B. Alan Taylor

Use of Characteristic Roots for Solving Infinite State Markov Chains

In this chapter, our interest is in determining the stationary distribution of an irreducible positive recurrent Markov chain with an infinite state space. In particular, we consider the solution of such chains using roots or zeros. A root of an equation f (z) = 0 is a zero of the function f (z),and so for notational convenience we use the terms root and zero interchangeably. A natural class of chains that can be solved using roots are those with a transition matrix that has an almost Toeplitz structure. Specifically, the classes of M/G/1 type chains and G/M/1 type chains lend themselves to solution methods that utilize roots. In the M/G/1 case, it is natural to transform the stationary equations and solve for the stationary distribution using generating functions. However, in the G/M/1 case the stationary probability vector itself is given directly in terms of roots or zeros. Although our focus in this chapter is on the discrete-time case, we will show how the continuous-time case can be handled by the same techniques. The M/G/1 and G/M/1 classes can be solved using the matrix analytic method [Neuts, 1981, Neuts, 1989], and we will also discuss the relationship between the approach using roots and this method.

Splitting of closed ideals in ()-algebras of entire functions and the property ()

For a plurisubharmonic weight function p on cn let Ap(Cn) denote the (DFN)-algebra of all entire functions on cn which do not grow faster than a power of exp(p). We prove that the splitting of many finitely generated closed ideals of a certain type in Ap(Cn), the splitting of a weighted 0-complex related with p, and the linear topological invariant (DN) of the strong dual of Ap(Cn) are equivalent. Moreover, we show that these equivalences can be characterized by convexity properties of p, phrased in terms of greatest plurisubharmonic minorants. For radial weight functions p, this characterization reduces to a covexity property of the inverse of p. Using these criteria, we present a wide range of examples of weights p for which the equivalences stated above hold and also where they fail. For p a nonnegative plurisubharmonic (psh) function on cn) let Ap(Cn) denote the algebra of all entire functions f such that If(Z)l 0 depending on f. Algebras of this type arise at various places in complex analysis and functional analysis, e.g. as Fourier transforms of certain convolution algebras. The structure of their closed ideals has been studied for a long time, primarily in the work of Schwartz [25], Ehrenpreis [9], Malgrange [17], and Palamodov [23] in connection with the existence and approximation of (systems of) convolution equations. The question whether a certain parameter dependence of the right-hand side of such an equation is shared also by its solutions, is closely related with the question of the existence of a continuous linear right inverse. The existence of such a right inverse is equivalent to the splitting of the closed ideal I associated to the corresponding equation. Also, since the quotient space Ap(Cn)/I is quite often identified with the space Ap(V) of holomorphic functions on the variety V of I which satisfy the restricted growth conditions, the latter question is equivalent to the existence of a linear extension operator from Ap(V) to Ap(Cn). Answers to these questions for various algebras have been given e.g. by Grothendieck (see lVeves [28]), Cohoon [7], Djakov and Mityagin [8], and Vogt [33]. The fact that for P(z) = IZl8, s > 1 all closed ideals in Ap(C) are cornplemented, was observed by Taylor [27]. Then Meise [19] extended Taylors resultsX using a more functional analytic approach. He showed that the structural property (DN) of the strong dual Ap(Cn)b of Ap(Cn) implies that all slowly decreasing ideals Received by the editors July 29, 1986. 1980 Mathematics Subject Classification (1985 Reon). Primary 32E25, 46E25; Secondary 46A12, 32Al .

Continuous linear right inverses for partial differential operators of order 2 and fundamental solutions in half spaces

SummaryLetP be a complex polynomial inn variables of degree 2 andP(D) the corresponding partial differential operator with constant coefficients. It is shown thatP (D) :C∞(ℝn) →C∞(ℝn) admits a continuous linear right inverse if and only if after a separation of variables and up to a complex factor for some c ∈ ℂ the polynomialP has the form

Continuous Linear Right Inverses for Partial Differential Operators on Non ‐ Quasianalytic Classes and on Ultradistributions

P(x_1 ,...,x_n ) = Q(x_1 ,...,x_r ) + L(x_{r + 1} ,...,x_n ) + c

Linear Extension Operators for Ultradifferentiable Functions of Beurling Type on Compact Sets

where eitherr=1 andL≡0 orr>1,Q andL are real andQ is indefinite. The proof of this characterization is based on the general solution of the right inverse problem for such operators and the fact that for each operatorP(D) of the given form and each characteristic vectorN there exists a fundamental solution forP(D) supported by {x ∈ ℝn : 〈x, N〉 ⪰ 0 #x007D;, which can be constructed explicitely using partial Fourier transform. The existence of sufficiently many fundamental solutions with support in closed half spaces implies that some right inverse can be given by a concrete formula. An example shows that the present characterization is restricted to operators of order 2.