Families of Orthogonal Polynomials, Operators and Properties John - - PowerPoint PPT Presentation

Families of Orthogonal Polynomials, Operators and Properties John Musonda

Department of Mathematics, University of Zambia Division of Applied Mathematics, M¨ alardalen University

First Network Meeting for Sida- and ISP-funded PhD Students in Mathematics Stockholm 7–8 March 2017

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My Advisors

• Prof. Sergei Silvestrov
• Prof. Anatoliy Malyarenko
• Dr. Isaac Tembo

Main advisor Assistant advisor Assistant advisor M¨ alardalen University M¨ alardalen University University of Zambia

• Prof. em. Sten Kaijser

Cooperating co-supervisor Uppsala University

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Research Topic

My research is in analysis but also borders algebra. The project deals with families of orthogonal polynomials, (pm, pn) = 0 for m = n

• perators associated with these polynomials,

connections between polynomials in terms of these operators, Analytic and algebraic properties of the operators on the Hilbert spaces in which the polynomials are orthogonal extension to corresponding Banach spaces commutation relations of the operators, and ordering formulas representations of the commutations relations by operators spectral analysis for finding other orthogonal polynomials

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Definitions and Method

Orthogonality on the real line R: For m = n,

• R

pm(x)pn(x)ω(x)dx = 0. (1) Orthogonality on the strip S = {z ∈ C : −1 ≤ Im(z) ≤ 1}: For m = n,

• R

pm(x + i)pn(x + i) + pm(x − i)pn(x − i) 2 ω(x)dx = 0 (2) where ω is some function on R which is nonnegative and locally integrable, i.e., ω ≥ 0, 0 <

• A

ω(x)dx < ∞, 0 <

• A

xnω(x)dx < ∞ (3) for some A ⊂ R. This ω is called the weight function. Gram-Schmidt procedure applied to 1, x, x2, x3, · · ·

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Starting Point

In 2012, Professor Sten Kaijser supervised my master’s thesis at Uppsala University, Sweden. 3 systems of orthogonal polynomials were studied. In the process, 3 basic operators were developed.

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Motivation

σ τ ρ σ0 = 1 τ0 = 1 ρ0 = 1 σ1 = x τ1 = x ρ1 = x σ2 = x2 τ2 = x2 − 1 ρ2 = x2 − 2 σ3 = x3 − 2x τ3 = x3 − 5x ρ3 = x3 − 8x σ4 = x4 − 8x2 τ4 = x4 − 14x2 + 9 ρ4 = x4 − 20x2 + 24 . . . . . . . . . First two columns were studied by Tsehaye K. Araaya (PhD -ISP). Comparing columns 1 and 3, we see that xρn = σn+1. (4) This is the motivation to define the operator Q by Qf = xf . (5)

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Motivation

{τn} are orthogonal with respect to ω(x) = 1/(2 cosh π

2 x) 1 Probability density function 2 Up to a dilation its own Fourier transform 3 Essentially Poisson kernel for S = {z ∈ C : −1 ≤ Im(z) ≤ 1}.

That is, for a harmonic and continuous function f in S, f (0) = ∞

−∞

f (x + i) + f (x − i) 2 ω(x)dx. (6)

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Motivation

{τn} are orthogonal with respect to ω(x) = 1/(2 cosh π

2 x) 1 Probability density function 2 Up to a dilation its own Fourier transform 3 Essentially Poisson kernel for S = {z ∈ C : −1 ≤ Im(z) ≤ 1}.

That is, for a harmonic and continuous function f in S, f (0) = ∞

−∞

f (x + i) + f (x − i) 2 ω(x)dx. (6)

This is the motivation to define the operator R by Rf (x) = f (x + i) + f (x − i) 2 , (7) and for symmetry, we also consider the operator Jf (x) = f (x + i) − f (x − i) 2i . (8)

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Three Basic Operators

Rf (x) = f (x + i) + f (x − i) 2 Jf (x) = f (x + i) − f (x − i) 2i Qf (x) = xf (x) σn

R

− → τn

R

− → ρn (9) σn

J

− → nτn−1

J

− → n(n − 1)ρn−2 (10) σn+1

Q

← − ρn (11)

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Commutation Relations

Definition Let A be an algebra. Two elements a, b ∈ A commute if ab = ba. The centralizer of a ∈ A is the set Cen(a) = {b ∈ A : ab = ba}. The center of A is the set Z(A) = {a ∈ A : ab = ba ∀b ∈ A} .

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Commutation Relations

Definition Let A be an algebra. Two elements a, b ∈ A commute if ab = ba. The centralizer of a ∈ A is the set Cen(a) = {b ∈ A : ab = ba}. The center of A is the set Z(A) = {a ∈ A : ab = ba ∀b ∈ A} . Proposition The operators J, R and Q satisfy the commutation relations QJ − JQ = −R, (12) QR − RQ = J, (13) JR − RJ = 0. (14) Proof. Use the definition of the operators involved

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First Article

Three systems of orthogonal polynomials Connections between them in terms of operators J, R, Q. Operators B = R−1 and S = JR−1 represented as singular integral operators of convolution type Bf (z) =

• R

f (t)dt 2 cosh π

2 (z − t),

Sf (x) = lim

ε→0

• |x−t|>ε

f (t)dt 2 sinh π

2 (x − t)

Boundedness of B and S on L2- and H2-spaces

Fourier transforms in the translation invariant case

• rthogonal polynomials in the weighted case

Proved that both operators are bounded on these spaces and estimates of the norms are obtained

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Second Article

Boundedness of B and S on Lp(R) and Lp(ω), where ω(x) = 1/(2 cosh π

2 x) and 1 < p < ∞.

Proved that both operators are bounded on these spaces and estimates of the norms are obtained Achieved by first proving boundedness for p = 2 and weak boundedness for p = 1, and then using interpolation to obtain boundedness for 1 < p ≤ 2. To obtain boundedness also for 2 ≤ p < ∞, we used duality in the translation invariant case, while the weighted case was partly based on the expositions on the conjugate function

• perator in [M. Riesz, Sur les fonctions conjugu´

ees, Mathematische Zeitschrift 27 (1928), 218–244].

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Third Article

Investigated the algebra generated by J, R, Q such that QJ − JQ = −R, (15) QR − RQ = J, (16) JR − RJ = 0. (17) Reordering formulas for functions of J, R, Q. [Q, p(J, R)] = −R ∂p(J, R) ∂J + J ∂p(J, R) ∂R . (18) Centralizers of J, R, Q, and thus the center of algebra as an application of the deduced reordering formulas. Ordered representations of the commutations relations by differential, integral and difference operators spectral analysis of the resulting operators for finding other families of orthogonal polynomials

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Impact and Applications of My Research

The operator S is essentially the Hilbert transform. Hf (x) = 1 πx ∗ f , Sf (x) = 1 2 sinh π

2 x ∗ f

Singular integrals are used to construct analytic disks. Analytic disks are important in cosmology and other parts of physics. All three systems belong to the class of Meixner-Pollaczek

• polynomials. These are important tools in investigating

geoscientific problems Connection of the systems to J, R and Q remarkable. QJ − JQ = −R, (19) QR − RQ = J, (20) JR − RJ = 0. (21) This is a three-dimensional Lie algebra, and appears quite

• ften in physics and engineering engineering.

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Tack s˚ a mycket! Thank you!

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