Abstract

Along with a review of some of the mathematical foundations of computed tomography, the article contains new results on derivation of reconstruction formulas in a general setting encompassing all standard formulas; discussion and examples of the role of the point spread function with recipes for producing suitable ones; formulas for, and examples of, the reconstruction of certain functions of the attenuation coefficient, e.g., sharpened versions of it, some of them with the property that reconstruction at a point requires only the attenuation along rays meeting a small neighborhood of the point.

© 1985 Optical Society of America

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References

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  1. K. T. Smith, “Inversion of the X-Ray Transform,” SIAM-AMS Proc. 14, 41 (1984).
  2. K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and Mathematical Aspects of Reconstructing Objects from Radiographs,” BAMS1227 (1977)
  3. J. V. Leahy, K. T. Smith, D. C. Solmon, “Uniqueness, Nonuniqueness, and Inversion in the X-ray and Radon Problems,” at International Symposium on Ill-Posed Problems, U. Delaware, Newark, 1979 (to appear).
  4. C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The Divergent Beam X-Ray Transform,” Rocky Mount. J. Math. 253 (1980).
  5. B. E. Petersen, K. T. Smith, D. C. Solmon, “Sums of Plane Waves and the Range of the Radon Transform,” Math. Ann. 163 (1979).
  6. J. Boman, “On the Closure of Sums of Plane Waves and the Range of the X-Ray Transform, I, II,”. Mathematics Department, U. Stockholm (1981), (1982). To appear in Ann. Inst. Fourier. Grenoble.
  7. D. V. Finch, D. C. Solmon, “Sums of Homogeneous Functions and the Range of the Divergent Beam X-Ray Transform,” Numer. Functional Anal. Optim. 5, 363 (1983).
    [CrossRef]
  8. B. F. Logan, L. A. Shepp, “Optimal Reconstruction of a Function from its Projections,” Duke Math. J. XX, 645 (1975).
    [CrossRef]
  9. C. Hamaker, D. C. Solmon, “The Angles Between the Null Spaces of X-Rays,” J. Math. Anal. Appl. XX, 1 (1978).
    [CrossRef]
  10. M. E. Davison, F. A. Grunbaum, “Tomographic Reconstruction with Arbitrary Directions,” Commun. Pure. Appl. Math. XX, 77 (1981).
    [CrossRef]
  11. R. B. Marr, “An Overview of Image Reconstruction,” at International Symposium of Ill-Posed Problems, U. Delaware, Newark, 1979. (To appear.)
  12. K. J. Falconer, “Consistency Conditions for a Finite Set of Projections of a Function,” Math. Proc. Cambridge Philos. Soc. XX, 61 (1979).
    [CrossRef]
  13. M. Riesz, “Intégrales de Riemann-Liouville et Potentiels,” Acta Szeged. XX, 1 (1938).
  14. J. Radon, “Über die Bestimmung von Functionen durch ihre Integralwerte langs gewissen Mannigfaltigkeiten,” Ber. Verh. Sach. Akad. Wiss. Leipzig. Math.-Nat. kl. XX, 262, (1977).
  15. N. Ramachandran, A. V. Lakshminarayanan, “Three-Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms,” Proc. Nat. Acad. Sci. U.S.A. 2236 (1971).
  16. A. V. Lakshminarayanan, “Reconstruction from Divergent X-Ray Data,” SUNY Tech. Report 92, Computer Sciences Department, Buffalo, N.Y. (1975).
  17. H. J. Scudder, “Introduction to Computer Aided Tomography,” Proc. IEEE, 66, 628 (1978).
    [CrossRef]
  18. K. T. Smith, “Reconstruction Formulas in Computed Tomography. Computed Tomography,” in Proceedings Symposium on Applied Mathematics No. 27, L. A. Shepp, Ed. (American Mathematical Society, Providence, R.I., 1983).
    [CrossRef]
  19. L. Schwartz, “Théorie des Distributions, I, II,” Ac. Sci. Ind. 1091, 1122 (1950–51).
  20. A. P. Calderón, A. Zygmund, “On the Existence of Certain Singular Integrals,” Acta Math. 88, 85 (1952).
    [CrossRef]
  21. O. Frostman, “Potentiels d’équilibre et capacité des ensembles,” Thesis, Lund (1935).
  22. N. Aronszajn, K. T. Smith, “Theory of Bessel Potentials, Part I,” Ann. Inst. Fourier Grenoble XI, 385 (1961).
    [CrossRef]

