Laguerre Polynomials and Interlacing of Zeros Kathy Driver - - PowerPoint PPT Presentation

laguerre polynomials and interlacing of zeros
SMART_READER_LITE
LIVE PREVIEW

Laguerre Polynomials and Interlacing of Zeros Kathy Driver - - PowerPoint PPT Presentation

Laguerre Polynomials and Interlacing of Zeros Kathy Driver University of Cape Town SANUM Conference Stellenbosch University Kathy Driver Laguerre Polynomials 22-24 March 2016 1 / 16 Laguerre Polynomials and Interlacing of Zeros Kathy


slide-1
SLIDE 1

Laguerre Polynomials and Interlacing of Zeros

Kathy Driver University of Cape Town SANUM Conference Stellenbosch University

Kathy Driver Laguerre Polynomials 22-24 March 2016 1 / 16

slide-2
SLIDE 2

Laguerre Polynomials and Interlacing of Zeros

Kathy Driver Laguerre Polynomials 22-24 March 2016 2 / 16

slide-3
SLIDE 3

Laguerre Polynomials and Interlacing of Zeros

Joint work with Martin Muldoon

Kathy Driver Laguerre Polynomials 22-24 March 2016 2 / 16

slide-4
SLIDE 4

Laguerre Polynomials and Interlacing of Zeros

Joint work with Martin Muldoon Overview of Talk

Kathy Driver Laguerre Polynomials 22-24 March 2016 2 / 16

slide-5
SLIDE 5

Laguerre Polynomials and Interlacing of Zeros

Joint work with Martin Muldoon Overview of Talk Sharpness of t- interval where zeros of Laguerre polynomials L(α)

n

and L(α+t)

n−k

are interlacing. α > −1, t > 0. Askey Conjecture

Kathy Driver Laguerre Polynomials 22-24 March 2016 2 / 16

slide-6
SLIDE 6

Laguerre Polynomials and Interlacing of Zeros

Joint work with Martin Muldoon Overview of Talk Sharpness of t- interval where zeros of Laguerre polynomials L(α)

n

and L(α+t)

n−k

are interlacing. α > −1, t > 0. Askey Conjecture Breakdown of interlacing of zeros of L(α)

n

and L(α)

n−1 when −2 < α < −1.

Add one point to restore interlacing. Quasi-orthogonal order 1 case.

Kathy Driver Laguerre Polynomials 22-24 March 2016 2 / 16

slide-7
SLIDE 7

Laguerre Polynomial L(α)

n

Kathy Driver Laguerre Polynomials 22-24 March 2016 3 / 16

slide-8
SLIDE 8

Laguerre Polynomial L(α)

n

Laguerre polynomial L(α)

n

defined by L(α)

n (x) = n

  • k=0

n + α n + k (−x)k k! . (1)

Kathy Driver Laguerre Polynomials 22-24 March 2016 3 / 16

slide-9
SLIDE 9

Laguerre Polynomial L(α)

n

Laguerre polynomial L(α)

n

defined by L(α)

n (x) = n

  • k=0

n + α n + k (−x)k k! . (1) For α > −1, sequence {L(α)

n (x), n = 0, 1, . . . } orthogonal with respect to

xαe−x on (0, ∞). All n zeros of Lα

n are real, distinct and positive. ∞

  • Ln(x)Lm(x)xαe−x dx = 0

for n = m,

  • LnLnxαe−x dx = 0

α > −1 necessary for convergence of integral each m, n.

Kathy Driver Laguerre Polynomials 22-24 March 2016 3 / 16

slide-10
SLIDE 10

Interlacing of zeros

Orthogonal sequence {pn}∞

n=0, zeros of pn and pn−1 are interlacing:

x1,n < x1,n−1 < x2,n < x2,n−1 · · · < xn−1,n < xn−1,n−1 < xn,n

Kathy Driver Laguerre Polynomials 22-24 March 2016 4 / 16

slide-11
SLIDE 11

Interlacing of zeros

Orthogonal sequence {pn}∞

n=0, zeros of pn and pn−1 are interlacing:

x1,n < x1,n−1 < x2,n < x2,n−1 · · · < xn−1,n < xn−1,n−1 < xn,n p, q real polynomials, real, simple, disjoint zeros,deg(p)> deg(q), zeros of p and q interlace if each zero of q lies between two successive zeros of p and at most one zero of q between any two successive zeros of p.

Kathy Driver Laguerre Polynomials 22-24 March 2016 4 / 16

slide-12
SLIDE 12

Askey Conjecture 1989

Zeros of orthogonal Jacobi polynomials P(α,β)

n

and P(α+2,β)

n

interlace

Kathy Driver Laguerre Polynomials 22-24 March 2016 5 / 16

slide-13
SLIDE 13

Askey Conjecture 1989

Zeros of orthogonal Jacobi polynomials P(α,β)

n

and P(α+2,β)

n

interlace Electrostatic interpretation of zeros of Jacobi polynomials, increasing one parameter means increasing charge at one endpoint.

