Total Positivity. Vol.1 by Karrlin, S.

By Karrlin, S.

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1, to be inadequate. A more precise method was needed for comparing the number of sign changes of a given function and the number of zeros (counted appropriately) of the transformed function. This led to the formulation of a hierarchy of sign-regular notions that in essence permit the variables x and y in A, to tend to the boundary in sach a way that the values of the compound kernel do not tend to zero. With this in mind I introduced the concepts of extended total positivity, regular total positivity, and several other intermediate notions; their interrelationships and interpretations are the subject of Chapter 2.

3. Variation-Diminishing Property An important feature of totally positive and sign-regular kernels is their variationdiminishing property (abbreviated VDP). Let f(t) be defined in I , where I is an ordered set of the real line. ,x,) is the number of sign changes of the indicated sequence, zero terms being discarded. u,,) denotes the maximum number of sign changes of the indicated sequence, the zero terms being permitted to take on arbitrary signs. Let K(x, y) defined on X x Y be Borel-measurable, and assume for simplicity that the integral j, K(x, y) dp(y) exists for every x in X .

G. Krein, Natanson, and their students. In this country and in England useful contributions have been made by Rosenbloom [1951], some related investigations on the moment problem by Rogosinski [1958], and the work of Karlin and Studden [1966]. For excellent reviews of some of the earlier work. lt [1956], Jackson as well as pert~nentreferences, the reader may c o n s ~ ~Achieser [1930], Natanson [1949], and Timan [1963]. An elegant summary of work prior to 1944 on several aspects of the theory of convex cones generated by T-systems is given by Popovicii~[1945].

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