1984 (1)

K. T. Smith, “Inversion of the X-Ray Transform,” SIAM-AMS Proc. 14, 41 (1984).

1983 (1)

D. V. Finch, D. C. Solmon, “Sums of Homogeneous Functions and the Range of the Divergent Beam X-Ray Transform,” Numer. Functional Anal. Optim. 5, 363 (1983).
[CrossRef]

1981 (1)

M. E. Davison, F. A. Grunbaum, “Tomographic Reconstruction with Arbitrary Directions,” Commun. Pure. Appl. Math. XX, 77 (1981).
[CrossRef]

1980 (1)

C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The Divergent Beam X-Ray Transform,” Rocky Mount. J. Math. 253 (1980).

1979 (2)

B. E. Petersen, K. T. Smith, D. C. Solmon, “Sums of Plane Waves and the Range of the Radon Transform,” Math. Ann. 163 (1979).

K. J. Falconer, “Consistency Conditions for a Finite Set of Projections of a Function,” Math. Proc. Cambridge Philos. Soc. XX, 61 (1979).
[CrossRef]

1978 (2)

H. J. Scudder, “Introduction to Computer Aided Tomography,” Proc. IEEE, 66, 628 (1978).
[CrossRef]

C. Hamaker, D. C. Solmon, “The Angles Between the Null Spaces of X-Rays,” J. Math. Anal. Appl. XX, 1 (1978).
[CrossRef]

1977 (2)

J. Radon, “Über die Bestimmung von Functionen durch ihre Integralwerte langs gewissen Mannigfaltigkeiten,” Ber. Verh. Sach. Akad. Wiss. Leipzig. Math.-Nat. kl. XX, 262, (1977).

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and Mathematical Aspects of Reconstructing Objects from Radiographs,” BAMS1227 (1977)

1975 (1)

B. F. Logan, L. A. Shepp, “Optimal Reconstruction of a Function from its Projections,” Duke Math. J. XX, 645 (1975).
[CrossRef]

1971 (1)

N. Ramachandran, A. V. Lakshminarayanan, “Three-Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms,” Proc. Nat. Acad. Sci. U.S.A. 2236 (1971).

1961 (1)

N. Aronszajn, K. T. Smith, “Theory of Bessel Potentials, Part I,” Ann. Inst. Fourier Grenoble XI, 385 (1961).
[CrossRef]

1952 (1)

A. P. Calderón, A. Zygmund, “On the Existence of Certain Singular Integrals,” Acta Math. 88, 85 (1952).
[CrossRef]

1938 (1)

M. Riesz, “Intégrales de Riemann-Liouville et Potentiels,” Acta Szeged. XX, 1 (1938).

Aronszajn, N.

N. Aronszajn, K. T. Smith, “Theory of Bessel Potentials, Part I,” Ann. Inst. Fourier Grenoble XI, 385 (1961).
[CrossRef]

Boman, J.

J. Boman, “On the Closure of Sums of Plane Waves and the Range of the X-Ray Transform, I, II,”. Mathematics Department, U. Stockholm (1981), (1982). To appear in Ann. Inst. Fourier. Grenoble.

Calderón, A. P.

A. P. Calderón, A. Zygmund, “On the Existence of Certain Singular Integrals,” Acta Math. 88, 85 (1952).
[CrossRef]

Davison, M. E.

M. E. Davison, F. A. Grunbaum, “Tomographic Reconstruction with Arbitrary Directions,” Commun. Pure. Appl. Math. XX, 77 (1981).
[CrossRef]

Falconer, K. J.