Kathy Driver Laguerre Polynomials 22-24 March 2016 5 / 16

slide-14
SLIDE 14

Askey Conjecture 1989

Zeros of orthogonal Jacobi polynomials P(α,β)

n

and P(α+2,β)

n

interlace Electrostatic interpretation of zeros of Jacobi polynomials, increasing one parameter means increasing charge at one endpoint. Analysis of Mixed TTRR’s gives upper and lower bounds for zeros of classical OP’s

Kathy Driver Laguerre Polynomials 22-24 March 2016 5 / 16

slide-15
SLIDE 15

Askey Conjecture 1989

Zeros of orthogonal Jacobi polynomials P(α,β)

n

and P(α+2,β)

n

interlace Electrostatic interpretation of zeros of Jacobi polynomials, increasing one parameter means increasing charge at one endpoint. Analysis of Mixed TTRR’s gives upper and lower bounds for zeros of classical OP’s Askey Conjecture. Driver, Jordaan and Mbuyi: Zeros of Jacobi P(α,β)

n

and P(α+k,β−l)

n

interlace if k, l ≤ 2 provided β − l remains > −1.

Kathy Driver Laguerre Polynomials 22-24 March 2016 5 / 16

slide-16
SLIDE 16

Askey Conjecture 1989

Zeros of orthogonal Jacobi polynomials P(α,β)

n

and P(α+2,β)

n

interlace Electrostatic interpretation of zeros of Jacobi polynomials, increasing one parameter means increasing charge at one endpoint. Analysis of Mixed TTRR’s gives upper and lower bounds for zeros of classical OP’s Askey Conjecture. Driver, Jordaan and Mbuyi: Zeros of Jacobi P(α,β)

n

and P(α+k,β−l)

n

interlace if k, l ≤ 2 provided β − l remains > −1. Ismail, Dimitrov and Rafaeli: Askey Conjecture sharp.

Kathy Driver Laguerre Polynomials 22-24 March 2016 5 / 16

slide-17
SLIDE 17

Zeros of Laguerre polynomials, different parameters

Kathy Driver Laguerre Polynomials 22-24 March 2016 6 / 16

slide-18
SLIDE 18

Zeros of Laguerre polynomials, different parameters

Kerstin Jordaan- KD 2011 Indag Math

Kathy Driver Laguerre Polynomials 22-24 March 2016 6 / 16

slide-19
SLIDE 19

Zeros of Laguerre polynomials, different parameters

Kerstin Jordaan- KD 2011 Indag Math Parameter difference = integer, assume no common zeros Zeros Lα

n and Lα+k n−2 interlace, k ∈ {1, 2, 3, 4}

Kathy Driver Laguerre Polynomials 22-24 March 2016 6 / 16

slide-20
SLIDE 20

Zeros of Laguerre polynomials, different parameters

Kerstin Jordaan- KD 2011 Indag Math Parameter difference = integer, assume no common zeros Zeros Lα

n and Lα+k n−2 interlace, k ∈ {1, 2, 3, 4}

At least one ”gap interval”, no zero of Lα+t

n−2, changes with t.

Markov monotonicity argument breaks down

Kathy Driver Laguerre Polynomials 22-24 March 2016 6 / 16

slide-21
SLIDE 21

Zeros of Laguerre polynomials, different parameters

Kerstin Jordaan- KD 2011 Indag Math Parameter difference = integer, assume no common zeros Zeros Lα

n and Lα+k n−2 interlace, k ∈ {1, 2, 3, 4}

At least one ”gap interval”, no zero of Lα+t

n−2, changes with t.

Markov monotonicity argument breaks down Conjecture Kerstin Jordaan -KD 2011 Zeros Lα

n and Lα+t n−2 interlace for 0 ≤ t ≤ 4 if the two polynomials have no

common zeros

Kathy Driver Laguerre Polynomials 22-24 March 2016 6 / 16

slide-22
SLIDE 22

Sharp interval. Zeros of Laguerre polynomials

Martin Muldoon and KD Journal of Approximation Theory 2013

Kathy Driver Laguerre Polynomials 22-24 March 2016 7 / 16

slide-23
SLIDE 23

Sharp interval. Zeros of Laguerre polynomials

Martin Muldoon and KD Journal of Approximation Theory 2013 Each t, 0 ≤ t ≤ 2k, excluding t for which common zeros occur, zeros of L(α)

n

and L(α+t)

n−k

interlace

Kathy Driver Laguerre Polynomials 22-24 March 2016 7 / 16

slide-24
SLIDE 24

Sharp interval. Zeros of Laguerre polynomials

Martin Muldoon and KD Journal of Approximation Theory 2013 Each t, 0 ≤ t ≤ 2k, excluding t for which common zeros occur, zeros of L(α)

n

and L(α+t)

n−k

interlace Interval 0 ≤ t ≤ 2k is largest possible for which interlacing holds each n, α, k

