Infinite Dimensional Analysis, Quantum Probability and Related Topics, Sep 1, 2006
We discover a family of probability measures μa, 0 &a... more We discover a family of probability measures μa, 0 < a ≤ 1, [Formula: see text] which contains the arcsine distribution (a = 1) and semi-circle distribution (a = 1/2). We show that the multiplicative renormalization method can be used to produce orthogonal polynomials, called Chebyshev polynomials with one parameter a, which reduce to Chebyshev polynomials of the first and second kinds when a = 1 and 1/2 respectively. Moreover, we derive the associated Jacobi–Szegö parameters. This one-parameter family of probability measures coincides with the vacuum distribution of the field operator of the interacting Fock spaces related to the Anderson model.
Brownian Motion.- Constructions of Brownian Motion.- Stochastic Integrals.- An Extension of Stoch... more Brownian Motion.- Constructions of Brownian Motion.- Stochastic Integrals.- An Extension of Stochastic Integrals.- Stochastic Integrals for Martingales.- The Ito Formula.- Applications of the Ito Formula.- Multiple Wiener-Ito Integrals.- Stochastic Differential Equations.- Some Applications and Additional Topics.
In this paper, we present a more direct way to compute the Szeg ö-Jacobi parameters from a genera... more In this paper, we present a more direct way to compute the Szeg ö-Jacobi parameters from a generating function than that in [5] and [6 ]. Our study is motivated by the notions of one-mode interacting Fock spaces defined in [1] and Segal-Bargmann transform associated with non-Gaussian probability measu r s introduced in [2]. Moreover, we examine the relationships between the representat ions of orthogonal polynomials in terms of differential or difference operators and o ur generating functions. The connections provide practical criteria to determine when f unctions of a certain form are orthogonal polynomials. 2000 AMS Mathematics Subject Classification:42C05, 46L53.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1998
The S-transform is studied as a mapping from a space of tensors to a space of functions over a co... more The S-transform is studied as a mapping from a space of tensors to a space of functions over a complex space. The range of this transform is characterized in terms of analyticity and growth. These results are applied to a broad class of generalized functions in white noise analysis. These correspond to completions of the Gaussian L2-space which preserve orthogonality of Hermite polynomials. The S-transform is defined for the new generalized functions, and the range of this S-transform is identified in terms of analyticity and growth. Examples of the new spaces of generalized functions are given; these include distributions considered by Kondratiev and Streit, as well as new classes of distributions whose S-transforms have growth bounded by iterated exponentials.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2006
We discover a family of probability measures μa, 0 < a ≤ 1, [Formula: see text] which contains... more We discover a family of probability measures μa, 0 < a ≤ 1, [Formula: see text] which contains the arcsine distribution (a = 1) and semi-circle distribution (a = 1/2). We show that the multiplicative renormalization method can be used to produce orthogonal polynomials, called Chebyshev polynomials with one parameter a, which reduce to Chebyshev polynomials of the first and second kinds when a = 1 and 1/2 respectively. Moreover, we derive the associated Jacobi–Szegö parameters. This one-parameter family of probability measures coincides with the vacuum distribution of the field operator of the interacting Fock spaces related to the Anderson model.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1999
Let {α(n); n≥0} be a sequence of positive numbers satisfying certain conditions. A Gel'fand t... more Let {α(n); n≥0} be a sequence of positive numbers satisfying certain conditions. A Gel'fand triple [Formula: see text] associated with the sequence {α (n); n ≥ 0} has been introduced on a white noise space (ℰ′,μ) by Cochran, Kuo and Sengupta. In this paper we obtain additional conditions on the sequence {α(n); n≥0} in order to carry out white noise distribution theory on the space (ℰ′,μ). Moreover, we show that the Bell numbers satisfy these additional conditions.
In this section we analyze further the structure of (L2) = L2(N*,B,μ). We shall establish the wel... more In this section we analyze further the structure of (L2) = L2(N*,B,μ). We shall establish the well-known Wiener—Ito decomposition theorem which states that (L2) has a direct sum decomposition into homogeneous chaos’ (see below). This theorem is due to Wiener (1938) and Ito (1951) in the case of the Wiener space (or the white noise space). In its general form this result has been proved first by Segal (1956). We refer the reader also to Hida (1980a) and Simon (1974). We shall prove the theorem here with the help of two important transformations, denoted by J and j, which will be introduced first. At the end of this chapter we shall specialize to the white noise space. In this case we shall provide the very powerful representation of elements in (L2) in terms of Wick powers of distributions. The ideas are closely related to “folklore wisdom” in quantum physics (e.g., Glimm and Jaffe (1981), Simon (1974)). The presentation of some parts of this chapter owes much to the article by Nelson (1973c).