K. J. Falconer, “Consistency Conditions for a Finite Set of Projections of a Function,” Math. Proc. Cambridge Philos. Soc. XX, 61 (1979).
[CrossRef]

Finch, D. V.

D. V. Finch, D. C. Solmon, “Sums of Homogeneous Functions and the Range of the Divergent Beam X-Ray Transform,” Numer. Functional Anal. Optim. 5, 363 (1983).
[CrossRef]

Frostman, O.

O. Frostman, “Potentiels d’équilibre et capacité des ensembles,” Thesis, Lund (1935).

Grunbaum, F. A.

M. E. Davison, F. A. Grunbaum, “Tomographic Reconstruction with Arbitrary Directions,” Commun. Pure. Appl. Math. XX, 77 (1981).
[CrossRef]

Hamaker, C.

C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The Divergent Beam X-Ray Transform,” Rocky Mount. J. Math. 253 (1980).

C. Hamaker, D. C. Solmon, “The Angles Between the Null Spaces of X-Rays,” J. Math. Anal. Appl. XX, 1 (1978).
[CrossRef]

Lakshminarayanan, A. V.

N. Ramachandran, A. V. Lakshminarayanan, “Three-Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms,” Proc. Nat. Acad. Sci. U.S.A. 2236 (1971).

A. V. Lakshminarayanan, “Reconstruction from Divergent X-Ray Data,” SUNY Tech. Report 92, Computer Sciences Department, Buffalo, N.Y. (1975).

Leahy, J. V.

J. V. Leahy, K. T. Smith, D. C. Solmon, “Uniqueness, Nonuniqueness, and Inversion in the X-ray and Radon Problems,” at International Symposium on Ill-Posed Problems, U. Delaware, Newark, 1979 (to appear).

Logan, B. F.

B. F. Logan, L. A. Shepp, “Optimal Reconstruction of a Function from its Projections,” Duke Math. J. XX, 645 (1975).
[CrossRef]

Marr, R. B.

R. B. Marr, “An Overview of Image Reconstruction,” at International Symposium of Ill-Posed Problems, U. Delaware, Newark, 1979. (To appear.)

Petersen, B. E.

B. E. Petersen, K. T. Smith, D. C. Solmon, “Sums of Plane Waves and the Range of the Radon Transform,” Math. Ann. 163 (1979).

Radon, J.

J. Radon, “Über die Bestimmung von Functionen durch ihre Integralwerte langs gewissen Mannigfaltigkeiten,” Ber. Verh. Sach. Akad. Wiss. Leipzig. Math.-Nat. kl. XX, 262, (1977).

Ramachandran, N.

N. Ramachandran, A. V. Lakshminarayanan, “Three-Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms,” Proc. Nat. Acad. Sci. U.S.A. 2236 (1971).

Riesz, M.

M. Riesz, “Intégrales de Riemann-Liouville et Potentiels,” Acta Szeged. XX, 1 (1938).

Schwartz, L.

L. Schwartz, “Théorie des Distributions, I, II,” Ac. Sci. Ind. 1091, 1122 (1950–51).

Scudder, H. J.

H. J. Scudder, “Introduction to Computer Aided Tomography,” Proc. IEEE, 66, 628 (1978).
[CrossRef]

Shepp, L. A.

B. F. Logan, L. A. Shepp, “Optimal Reconstruction of a Function from its Projections,” Duke Math. J. XX, 645 (1975).
[CrossRef]

Smith, K. T.

K. T. Smith, “Inversion of the X-Ray Transform,” SIAM-AMS Proc. 14, 41 (1984).

C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The Divergent Beam X-Ray Transform,” Rocky Mount. J. Math. 253 (1980).

B. E. Petersen, K. T. Smith, D. C. Solmon, “Sums of Plane Waves and the Range of the Radon Transform,” Math. Ann. 163 (1979).

