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Topics in Generalized Differentiation Topics in Generalized Differentiation J. Marshall Ash Abstract The course will be built around three topics: (1) Prove the almost everywhere equivalence of the Lp n-th sym- metric quantum derivative and the Lp Peano derivative. (2) Develop a theory of non-linear generalized derivatives, for ex- ample of the form anf(x + bnh)f(x + cnh). X (3) Classify which generalized derivatives of the form anf(x + bnh) satisfy the mean value theorem. X 1 Lecture 1 I will discuss three types of difference quotients. The first are the additive linear ones. These have been around for a long time. One can see their shadow already in the notation Leibnitz used for the dth derivative, dd . dxd For an example with d = 2, let h = dx, and consider the Schwarz generalized second derivative d2f (x) f (x + h) 2f (x) + f (x h) lim = lim − − . (1) h 0 h2 h 0 h2 → → The difference quotient associated with the generalized additive linear deriva- tive has the form Duwf (x)= lim Duwf (x,h) h 0 → ∆ f (x,h) = lim uw h 0 hd (2) → d+e w f (x + u h) = lim i=0 i i . h 0 hd → P 57 58 Seminar of Mathematical Analysis Here f will be a real-valued function of a real variable. The wi’s are the weights and the u h’s are the base points, u > u > > u . This is a i 0 1 ··· d+e generalized dth derivative when w and u satisfy d+e 0 j =0, 1,...,d 1 j − w u = . (3) i i i=0 d! j = d X The scale of such a derivative will be the closest that two base points come when h is set equal to 1. Thus the scale of the Schwarz generalized second derivative is min (x + 1) x , (x + 1) (x 1) , x (x 1) =1. {| − | | − − | | − − |} That is why we will systematically write, for example, the third symmetric Riemann derivative as f x + 3 h 3f x + 1 h +3f x 1 h f x 3 h lim 2 − 2 − 2 − − 2 (4) h 0 h3 → instead of the completely equivalent f (x +3h) 3f (x + h)+3f (x h) f (x 3h) lim − − − − . h 0 8h3 → In other words, there are an infinite number of ways to write the difference quotient for a single generalized derivative, but only at most 2 if the scale is fixed at 1. (Note that the substitution h h does not change (1), it does → − f(x+h) f(x) f(x h) f(x) change the standard first difference quotient, − = − − .) h 6 h Henceforth the scale will always be taken to be 1. − We will systematically abuse notation, referring to any of the difference quotient, or its numerator, or its limit, as the derivative. Thus, the second symmetric Riemann derivative (also called the Schwarz derivative) will refer either to the quotient f (x + h) 2f (x) + f (x h) − − , h2 or to its numerator f (x + h) 2f (x) + f (x h) , − − or to its limit as given in equation (1) above. J. M. Ash 59 The third symmetric Riemann derivative, given by the quotient (4) is an instance of the kth symmetric Riemann derivative. There is one of these as well as a forward Riemann derivative for every natural number k. The first three forward Riemann derivatives are given by f (x + h) f (x) − , h f (x +2h) 2f (x + h) + f (x) − , h2 and f (x +3h) 3f (x +2h)+3f (x + h) f (x) − − . h3 Zygmund and Marcinkiewicz in the middle 1930s did some of the best early work on generalized derivatives. Here is an important family of generalized derivatives that they introduced. They began with a family of differences. ∆˜ = f (x + h) 20f (x) , 1 − ∆˜ =[f (x +2h) f (x)] 21 [f (x + h) f (x)] 2 − − − = f (x +2h) 2f (x + h) + f (x) , − ∆˜ =[f (x +4h) 2f (x +2h) + f (x)] 3 − 22 [f (x +2h) 2f (x + h) + f (x)] (5) − − = f (x +4h) 6f (x +2h)+8f (x + h) 3f (x) . − − The next one is formed by taking the third difference at 2h and sub- tracting 23 times the third difference from it; and thus has base points 8h, 4h, 2h,h, and 0. It should be clear how to inductively create the entire family of these differences. The dth difference corresponds to a dth deriva- tive, after multiplication by a constant. Other generalized derivatives show up in numerical analysis. One measure for evaluating how good a derivative is comes from considering Taylor expansions. If f is sufficiently smooth, we write h2 h3 f (x + h) = f (x) + f ′ (x) h + f ′′ (x) + f ′′′ (x) + ... 2 3! 