Toeplitz forms and Szego's limit theorems

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Problem 7.43 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.3 "Generation of Positive Quadratic Forms"

Let ϑ\vartheta be real and let the complex numbers

c−n,c−n+1,…,c−1,c0,c1,…,cn−1,cn,c−ν=cˉν,ν=0,1,2,…,n c_{-n}, c_{-n+1}, \ldots , c_{-1}, c_{0}, c_{1}, \ldots , c_{n-1}, c_{n}, \qquad c_{-\nu } = \bar{c}_{\nu }, \quad \nu = 0, 1, 2, \ldots , n

have the following property: If

g(ϑ)=α0+2(α1cos⁡ϑ+β1sin⁡ϑ+α2cos⁡2ϑ+β2sin⁡2ϑ+⋯+αncos⁡nϑ+βnsin⁡nϑ) g(\vartheta ) = \alpha _{0} + 2(\alpha _{1} \cos \vartheta + \beta _{1} \sin \vartheta + \alpha _{2} \cos 2\vartheta + \beta _{2} \sin 2\vartheta + \cdots + \alpha _{n} \cos n\vartheta + \beta _{n} \sin n\vartheta ) =∑ν=−nnγνe−iνϑ,γν=γˉ−ν=αν+iβν,ν=0,1,2,…,n;β0=0, = \sum _{\nu =-n}^{n} \gamma _{\nu } e^{-i\nu \vartheta }, \qquad \gamma _{\nu } = \bar{\gamma }_{-\nu } = \alpha _{\nu } + i\beta _{\nu }, \quad \nu = 0, 1, 2, \ldots , n; \quad \beta _{0} = 0,

denotes an arbitrary trigonometric polynomial of degree at most nn, that does not vanish identically and does not have a negative value for any value of ϑ\vartheta, let

∑ν=−nncνγνe−iνϑ≥0(or>0). \sum _{\nu =-n}^{n} c_{\nu } \gamma _{\nu } e^{-i\nu \vartheta } \geq 0 \qquad (\text{or} > 0).

Show that for this to be true, it is necessary and sufficient that the Hermitian form

∑λ=0n∑μ=0ncμ−λxλxˉμ \sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} c_{\mu -\lambda } x_{\lambda } \bar{x}_{\mu }

is positive (or positive definite, respectively).

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Problem 7.67 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.5 "Determinants of Systems of Functions"

Let the function f(ϑ)f(\vartheta ) be properly integrable in the interval 0≤ϑ<2π0 \leq \vartheta < 2\pi. Let its Fourier series (Section 4 of Part VI) be

f(ϑ)∼a0+2∑n=1∞(ancos⁡nϑ+bnsin⁡nϑ). f(\vartheta ) \sim a_{0} + 2 \sum _{n=1}^{\infty } (a_{n} \cos n\vartheta + b_{n} \sin n\vartheta ).

The Hermite forms

Tn(f)=∑λ=0n∑μ=0ncμ−λxλxˉμ T_{n}(f) = \sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} c_{\mu -\lambda } x_{\lambda } \bar{x}_{\mu }

formed from the constants c0=a0c_{0} = a_{0}, cn=an+ibnc_{n} = a_{n} + ib_{n}, c−n=cˉnc_{-n} = \bar{c}_{n}, n=1,2,3,…n = 1, 2, 3, \ldots, are called the Toeplitz forms associated with f(ϑ)f(\vartheta ). (Cf. Problem 43).

Form the Toeplitz forms that belong to the following functions:

f(ϑ)=c=const.,f(ϑ)=a0+2(a1cos⁡ϑ+b1sin⁡ϑ), f(\vartheta ) = c = \text{const.}, \qquad f(\vartheta ) = a_{0} + 2(a_{1} \cos \vartheta + b_{1} \sin \vartheta ), f(ϑ)=1−r21−2rcos⁡ϑ+r2,0<r<1. f(\vartheta ) = \frac{1-r^{2}}{1-2r\cos \vartheta +r^{2}}, \qquad 0 < r < 1.
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Problem 7.68 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.5 "Determinants of Systems of Functions"