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and Mathematical Aspects of Reconstructing Objects from Radiographs,” BAMS1227 (1977)

N. Aronszajn, K. T. Smith, “Theory of Bessel Potentials, Part I,” Ann. Inst. Fourier Grenoble XI, 385 (1961).
[CrossRef]

K. T. Smith, “Reconstruction Formulas in Computed Tomography. Computed Tomography,” in Proceedings Symposium on Applied Mathematics No. 27, L. A. Shepp, Ed. (American Mathematical Society, Providence, R.I., 1983).
[CrossRef]

J. V. Leahy, K. T. Smith, D. C. Solmon, “Uniqueness, Nonuniqueness, and Inversion in the X-ray and Radon Problems,” at International Symposium on Ill-Posed Problems, U. Delaware, Newark, 1979 (to appear).

Solmon, D. C.

D. V. Finch, D. C. Solmon, “Sums of Homogeneous Functions and the Range of the Divergent Beam X-Ray Transform,” Numer. Functional Anal. Optim. 5, 363 (1983).
[CrossRef]

C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The Divergent Beam X-Ray Transform,” Rocky Mount. J. Math. 253 (1980).

B. E. Petersen, K. T. Smith, D. C. Solmon, “Sums of Plane Waves and the Range of the Radon Transform,” Math. Ann. 163 (1979).

C. Hamaker, D. C. Solmon, “The Angles Between the Null Spaces of X-Rays,” J. Math. Anal. Appl. XX, 1 (1978).
[CrossRef]

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and Mathematical Aspects of Reconstructing Objects from Radiographs,” BAMS1227 (1977)

J. V. Leahy, K. T. Smith, D. C. Solmon, “Uniqueness, Nonuniqueness, and Inversion in the X-ray and Radon Problems,” at International Symposium on Ill-Posed Problems, U. Delaware, Newark, 1979 (to appear).

Wagner, S. L.

C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The Divergent Beam X-Ray Transform,” Rocky Mount. J. Math. 253 (1980).

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and Mathematical Aspects of Reconstructing Objects from Radiographs,” BAMS1227 (1977)

Zygmund, A.

A. P. Calderón, A. Zygmund, “On the Existence of Certain Singular Integrals,” Acta Math. 88, 85 (1952).
[CrossRef]

Ac. Sci. Ind. (1)

L. Schwartz, “Théorie des Distributions, I, II,” Ac. Sci. Ind. 1091, 1122 (1950–51).

Acta Math. (1)

A. P. Calderón, A. Zygmund, “On the Existence of Certain Singular Integrals,” Acta Math. 88, 85 (1952).
[CrossRef]

Acta Szeged. (1)

M. Riesz, “Intégrales de Riemann-Liouville et Potentiels,” Acta Szeged. XX, 1 (1938).

Ann. Inst. Fourier Grenoble (1)

N. Aronszajn, K. T. Smith, “Theory of Bessel Potentials, Part I,” Ann. Inst. Fourier Grenoble XI, 385 (1961).
[CrossRef]

BAMS (1)

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and Mathematical Aspects of Reconstructing Objects from Radiographs,” BAMS1227 (1977)

Ber. Verh. Sach. Akad. Wiss. Leipzig. Math.-Nat. kl. (1)

J. Radon, “Über die Bestimmung von Functionen durch ihre Integralwerte langs gewissen Mannigfaltigkeiten,” Ber. Verh. Sach. Akad. Wiss. Leipzig. Math.-Nat. kl. XX, 262, (1977).

Commun. Pure. Appl. Math. (1)

M. E. Davison, F. A. Grunbaum, “Tomographic Reconstruction with Arbitrary Directions,” Commun. Pure. Appl. Math. XX, 77 (1981).
[CrossRef]

Duke Math. J. (1)

B. F. Logan, L. A. Shepp, “Optimal Reconstruction of a Function from its Projections,” Duke Math. J. XX, 645 (1975).
[CrossRef]

J. Math. Anal. Appl. (1)

C. Hamaker, D. C. Solmon, “The Angles Between the Null Spaces of X-Rays,” J. Math. Anal. Appl. XX, 1 (1978).
[CrossRef]

Math. Ann. (1)

B. E. Petersen, K. T. Smith, D. C. Solmon, “Sums of Plane Waves and the Range of the Radon Transform,” Math. Ann. 163 (1979).

