Polynomial operators.

*(English)*Zbl 0762.46037
Topics in mathematical analysis, Vol. Dedicated Mem. of A. L. Cauchy, Ser. Pure Math. 11, 410-444 (1989).

[For the entire collection see Zbl 0721.00014.]

The author gives a survey of basic properties of polynomials in (topological) vector spaces starting with works at the second quarter of this century (Part I). Part II contains remarks concerning Fréchet’s and Cauchy’s functional equations and remarks concerning polynomials defined on groups and semi-groups.

In details, let \(X\) and \(Y\) be vector spaces over a commutative field \(A\) of characteristic zero. Consider different definitions of polynomials:

1) An operator \(P\) from \(X\) into \(Y\) is called a polynomial of degree \(m\) if \(P=P_ 0+\dots+P_ m\), where \(P_ k\) is a \(k\)-homogeneous polynomial \((P_ m\neq 0)\) defined via a \(k\)-linear mapping [due to S. Banach (1938) and A. D. Michal (1934)].

2) An operator \(P\) from \(X\) into \(Y\) called a \(G\)-polynomial of degree \(m\), if there are associated functions \(q_ 0,\dots,q_ m\) (\(q_ m\neq 0\)) such that \[ P(x+\lambda z)=q_ 0(x,z)+\lambda q_ 1(x,z)+\dots+\lambda^ m q_ m(x,z) \] for all \(x,z\in X\) and \(\lambda\in A\) [due to Gateaux (1922), slightly generalized]. Among others, it is shown that both definitions are equivalent, the sums in the definitions are unique, and the polynomials satisfy the following functional equation. \({\overset{m+1} {\underset{x_ 1,\dots,x_{m+1}}\Delta}}F=0\), where \({\overset {k}{\underset{x_ 1,\dots,x_ k}\Delta}}\) is the iterated difference operator \(\bigl({\overset {1}{\underset{x_ 1} \Delta}}F)(x):=F(x+x_ 1)- F(x)\).

3) Let \(X\) and \(Y\) be locally convex Hausdorff spaces, in additional. \(P\) from \(X\) into \(Y\) is an \(F\)-polynomial of degree \(m\) if \(P\) is Gâteaux differentiable on \(X\) and if its \((m+1)\)st difference with arbitrary increments vanishes identically, while its \(m\)th difference does not. This definition is also equivalent to 1) and 2).

If \(X\) and \(Y\) are Banach spaces then the well-known relation between the uniform norms of a \(k\)-homogeneous polynomial \(P_ k\) and the associated symmetric \(k\)-linear mapping \(P_ k^*\) and their coincidence on Hilbert spaces [due to A. E. Taylor (1938), J. Kopec and J. Musielak (1955) and due to S. Banach (1938)]. Moreover, the author shows that the principle of uniform boundedness (resonance theorem) can be extended to multilinear mappings and homogeneous polynomials.

In section 4 it is shown that there are unique extensions of multilinear mappings and polynomials between real normed spaces to their complexifications. This is based on a work of A. E. Taylor (1938).

Part II: G. Van der Lijn (1939, 1940) has used the functional equation \({\overset {m+1} {\underset {x_ 1,\dots,x_{m+1}} \Delta}}F=0\) to define polynomials of degree \(n\) at most between abelian groups and obtained several results analogous to those discussed above.

D. Z. Djokvic (1969) deals with functions from an abelian semigroup into an abelian group and obtains a representation theorem for the difference operator of order \(n\) with different increments in terms of the shift operator together with difference operators of order \(n\) with the same increment.

The last section (Section 6) contains remarks on stability of Cauchy’s functional equation [investigated by the author (1941)] and Fréchet’s functional equation [studied by H. Whitney (1957, 1959) and the author (1961, 1983)].

The author gives a survey of basic properties of polynomials in (topological) vector spaces starting with works at the second quarter of this century (Part I). Part II contains remarks concerning Fréchet’s and Cauchy’s functional equations and remarks concerning polynomials defined on groups and semi-groups.

In details, let \(X\) and \(Y\) be vector spaces over a commutative field \(A\) of characteristic zero. Consider different definitions of polynomials:

1) An operator \(P\) from \(X\) into \(Y\) is called a polynomial of degree \(m\) if \(P=P_ 0+\dots+P_ m\), where \(P_ k\) is a \(k\)-homogeneous polynomial \((P_ m\neq 0)\) defined via a \(k\)-linear mapping [due to S. Banach (1938) and A. D. Michal (1934)].

2) An operator \(P\) from \(X\) into \(Y\) called a \(G\)-polynomial of degree \(m\), if there are associated functions \(q_ 0,\dots,q_ m\) (\(q_ m\neq 0\)) such that \[ P(x+\lambda z)=q_ 0(x,z)+\lambda q_ 1(x,z)+\dots+\lambda^ m q_ m(x,z) \] for all \(x,z\in X\) and \(\lambda\in A\) [due to Gateaux (1922), slightly generalized]. Among others, it is shown that both definitions are equivalent, the sums in the definitions are unique, and the polynomials satisfy the following functional equation. \({\overset{m+1} {\underset{x_ 1,\dots,x_{m+1}}\Delta}}F=0\), where \({\overset {k}{\underset{x_ 1,\dots,x_ k}\Delta}}\) is the iterated difference operator \(\bigl({\overset {1}{\underset{x_ 1} \Delta}}F)(x):=F(x+x_ 1)- F(x)\).

3) Let \(X\) and \(Y\) be locally convex Hausdorff spaces, in additional. \(P\) from \(X\) into \(Y\) is an \(F\)-polynomial of degree \(m\) if \(P\) is Gâteaux differentiable on \(X\) and if its \((m+1)\)st difference with arbitrary increments vanishes identically, while its \(m\)th difference does not. This definition is also equivalent to 1) and 2).

If \(X\) and \(Y\) are Banach spaces then the well-known relation between the uniform norms of a \(k\)-homogeneous polynomial \(P_ k\) and the associated symmetric \(k\)-linear mapping \(P_ k^*\) and their coincidence on Hilbert spaces [due to A. E. Taylor (1938), J. Kopec and J. Musielak (1955) and due to S. Banach (1938)]. Moreover, the author shows that the principle of uniform boundedness (resonance theorem) can be extended to multilinear mappings and homogeneous polynomials.

In section 4 it is shown that there are unique extensions of multilinear mappings and polynomials between real normed spaces to their complexifications. This is based on a work of A. E. Taylor (1938).

Part II: G. Van der Lijn (1939, 1940) has used the functional equation \({\overset {m+1} {\underset {x_ 1,\dots,x_{m+1}} \Delta}}F=0\) to define polynomials of degree \(n\) at most between abelian groups and obtained several results analogous to those discussed above.

D. Z. Djokvic (1969) deals with functions from an abelian semigroup into an abelian group and obtains a representation theorem for the difference operator of order \(n\) with different increments in terms of the shift operator together with difference operators of order \(n\) with the same increment.

The last section (Section 6) contains remarks on stability of Cauchy’s functional equation [investigated by the author (1941)] and Fréchet’s functional equation [studied by H. Whitney (1957, 1959) and the author (1961, 1983)].

Reviewer: H.-A.Braunß (Potsdam)