Kathy Driver Laguerre Polynomials 22-24 March 2016 7 / 16

slide-25
SLIDE 25

Proofs of interlacing and sharpness results

t interval 0 ≤ t ≤ 2k largest possible for interlacing of zeros of L(α)

n ,

L(α+t)

n−k , each n ∈ N, 0 < k ≤ n − 2, and fixed α ≥ 0, (excluding t for

which common zeros occur)

Kathy Driver Laguerre Polynomials 22-24 March 2016 8 / 16

slide-26
SLIDE 26

Proofs of interlacing and sharpness results

t interval 0 ≤ t ≤ 2k largest possible for interlacing of zeros of L(α)

n ,

L(α+t)

n−k , each n ∈ N, 0 < k ≤ n − 2, and fixed α ≥ 0, (excluding t for

which common zeros occur) Interlacing proof involves monotonicity properties of common zeros of L(α)

n

and L(α+t)

n−k

and Sturm Comparison Theorem.

Kathy Driver Laguerre Polynomials 22-24 March 2016 8 / 16

slide-27
SLIDE 27

Proofs of interlacing and sharpness results

t interval 0 ≤ t ≤ 2k largest possible for interlacing of zeros of L(α)

n ,

L(α+t)

n−k , each n ∈ N, 0 < k ≤ n − 2, and fixed α ≥ 0, (excluding t for

which common zeros occur) Interlacing proof involves monotonicity properties of common zeros of L(α)

n

and L(α+t)

n−k

and Sturm Comparison Theorem. Sharpness Inequalities satisfied by zeros of L(α)

n

and zeros of Bessel function Asymptotic behaviour of zeros of L(α)

n

using Airy function

Kathy Driver Laguerre Polynomials 22-24 March 2016 8 / 16

slide-28
SLIDE 28

Zeros of L0

6(x) and Lt 5(x) as a function of t

Kathy Driver Laguerre Polynomials 22-24 March 2016 9 / 16

slide-29
SLIDE 29

Zeros of L0

6(x) and Lt 4(x) as a function of t

Kathy Driver Laguerre Polynomials 22-24 March 2016 10 / 16

slide-30
SLIDE 30

Zeros of L0

6(x) and Lt 3(x) as a function of t

Kathy Driver Laguerre Polynomials 22-24 March 2016 11 / 16

slide-31
SLIDE 31

Laguerre sequences {L(α)

n }∞ n=0, −2 < α < −1

As α decreases below −1, one positive zero of L(α)

n

leaves (0, ∞) each time α passes through −1, −2, . . . , −n.

Kathy Driver Laguerre Polynomials 22-24 March 2016 12 / 16

slide-32
SLIDE 32

Laguerre sequences {L(α)

n }∞ n=0, −2 < α < −1

As α decreases below −1, one positive zero of L(α)

n

leaves (0, ∞) each time α passes through −1, −2, . . . , −n. Sequences {L(α)

n }∞ n=0, −2 < α < −1 only Laguerre sequences (except

  • rthogonal)

Kathy Driver Laguerre Polynomials 22-24 March 2016 12 / 16

slide-33
SLIDE 33

Laguerre sequences {L(α)

n }∞ n=0, −2 < α < −1

As α decreases below −1, one positive zero of L(α)

n

leaves (0, ∞) each time α passes through −1, −2, . . . , −n. Sequences {L(α)

n }∞ n=0, −2 < α < −1 only Laguerre sequences (except

  • rthogonal)

for which ALL zeros of L(α)

n

are real.

Kathy Driver Laguerre Polynomials 22-24 March 2016 12 / 16

slide-34
SLIDE 34

Laguerre sequences {L(α)

n }∞ n=0, −2 < α < −1

As α decreases below −1, one positive zero of L(α)

n

leaves (0, ∞) each time α passes through −1, −2, . . . , −n. Sequences {L(α)

n }∞ n=0, −2 < α < −1 only Laguerre sequences (except

  • rthogonal)

for which ALL zeros of L(α)

n

are real. Brezinski, Driver, Redivo-Zaglia 2004 If −2 < α < −1, zeros of L(α)

n

are real, simple, n − 1 are positive, 1 negative

Kathy Driver Laguerre Polynomials 22-24 March 2016 12 / 16

slide-35
SLIDE 35

Zeros of Laguerre polynomials L(α)

n , −2 < α < −1

Question Muldoon-Driver Interlacing of zeros of L(α)

n

and zeros of L(α)

n−1 when −2 < α < −1 ??