Math. Proc. Cambridge Philos. Soc. (1)

K. J. Falconer, “Consistency Conditions for a Finite Set of Projections of a Function,” Math. Proc. Cambridge Philos. Soc. XX, 61 (1979).
[CrossRef]

Numer. Functional Anal. Optim. (1)

D. V. Finch, D. C. Solmon, “Sums of Homogeneous Functions and the Range of the Divergent Beam X-Ray Transform,” Numer. Functional Anal. Optim. 5, 363 (1983).
[CrossRef]

Proc. IEEE (1)

H. J. Scudder, “Introduction to Computer Aided Tomography,” Proc. IEEE, 66, 628 (1978).
[CrossRef]

Proc. Nat. Acad. Sci. U.S.A. (1)

N. Ramachandran, A. V. Lakshminarayanan, “Three-Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms,” Proc. Nat. Acad. Sci. U.S.A. 2236 (1971).

Rocky Mount. J. Math. (1)

C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The Divergent Beam X-Ray Transform,” Rocky Mount. J. Math. 253 (1980).

SIAM-AMS Proc. (1)

K. T. Smith, “Inversion of the X-Ray Transform,” SIAM-AMS Proc. 14, 41 (1984).

Other (6)

J. V. Leahy, K. T. Smith, D. C. Solmon, “Uniqueness, Nonuniqueness, and Inversion in the X-ray and Radon Problems,” at International Symposium on Ill-Posed Problems, U. Delaware, Newark, 1979 (to appear).

J. Boman, “On the Closure of Sums of Plane Waves and the Range of the X-Ray Transform, I, II,”. Mathematics Department, U. Stockholm (1981), (1982). To appear in Ann. Inst. Fourier. Grenoble.

A. V. Lakshminarayanan, “Reconstruction from Divergent X-Ray Data,” SUNY Tech. Report 92, Computer Sciences Department, Buffalo, N.Y. (1975).

R. B. Marr, “An Overview of Image Reconstruction,” at International Symposium of Ill-Posed Problems, U. Delaware, Newark, 1979. (To appear.)

K. T. Smith, “Reconstruction Formulas in Computed Tomography. Computed Tomography,” in Proceedings Symposium on Applied Mathematics No. 27, L. A. Shepp, Ed. (American Mathematical Society, Providence, R.I., 1983).
[CrossRef]

O. Frostman, “Potentiels d’équilibre et capacité des ensembles,” Thesis, Lund (1935).

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Figures (3)

Fig. 1
Fig. 1

Reconstructions with Λ. Top left: mathematical phantom. Top right: reconstruction of the attenuation coefficient f. Bottom: central portion of Λf, reconstructed with the x-ray beam coned to that portion alone. With the interior background at 100 (to exhibit percentages), the attenuation coefficients are: central dark ellipses, 99; dark ellipse at 11 o’clock, 99.5; light circle at 10 o’clock, 101; three ellipses at 2 o’clock, 101, 102, 100.5; two tiny cirlces above and below the center, 100.25 and 99.75; skull, 200. The reconstructions use 128 x-ray directions and 256 detectors and are displayed on a 128 × 128 matrix. With 128 as the diameter of the circle reconstructed, the tiny circles have a radius of 0.96 (i.e., ρ = 0.96), and R 147 ~ 1.5ρ. All features in the phantom are elliptical, even though some do not look elliptical in the original—an effect of the discrete representations on the 128 × 128 matrix.

Fig. 2
Fig. 2

Effects of the point spread radius. The object is an exploded sodium—sulfur battery. After the explosion the core was an uneven mixture of sodium and less dense materials. The rings (inside to out) were steel, ceramic, an uneven mixture of sulfur and sodium polysulfides, and steel. The explosion was caused by the V-shaped crack at 2 o’clock in the ceramic ring. The four reconstructions put the kernel minimum at the first four detectors, giving point spread radii of 0.42, 0.85, 1.27, and 1.70 (1.5 would be preferred according to the criteria above). They use 128 x-ray directions and 452 detectors of 0.127-mm diameter.

