Exporing the Lanczos' Generalized Derivative

Developed in the 1950's by the Hungarian mathematician Cornelius Lanczos, the Lanczos' Generalized Derivative (LGD) the function f at x is written as

LGD_definition.gif (1221 bytes),

provided that this limit exists. The motivation for this formula comes from least squares regression. It can be shown that this formula yields a proper extension of the derivative. In other words, if a function has a normal derivate at a point, then the function possesses an LGD and the two quantities are equal. However, a function can possess an LGD at a point even if the normal derivative is not defined. An example is the piecewise linear function

piecewise_formula.gif (799 bytes),

whose graph is shown below.

talk_image.jpg (16384 bytes)

For this function, LGD(f)(0)=.5, a fact that allows one to draw a "pseudo-tangent" line of sorts at the origin. (The line in red above.)

Since the LGD is based upon the ideas of linear regression, one can derive a quantity known as "instantaneous correlation," which measures the ability of pseudo-tangent lines to approximate the original function near the point at which the LGD is based. For example, the instantaneous  correlation of the pseudo-tangent line above works about to be approximately .554. Not surprisingly, if a function has a normal, nonzero derivative at a point, the correlation is 1 or -1, a fact demonstrating perfect positive or negative association, respectively.

Many interesting functions possess LGD's. For example, the function shown below was constructed by Karl Weierstrass and is continuous but has a normal derivative nowhere.

weierstrass_function.jpg (20480 bytes)

Thus function has period one and possesses a LGD at x if x has a finite ternary expansion. If x has a repeating ternary expansion that cannot be written as a finite one, then LGD(f)(x) does not exist.

Many open question remain concerning the LGD. Here are but a few.