Abstract

We investigate relationships between the structure of a real object of compact support and the realness of the zeros of its Fourier transform. We introduce a necessary condition, and some sufficient conditions are presented for one-dimensional functions. A very useful property of the sinc function is also presented.

© 1999 Optical Society of America

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References

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  1. M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 499–529.
  2. M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
    [CrossRef]
  3. P.-T. Chen, M. A. Fiddy, “Image reconstruction from power spectral data with use of point-zero locations,” J. Opt. Soc. Am. A 11, 2210–2214 (1994).
    [CrossRef]
  4. P.-T. Chen, M. A. Fiddy, C.-W. Liao, D. A. Pommet, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 13, 1524–1531 (1996).
    [CrossRef]
  5. C.-W. Liao, M. A. Fiddy, C. L. Byrne, “Imaging from the zero locations of far-field intensity data,” J. Opt. Soc. Am. A 14, 3155–3161 (1997).
    [CrossRef]
  6. A. J. Noushin, M. A. Fiddy, D. A. Pommet, “Optical imaging from scattering data: imaging from Fourier-intensity data,” in Computational, Experimental, and Numerical Methods for Solving Ill-Posed Inverse Imaging Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds. Proc. SPIE3171, 96–103 (1997).
    [CrossRef]
  7. R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980).
  8. A. J. Noushin, “Real functions of compact support with real zeros in the Fourier domain” (University of Massachusetts Lowell, Lowell, Mass., 1999).
  9. M. A. Fiddy, M. J. Huiser, M. Nieto-Vesperinas, G. Ross, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
    [CrossRef]
  10. G. Polya, “Uber Trigonometrische Integrale mit nur Reellen Nullstellen,” J. Reine Agnew. Math. 158, 6–18 (1927).
  11. E. Hille, Analytic Function Theory (Chelsea, New York, 1959).
  12. A. D. Wunsch, Complex Variables with Applications (Addison-Wesley, Reading, Mass., 1994).

1997 (1)

1996 (1)

1994 (1)

1985 (1)

1982 (1)

M. A. Fiddy, M. J. Huiser, M. Nieto-Vesperinas, G. Ross, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
[CrossRef]

1927 (1)

G. Polya, “Uber Trigonometrische Integrale mit nur Reellen Nullstellen,” J. Reine Agnew. Math. 158, 6–18 (1927).

Byrne, C. L.

Chen, P.-T.

Fiddy, M. A.

C.-W. Liao, M. A. Fiddy, C. L. Byrne, “Imaging from the zero locations of far-field intensity data,” J. Opt. Soc. Am. A 14, 3155–3161 (1997).
[CrossRef]

P.-T. Chen, M. A. Fiddy, C.-W. Liao, D. A. Pommet, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 13, 1524–1531 (1996).
[CrossRef]

P.-T. Chen, M. A. Fiddy, “Image reconstruction from power spectral data with use of point-zero locations,” J. Opt. Soc. Am. A 11, 2210–2214 (1994).
[CrossRef]

M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
[CrossRef]

M. A. Fiddy, M. J. Huiser, M. Nieto-Vesperinas, G. Ross, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
[CrossRef]

A. J. Noushin, M. A. Fiddy, D. A. Pommet, “Optical imaging from scattering data: imaging from Fourier-intensity data,” in Computational, Experimental, and Numerical Methods for Solving Ill-Posed Inverse Imaging Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds. Proc. SPIE3171, 96–103 (1997).
[CrossRef]

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 499–529.

Hille, E.

E. Hille, Analytic Function Theory (Chelsea, New York, 1959).

Huiser, M. J.

M. A. Fiddy, M. J. Huiser, M. Nieto-Vesperinas, G. Ross, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
[CrossRef]

Liao, C.-W.

Nieto-Vesperinas, M.

M. A. Fiddy, M. J. Huiser, M. Nieto-Vesperinas, G. Ross, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
[CrossRef]

Noushin, A. J.

A. J. Noushin, M. A. Fiddy, D. A. Pommet, “Optical imaging from scattering data: imaging from Fourier-intensity data,” in Computational, Experimental, and Numerical Methods for Solving Ill-Posed Inverse Imaging Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds. Proc. SPIE3171, 96–103 (1997).
[CrossRef]

A. J. Noushin, “Real functions of compact support with real zeros in the Fourier domain” (University of Massachusetts Lowell, Lowell, Mass., 1999).

Polya, G.

G. Polya, “Uber Trigonometrische Integrale mit nur Reellen Nullstellen,” J. Reine Agnew. Math. 158, 6–18 (1927).

Pommet, D. A.

P.-T. Chen, M. A. Fiddy, C.-W. Liao, D. A. Pommet, “Blind deconvolution and phase retrieval from point zeros,” J. Opt. Soc. Am. A 13, 1524–1531 (1996).
[CrossRef]

A. J. Noushin, M. A. Fiddy, D. A. Pommet, “Optical imaging from scattering data: imaging from Fourier-intensity data,” in Computational, Experimental, and Numerical Methods for Solving Ill-Posed Inverse Imaging Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds. Proc. SPIE3171, 96–103 (1997).
[CrossRef]

Ross, G.

M. A. Fiddy, M. J. Huiser, M. Nieto-Vesperinas, G. Ross, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
[CrossRef]

Scivier, M. S.

Wunsch, A. D.

A. D. Wunsch, Complex Variables with Applications (Addison-Wesley, Reading, Mass., 1994).

Young, R. M.

R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980).