Fig. 3
Fig. 3

Reconstructions with Λ. Top: scanner reconstruction of a phantom made of various low contrast materials with a monkey spine near the center. Bottom: reconstruction of the central third of Λf with the beam coned to the central third. Resolution is limited by the small number (42) of detectors within the coned beam. The data and scanner reconstruction were supplied by B. Rutt, Dept. Radiology, UC—San Francisco. The scanner is a low dose machine providing about 0.01 R/scan, and consequently noisy data. It is surprising that Λf is not exceedingly noisy. (No noise elimination was used.)

Equations (80)

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D a f ( θ ) = 0 f ( a + t θ ) d t .
x , y = j = 1 n x j y j and | x | = x , x .
P θ f ( x ) = f ( x + t θ ) d t , x θ .
R θ f ( t ) = x , θ = t f ( x ) d x .
L x f ( y ) = f [ x + t ( y / | y | ) ] d t , L ˜ x f ( y ) = | y , x | | y | n L x f ( y ) .
L x f ( θ ) = D x f ( θ ) + D x f ( θ ) = P θ f ( E θ x ) ,
f ˆ ( ξ ) = ( 2 π ) n / 2 exp ( i x , ξ ) f ( x ) d x .
R ˆ α ( ξ ) = ( 2 π ) n / 2 | ξ | α , 0 < α < n .
R α ( x ) = C ( n , α ) | x | α n , with C ( n , α ) = Γ [ n α ] / 2 2 α π n / 2 Γ ( α / 2 ) .
( Λ α f ) ˆ ( ξ ) = | ξ | α f ˆ ( ξ ) ,
Λ α ( R α * f ) = f .
Λ f = j = 1 n ( R 1 / x j ) * f / x j ,
[ 1 / C ( n , 1 ) ] R 1 * g ( x ) = S n 1 D x g ( θ ) = ( 1 2 ) S n 1 P θ g ( E θ x ) d θ ,
S n 1 D x g ( θ ) d θ = S n 1 0 g ( x + t θ ) d t d θ = g ( x + y ) | y | 1 n d y ,
g ( x ) = C ( n , 1 ) Λ S n 1 D x g ( θ ) d θ = ( 1 2 ) C ( n , 1 ) Λ S n 1 L x g ( θ ) d θ ,
g ( x ) = ( 1 / 2 ) C ( n , 1 ) Λ S n 1 P θ g ( E θ x ) d θ = ( 1 / 2 ) C ( n , 1 ) S n 1 Λ P θ g ( E θ x ) d θ .
θ = ( a x ) / | a x | , d θ = ( 1 / R ) | x a , a | | x a | n d a .
L a g ( t y ) = L a g ( y ) , L x g ( a x ) = L a g ( x a )
g ( x ) = [ C ( n , 1 ) / 2 R ] Λ A L ˜ a g ( x a ) d a .
g ( x ) = ( 1 / 2 ) ( 2 π ) 1 n Λ n 1 S n 1 R θ g ( x , θ ) d θ .
P θ ( e * f ) = P θ e * P θ f ,
e * f ( x ) = S n 1 k * P θ f ( E θ x ) d θ , with k = ( 1 / 2 ) C ( n , 1 ) Λ P θ e .
e * f ( x ) = ( 1 / 2 R ) A S n 1 L ˜ a f ( θ ) k ( E θ x E θ a ) d θ d a