Kathy Driver Laguerre Polynomials 22-24 March 2016 13 / 16

slide-36
SLIDE 36

Zeros of Laguerre polynomials L(α)

n , −2 < α < −1

Question Muldoon-Driver Interlacing of zeros of L(α)

n

and zeros of L(α)

n−1 when −2 < α < −1 ??

Sequence not orthogonal but all zeros real–is there interlacing for any n ∈ N?

Kathy Driver Laguerre Polynomials 22-24 March 2016 13 / 16

slide-37
SLIDE 37

Zeros L(α)

n , L(α) n−1, −2 < α < −1

x1,n < 0 < x2,n < · · · < xn,n zeros of L(α)

n

x1,n−1 < 0 < x2,n−1 < · · · < xn−1,n−1 zeros of L(α)

n−1

Kathy Driver Laguerre Polynomials 22-24 March 2016 14 / 16

slide-38
SLIDE 38

Zeros L(α)

n , L(α) n−1, −2 < α < −1

x1,n < 0 < x2,n < · · · < xn,n zeros of L(α)

n

x1,n−1 < 0 < x2,n−1 < · · · < xn−1,n−1 zeros of L(α)

n−1

x1,n−1 < x1,n < 0 < x2,n < x2,n−1 < · · · < xn−1,n < xn−1,n−1 < xn,n

Kathy Driver Laguerre Polynomials 22-24 March 2016 14 / 16

slide-39
SLIDE 39

Zeros L(α)

n , L(α) n−1, −2 < α < −1

x1,n < 0 < x2,n < · · · < xn,n zeros of L(α)

n

x1,n−1 < 0 < x2,n−1 < · · · < xn−1,n−1 zeros of L(α)

n−1

x1,n−1 < x1,n < 0 < x2,n < x2,n−1 < · · · < xn−1,n < xn−1,n−1 < xn,n Positive zeros L(α)

n−1, L(α) n

interlacing, real zeros NOT

Kathy Driver Laguerre Polynomials 22-24 March 2016 14 / 16

slide-40
SLIDE 40

Zeros L(α)

n , L(α) n−1, −2 < α < −1

x1,n < 0 < x2,n < · · · < xn,n zeros of L(α)

n

x1,n−1 < 0 < x2,n−1 < · · · < xn−1,n−1 zeros of L(α)

n−1

x1,n−1 < x1,n < 0 < x2,n < x2,n−1 < · · · < xn−1,n < xn−1,n−1 < xn,n Positive zeros L(α)

n−1, L(α) n

interlacing, real zeros NOT Zeros xL(α)

n−1(x), L(α) n (x) interlace

Kathy Driver Laguerre Polynomials 22-24 March 2016 14 / 16

slide-41
SLIDE 41

Zeros L(α)

n , L(α) n−1, −2 < α < −1

x1,n < 0 < x2,n < · · · < xn,n zeros of L(α)

n

x1,n−1 < 0 < x2,n−1 < · · · < xn−1,n−1 zeros of L(α)

n−1

x1,n−1 < x1,n < 0 < x2,n < x2,n−1 < · · · < xn−1,n < xn−1,n−1 < xn,n Positive zeros L(α)

n−1, L(α) n

interlacing, real zeros NOT Zeros xL(α)

n−1(x), L(α) n (x) interlace

L(α)

n

and L(α)

n−1 are co-prime each n ∈ N

Kathy Driver Laguerre Polynomials 22-24 March 2016 14 / 16

slide-42
SLIDE 42

Zeros L(α)

n , L(α) n−1, −2 < α < −1

x1,n < 0 < x2,n < · · · < xn,n zeros of L(α)

n

x1,n−1 < 0 < x2,n−1 < · · · < xn−1,n−1 zeros of L(α)

n−1

x1,n−1 < x1,n < 0 < x2,n < x2,n−1 < · · · < xn−1,n < xn−1,n−1 < xn,n Positive zeros L(α)

n−1, L(α) n

interlacing, real zeros NOT Zeros xL(α)

n−1(x), L(α) n (x) interlace

L(α)

n

and L(α)

n−1 are co-prime each n ∈ N

Negative zero of L(α)

n

increases with n (Ismail, Zeng 2015)

Kathy Driver Laguerre Polynomials 22-24 March 2016 14 / 16

slide-43
SLIDE 43

Zeros of L0

6(x) and Lt 3(x) as functions of t

Zeros of L(−3/2)

2

(blue) and L(−3/2)

3

(red)

Kathy Driver Laguerre Polynomials 22-24 March 2016 15 / 16

slide-44
SLIDE 44

Thank you

Kathy Driver Laguerre Polynomials 22-24 March 2016 16 / 16