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 1 / 14

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 2 / 14

Research Topic My research is in analysis but also borders algebra. The project deals with families of orthogonal polynomials, ( p m , p n ) = 0 for m � = n operators 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 3 / 14

Definitions and Method Orthogonality on the real line R : For m � = n , � p m ( x ) p n ( x ) ω ( x ) dx = 0 . (1) R Orthogonality on the strip S = { z ∈ C : − 1 ≤ Im( z ) ≤ 1 } : For m � = n , p m ( x + i ) p n ( x + i ) + p m ( x − i ) p n ( x − i ) � ω ( x ) dx = 0 (2) 2 R where ω is some function on R which is nonnegative and locally integrable, i.e., � � x n ω ( x ) dx < ∞ (3) ω ≥ 0 , 0 < ω ( x ) dx < ∞ , 0 < A A for some A ⊂ R . This ω is called the weight function. Gram-Schmidt procedure applied to 1 , x , x 2 , x 3 , · · · 4 / 14

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. 5 / 14

Motivation σ τ ρ σ 0 = 1 τ 0 = 1 ρ 0 = 1 σ 1 = x τ 1 = x ρ 1 = x τ 2 = x 2 − 1 ρ 2 = x 2 − 2 σ 2 = x 2 σ 3 = x 3 − 2 x τ 3 = x 3 − 5 x ρ 3 = x 3 − 8 x σ 4 = x 4 − 8 x 2 τ 4 = x 4 − 14 x 2 + 9 ρ 4 = x 4 − 20 x 2 + 24 . . . . . . . . . First two columns were studied by Tsehaye K. Araaya (PhD -ISP). Comparing columns 1 and 3, we see that x ρ n = σ n +1 . (4) This is the motivation to define the operator Q by Qf = xf . (5) 6 / 14

Motivation { τ n } 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 ( x + i ) + f ( x − i ) f (0) = ω ( x ) dx . (6) 2 −∞ 7 / 14

Motivation { τ n } 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 ( x + i ) + f ( x − i ) f (0) = ω ( x ) dx . (6) 2 −∞ This is the motivation to define the operator R by Rf ( x ) = f ( x + i ) + f ( x − i ) , (7) 2 and for symmetry, we also consider the operator Jf ( x ) = f ( x + i ) − f ( x − i ) . (8) 2 i 7 / 14

Three Basic Operators Rf ( x ) = f ( x + i ) + f ( x − i ) 2 Jf ( x ) = f ( x + i ) − f ( x − i ) 2 i Qf ( x ) = xf ( x ) R R σ n − → τ n − → ρ n (9) J J σ n − → n τ n − 1 − → n ( n − 1) ρ n − 2 (10) Q σ n +1 ← − ρ n (11) 8 / 14

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 } . 9 / 14

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 9 / 14

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 f ( t ) dt f ( t ) dt � � Bf ( z ) = 2 ( z − t ) , Sf ( x ) = lim 2 cosh π 2 sinh π 2 ( x − t ) ε → 0 | x − t | >ε R Boundedness of B and S on L 2 - and H 2 -spaces Fourier transforms in the translation invariant case orthogonal polynomials in the weighted case Proved that both operators are bounded on these spaces and estimates of the norms are obtained 10 / 14

Second Article Boundedness of B and S on L p ( R ) and L p ( 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 operator in [M. Riesz, Sur les fonctions conjugu´ ees , Mathematische Zeitschrift 27 (1928), 218–244]. 11 / 14

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 ∂ p ( J , R ) . (18) ∂ J ∂ R 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 12 / 14

Impact and Applications of My Research The operator S is essentially the Hilbert transform. Hf ( x ) = 1 1 π x ∗ f , Sf ( x ) = 2 x ∗ f 2 sinh π 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 often in physics and engineering engineering. 13 / 14

Tack s˚ a mycket! Thank you! 14 / 14