If the function f(ϑ)f(\vartheta ) is positive in the interval 0≤ϑ<2π0 \leq \vartheta < 2\pi, then all its associated Toeplitz forms are positive definite. More precisely: If f(ϑ)f(\vartheta ) lies between the bounds m≤f(ϑ)≤Mm \leq f(\vartheta ) \leq M in the interval 0≤ϑ<2π0 \leq \vartheta < 2\pi, then we have

m≤Tn(f)≤M, m \leq T_{n}(f) \leq M,

provided that the variables x0,x1,…,xnx_{0}, x_{1}, \ldots , x_{n} satisfy the condition Tn(1)=1T_{n}(1) = 1. The equality sign can hold here only if f(ϑ)=const.f(\vartheta ) = \text{const.} at every point of continuity of f(ϑ)f(\vartheta ).

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Problem 7.69 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.5 "Determinants of Systems of Functions"

Calculate the determinant Dn(f)D_{n}(f) of the form Tn(f)T_{n}(f) for the functions in Problem 67 and show that, if f(ϑ)f(\vartheta ) is everywhere positive,

lim⁡n→∞Dn(f)n+1=e12π∫02πlog⁡f(ϑ) dϑ=G(f). \lim _{n \rightarrow \infty } \sqrt[n+1]{D_{n}(f)} = e^{\frac{1}{2\pi } \int _{0}^{2\pi } \log f(\vartheta ) \, d\vartheta } = \mathfrak {G}(f).
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Problem 7.70 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.5 "Determinants of Systems of Functions"

Let f(ϑ)f(\vartheta ) be a trigonometric polynomial of first order and let hh be a parameter. The determinant Dn(f−h)D_{n}(f-h) of the Toeplitz form Tn(f−h)T_{n}(f-h) associated with f(ϑ)−hf(\vartheta )-h is a polynomial of degree n+1n+1 in hh with only real zeros. Calculate these zeros.

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Problem 7.71 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.5 "Determinants of Systems of Functions"

(continued). The zeros h0n,h1n,…,hnnh_{0n}, h_{1n}, \ldots , h_{nn} of Dn(f−h)D_{n}(f-h) all lie between the minimum and the maximum of f(ϑ)f(\vartheta ), that is m≤hνn≤Mm \leq h_{\nu n} \leq M, if m≤f(ϑ)≤Mm \leq f(\vartheta ) \leq M in the interval 0,2π0, 2\pi. Moreover if F(h)F(h) denotes an arbitrary properly integrable function in the interval m≤h≤Mm \leq h \leq M, then we have

lim⁡n→∞F(h0n)+F(h1n)+⋯+F(hnn)n+1=12π∫02πF[f(ϑ)] dϑ. \lim _{n \rightarrow \infty } \frac{F(h_{0n})+F(h_{1n})+\cdots +F(h_{nn})}{n+1} = \frac{1}{2\pi } \int _{0}^{2\pi } F[f(\vartheta )] \, d\vartheta .
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Problem 7.72 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 7.5 "Determinants of Systems of Functions"

Let the Hermitian form

∑λ=0n∑μ=0ncλ−μxλxˉμ,c−ν=cˉν, \sum _{\lambda =0}^{n} \sum _{\mu =0}^{n} c_{\lambda -\mu } x_{\lambda } \bar{x}_{\mu }, \qquad c_{-\nu } = \bar{c}_{\nu },

be positive definite. Then all the zeros of the polynomial

∣c0c1c2…cnc−1c0c1…cn−1c−2c−1c0…cn−2\multicolumn5c\dotfillc−n+1c−n+2c−n+3…c11zz2…zn∣ \begin{vmatrix} c_{0} & c_{1} & c_{2} & \ldots & c_{n} \\ c_{-1} & c_{0} & c_{1} & \ldots & c_{n-1} \\ c_{-2} & c_{-1} & c_{0} & \ldots & c_{n-2} \\ \multicolumn {5}{c}{\dotfill } \\ c_{-n+1} & c_{-n+2} & c_{-n+3} & \ldots & c_{1} \\ 1 & z & z^{2} & \ldots & z^{n} \end{vmatrix}

lie in the interior of the unit circle.

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References

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.