, Aequationes Mat., 974),. [6] As, J. M., Jones, R., Optimal numerical differentiation using tree function evaluations, Mat. Comp., 3798), [7], Mean value teorems for generalized Riemann derivatives, Proc. Amer. Mat. Soc., 987), [8] As, J. M., Janson, S., and Jones, R., Optimal numerical differentiation using n function evaluations, Estratto da Calcolo, 2984), [9] Marcinkiewicz, J., and Zygmund, A., On te differentiability of functions and summability of trigonometric series, Fund. Mat., 26936), 43. Autor s address: J. Marsall As Department of Matematical Sciences DePaul University 232 Nort Kenmore Ave. Cicago, IL, 664, U. S. A. kmas@condor.depaul.edu