60 Seminar of Mathematical Analysis Substitute this into, say, the third difference (5) to get f (x +4h) 6f (x +2h) ∆˜ 3 = − +8f (x + h) 3f (x) − 2 h2 3 h3 1 f (x) +1 4f ′ (x) h +1 4 f ′′ (x) 2 +1 4 f ′′′ (x) 3! + ... · · · 2 · 3 6 f (x) 6 2f (x) h 6 22f (x) h 6 23f (x) h + ... = ′ ′′ 2 ′′′ 3! . − · − · − · 2 h2 − · 3 h3 +8 f (x) +8 1f ′ (x) h +8 1 f ′′ (x) +8 1 f ′′′ (x) + ... · · · 2 · 3! 3 f (x) − · 3 Observe that the right hand side simplifies to 4f ′′′ (x) h + ... so we get a generalized third derivative here when we divide by 4h3. (In other words, 1 ˜ 4 ∆3 corresponds to a generalized third derivative.) This calculation gives an explanation for the conditions (3). To make this more explicit we will consider a very basic generalized derivative, the dth Peano derivative. Say that f has an dth Peano derivative at x, denoted fd (x) if for x fixed there are numbers f0 (x) , f1 (x) ,...,fd 1 (x) such that − f (x + h) = f (x) + f (x) h + 0 1 ··· d 1 d h − h d + fd 1 (x) + fd (x) + o(h ) (6) − (d 1)! d! − as h 0. This notion, the notion of approximation near x to dth order, is → very basic indeed. In fact a serious argument can be made that this is a better definition of higher order differentiation than the standard one which defines the dth derivative iteratively with the dth derivative f (d) (x) being (d 1) defined as the ordinary first derivative of f − (x) . I will have more to say about this later on. Note that f0 (x) = f (x) if and only if f is continuous at x, and that f1 (x) and f ′ (x) are identical by definition. Also, (d) if f (x) exists, then fd (x) exists and equals fd (x) . (7) (This is the content of Peano’s powerful, but little known version of Taylor’s Theorem.) This implication is irreversible for every d 2. I will give a ≥ simple example of this in my next lecture. Here is a connection between the two generalizations considered so far. If fd (x) exists, then every Duwf (x) with w and u satisfying (3) exists and equals fd (x) . (8) J. M. Ash 61 This fact follows immediately from substituting (1) into (3) and interchang- ing the order of summation. This calculation is the basis for the condi- tions (3). The implication (8) is also irreversible; for example, D( 1 , 1 )(1, 1) 2 − 2 − x = 0 at x = 0, but x is not differentiable there. (This D is the | | | | uw symmetric first derivative.) Nevertheless, the implication (8) is “more re- versible” than is the implication (7). Much of what I have to say in these lectures will be about the possibilities of partial converses to implication (8), to generalizations of implication (8), and to partial converses of those generalizations. From a certain numerical analysis point of view, the ordinary first deriva- tive is “bad”, while the symmetric first derivative is “good”. More specifi- cally, for the ordinary first derivative we have the expansion h2 2 f (x + h) f (x) f (x) + f ′ (x) h + f2 (x) + o h f (x) − = 2 − h h = f ′ (x) + E, where h E = f (x) + o (h); 2 2 while for the symmetric first derivative of the same scale we have f x + h f x h 1 f x + h 2 − − 2 = 2 h h f x h − − 2 h h2 h3 3 1 f (x) +f ′ (x) 2 +f2 (x) 8 +f3 (x) 48 +o h = 2 3 h f (x) +f (x) h f (x) h +f (x) h +o h3 ( − ′ 2 − 2 8 3 48 ) = f ′ (x) + F, where h2 F = f (x) + o h2 . 3 24 Comparing the error terms E and F gives a measure of the goodness of approximation of the first derivative at x by these two difference quotients for fixed small h decidedly favoring the symmetric first derivative. There are many generalized first derivatives. One way to make new ones is to “slide”, that is to add a constant multiple of h to every base point. 1 Thus, sliding by 2 h, 1 1 1 1 1 1 x h,x + h x h + h,x + h + h = x,x + h , − 2 2 → − 2 2 2 2 { } 62 Seminar of Mathematical Analysis changes the first symmetric derivative into the ordinary first derivative. So sliding does not preserve the numerical version of “good” that we have just mentioned. Furthermore, the function x does have a first symmetric deriva- | | tive at x =0, but it fails to be differentiable there, so sliding does not pre- serve differentiability either. Sliding does preserve the property of being a derivative.
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