J. Opt. Soc. Am. A (4)

J. Reine Agnew. Math. (1)

G. Polya, “Uber Trigonometrische Integrale mit nur Reellen Nullstellen,” J. Reine Agnew. Math. 158, 6–18 (1927).

Opt. Acta (1)

M. A. Fiddy, M. J. Huiser, M. Nieto-Vesperinas, G. Ross, “Encoding of information in inverse optical problems,” Opt. Acta 29, 23–40 (1982).
[CrossRef]

Other (6)

M. A. Fiddy, “The role of analyticity in image recovery,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 499–529.

E. Hille, Analytic Function Theory (Chelsea, New York, 1959).

A. D. Wunsch, Complex Variables with Applications (Addison-Wesley, Reading, Mass., 1994).

A. J. Noushin, M. A. Fiddy, D. A. Pommet, “Optical imaging from scattering data: imaging from Fourier-intensity data,” in Computational, Experimental, and Numerical Methods for Solving Ill-Posed Inverse Imaging Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds. Proc. SPIE3171, 96–103 (1997).
[CrossRef]

R. M. Young, An Introduction to Nonharmonic Fourier Series (Academic, New York, 1980).

A. J. Noushin, “Real functions of compact support with real zeros in the Fourier domain” (University of Massachusetts Lowell, Lowell, Mass., 1999).

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Figures (2)

Fig. 1
Fig. 1

Area under the curve un cos πu is the nth derivative of the sinc function in which n is even. In this graph, |Am| <|Am+1|, where Am is the area under the curve in between two consecutive integer locations. For example, A3 is the shaded area from 2 to 3, as shown above.

Fig. 2
Fig. 2

Area under the curve un sin πu is the nth derivative of the sinc function in which n is odd. In this graph, |Am| <|Am+1|, where Am is the area under the curve in between two half-integer locations. For example, A3 is the shaded area from 5/2 to 7/2, as shown above.

Equations (57)

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F(z)=12π -+f(t)exp(izt)dt=12π -ππf(t)exp(izt)dt,
|F(z)|12π -ππ|f(t)exp(izt)|dt12π exp(π|y|)-ππ|f(t)|dt
-+|F(x)|2dx=2π-ππ|f(t)|2dx<.
F(z)=12π -aaf(t)exp(izt)dt.
F(λn)=-π+πf(t)exp(iλnt)dt=0,
n=0, ±1, ±2, ±3,.
F(z)=12π -ππ exp(izt)dt=sin πzπz,
-ππ exp(iαnt)dt=0n,
F(z)=N(z)sin πzD(z),
f(t)=P0tπ1-pqqp q2q2-p2×Ap cos(pt+ϕp),
Ap=q [(xq2+yq2-p2)2+4p2yq2]1/2xq2+yq2,
ϕp=q tan-12pyqxq2+yq2-p2.
f(t)=P0tπ1-pqp q2q2-p2×q xq2-p2xq2cos pt.
f(t)=[f(-t)]*,-<t<,
f(t)=O exp(-|t|b),t, b>2,
F(z)=12π1iπz[f(π)exp(iπz)-f(-π)exp(-iπz)]-1iπz -ππf(t)exp(izt)dt,
zn=2n π+i logf(π)f(-π)2π,
F(z)=(A+iB)exp[(α+iβ)z]zm1-zznexpzzn,
Im-(A-iB)f(t)exp(ixt)dtrealx.
A-f(t)sin xtdt+B-+f(t)cos xtdt=0realx.
F(z)=12π -ππ exp(izt)dt=sin πzπz
F(z)=πz cos πz-sin πzπz2,
F(z)=sin πzπz=12π -ππ exp(izt)dt
F(n)(z)=(i)n2π -ππtn exp(izt)dt.
zn+1(iπ)n F(n)(z)=0zun cos πudu,evenn,
zn+1(iπ)n+1 F(n)(z)=0zun sin πudu,oddn.
b0=01/2un cos πudu,
b1/2=1/21un cos πudu.
b0+b1/2<0.
Am=mm+1un cos πudu.
Am+Am-1<0,evenm,
Am+Am-1>0,oddm.
b1=03/2un sin πudu,
b0=01/2un sin πudu>0.
b1<0.
Am=m-1/2m+1/2un sin πudu.
Am+Am-1>0,evenm,
Am+Am-1<0,oddm.
mm+1un cos πudu+m-1mun cos πudu<0.
01(w+m)n cos πwdw+01(w+m-1)n cos πwdw<0
01[(w+m)n+(w+m-1)n]cos πwdw<0.
(-1)nF(2n)(x)=2z(2n+1) 0zu2n cos πudu
-ππf (t)exp(int)dt=0n,
-ππφ(t)exp(iλnt)dt=0n
|λn-n|12p,n=0, ±1, ±2, ±3,
fN(t)=n=0Nant2n,an0.
FN(z)=12π -ππf(t)exp(izt)dt=12π -ππ n=0Nant2n exp(izt)dt=12π n=0Nan-ππt2n exp(izt)dt,
FN(z)=n=0Ncn d2ndz2n sin πzπz,cn=(-1)n2π an.
0< |z-z0| <δ  |F(z)| 0.
fN(t)=tn=0Nantn,evenn,
12π -ππ(t)exp(ixt)dt<1πx,x0,
F(x)=12π -ππ[1+(t)]exp(ixt)dt=sin πxπx+12π -ππ(t)exp(ixt)dt.
12π -ππ(t)exp(ixt)dt<1πx,x>x0,
x0-ππ|(t)|dt<2,
-ππ(t)exp(ixt)dt-ππ|(t)exp(ixt)|dt=-ππ|(t)|dt.
12π -ππ|(t)|dt<1πx0,
12π -ππ(t)exp(ixt)dt<1πx0,0<x<x0.

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