Infinite Dimensional Analysis, Quantum Probability and Related Topics, Sep 1, 2006
We discover a family of probability measures μa, 0 &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;a... more We discover a family of probability measures μa, 0 &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt; a ≤ 1, [Formula: see text] which contains the arcsine distribution (a = 1) and semi-circle distribution (a = 1/2). We show that the multiplicative renormalization method can be used to produce orthogonal polynomials, called Chebyshev polynomials with one parameter a, which reduce to Chebyshev polynomials of the first and second kinds when a = 1 and 1/2 respectively. Moreover, we derive the associated Jacobi–Szegö parameters. This one-parameter family of probability measures coincides with the vacuum distribution of the field operator of the interacting Fock spaces related to the Anderson model.
Brownian Motion.- Constructions of Brownian Motion.- Stochastic Integrals.- An Extension of Stoch... more Brownian Motion.- Constructions of Brownian Motion.- Stochastic Integrals.- An Extension of Stochastic Integrals.- Stochastic Integrals for Martingales.- The Ito Formula.- Applications of the Ito Formula.- Multiple Wiener-Ito Integrals.- Stochastic Differential Equations.- Some Applications and Additional Topics.
In this paper, we present a more direct way to compute the Szeg ö-Jacobi parameters from a genera... more In this paper, we present a more direct way to compute the Szeg ö-Jacobi parameters from a generating function than that in [5] and [6 ]. Our study is motivated by the notions of one-mode interacting Fock spaces defined in [1] and Segal-Bargmann transform associated with non-Gaussian probability measu r s introduced in [2]. Moreover, we examine the relationships between the representat ions of orthogonal polynomials in terms of differential or difference operators and o ur generating functions. The connections provide practical criteria to determine when f unctions of a certain form are orthogonal polynomials. 2000 AMS Mathematics Subject Classification:42C05, 46L53.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1998
The S-transform is studied as a mapping from a space of tensors to a space of functions over a co... more The S-transform is studied as a mapping from a space of tensors to a space of functions over a complex space. The range of this transform is characterized in terms of analyticity and growth. These results are applied to a broad class of generalized functions in white noise analysis. These correspond to completions of the Gaussian L2-space which preserve orthogonality of Hermite polynomials. The S-transform is defined for the new generalized functions, and the range of this S-transform is identified in terms of analyticity and growth. Examples of the new spaces of generalized functions are given; these include distributions considered by Kondratiev and Streit, as well as new classes of distributions whose S-transforms have growth bounded by iterated exponentials.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2006
We discover a family of probability measures μa, 0 < a ≤ 1, [Formula: see text] which contains... more We discover a family of probability measures μa, 0 < a ≤ 1, [Formula: see text] which contains the arcsine distribution (a = 1) and semi-circle distribution (a = 1/2). We show that the multiplicative renormalization method can be used to produce orthogonal polynomials, called Chebyshev polynomials with one parameter a, which reduce to Chebyshev polynomials of the first and second kinds when a = 1 and 1/2 respectively. Moreover, we derive the associated Jacobi–Szegö parameters. This one-parameter family of probability measures coincides with the vacuum distribution of the field operator of the interacting Fock spaces related to the Anderson model.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1999
Let {α(n); n≥0} be a sequence of positive numbers satisfying certain conditions. A Gel'fand t... more Let {α(n); n≥0} be a sequence of positive numbers satisfying certain conditions. A Gel'fand triple [Formula: see text] associated with the sequence {α (n); n ≥ 0} has been introduced on a white noise space (ℰ′,μ) by Cochran, Kuo and Sengupta. In this paper we obtain additional conditions on the sequence {α(n); n≥0} in order to carry out white noise distribution theory on the space (ℰ′,μ). Moreover, we show that the Bell numbers satisfy these additional conditions.
In this section we analyze further the structure of (L2) = L2(N*,B,μ). We shall establish the wel... more In this section we analyze further the structure of (L2) = L2(N*,B,μ). We shall establish the well-known Wiener—Ito decomposition theorem which states that (L2) has a direct sum decomposition into homogeneous chaos’ (see below). This theorem is due to Wiener (1938) and Ito (1951) in the case of the Wiener space (or the white noise space). In its general form this result has been proved first by Segal (1956). We refer the reader also to Hida (1980a) and Simon (1974). We shall prove the theorem here with the help of two important transformations, denoted by J and j, which will be introduced first. At the end of this chapter we shall specialize to the white noise space. In this case we shall provide the very powerful representation of elements in (L2) in terms of Wick powers of distributions. The ideas are closely related to “folklore wisdom” in quantum physics (e.g., Glimm and Jaffe (1981), Simon (1974)). The presentation of some parts of this chapter owes much to the article by Nelson (1973c).
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