θ P θ f ( y ) K ( y ) d y = ( 1 / 2 R ) A L ˜ a f ( θ ) K ( E θ a ) d a
R 1 / x j = ( 1 n ) C ( n , 1 ) x j / | x | n + 1
υ p R 1 / x j , ϕ = lim 0 | x | > R 1 / x j ϕ ¯ d x , ϕ C 0 ,
( υ p R 1 / x j ) * g ( x ) = lim r 0 < | x y | < r R 1 / x j ( x y ) g ( y ) d y .
D j R j = υ p R 1 / x j , ( D j R 1 ) ˆ = ( 2 π ) n / 2 i ξ j | ξ | 1 .
| R 1 * ϕ ( x ) | C ( 1 + | x | ) 1 n [ ϕ L 1 + ( 1 + | x | ) n ϕ L ] .
R 1 * ϕ ( x ) C M | z | 1 | z | 1 n d z + C ϕ L 1 .
| x | 1 n 1 / 2 | z | 2 | θ z | 1 n ϕ ( | x | z ) | x | n d z .
C M | x | 1 n | w | 3 | w | 1 n d w .
| R 1 * g , ϕ | = | R 1 * g ( x ) ϕ ¯ ( x ) d x | C ( 1 + | x | ) 1 n g L 1 [ ϕ L 1 + ( 1 + | x | ) n ϕ L
D x r = { g L 2 : ( 1 + | x | ) 1 n g L 1 } ,
g D x r = g L 2 + ( 1 + | x | ) 1 n g L 1 .
e * g D x r C ( 1 + | x | ) n 1 e L 1 g D x r .
| e | * | g | ( 1 + | x | 2 ) ( 1 n ) / 2 d x ( 1 + | x y | 2 ) ( 1 n ) / 2 | g ( x y ) | ( 1 + | y | ) n 1 | e ( y ) | d y d x = e 1 * g 1 L 1 e 1 L 1 g 1 L 1 with e 1 = ( 1 + | x | ) n 1 e and g 1 = ( 1 + | x | 2 ) ( 1 n ) / 2 | g | .
| u , e r | u L 2 e r L 2 = r n / 2 u L 2 e L 2 .
u , e r = ( 2 π ) n / 2 exp ( i x , ξ ) u ˆ ( ξ ) e ¯ r ( x ) d ξ d x = e ˆ ( r ξ ) u ˆ ( ξ ) d ξ .
| R 1 * g , e r | C r n | x | 2 r 0 | R 1 * g ( x ) | d x + C r n g D x r 2 r 0 | z | r | x | 1 n d x .
| R 1 * g , e r | C r 1 n ( 1 + r 0 ) n g D x r , for r 1 .
| R 1 * g 1 , e r | C r 1 n g D x r , for r 1 .
| R 1 * g 2 , e r | C R 1 * | g 2 | ( 0 ) = C | y | 1 | y | 1 n | g ( y ) | d y C g D x r .
( R 1 * g ˆ ) , ϕ = | ξ | 1 g ˆ ϕ ¯ d ξ .
i ξ j ( R 1 * g ) ˆ = [ D j ( R 1 * g ) ] ˆ = [ ( D j R 1 ) * g ] ˆ = i ( ξ j / | ξ | ) g ˆ .
ϕ ( ξ ) = j = 1 n ξ j ϕ j ( ξ ) with ϕ j ( ξ ) = 1 ϕ / ξ j ( t ξ ) d t .
( R 1 * g ) ˆ , ϕ = j = 1 n ( R 1 * g ˆ ) , ξ j ϕ j = j = 1 n ξ j ( R 1 * g ) ˆ , ϕ j = j = 1 n ξ j | ξ | 1 g ˆ ϕ ¯ j d ξ = | ξ | 1 g ˆ ϕ ¯ j d ξ .
S n 1 P θ g , P θ ϕ d θ = R 1 * g , ϕ ,
| S n 1 P θ g , P θ ϕ d θ | C g D x r [ ϕ L 1 + ( 1 + | x | ) n ϕ L ] .
S n 1 θ | ξ | h ( ξ ) d ξ d θ = | S n 2 | R n h ( ξ ) d ξ .
S n 1 S n 1 θ h ˜ ( θ ) d ϕ d θ = | S n 2 | S n 1 h ˜ ( θ ) d θ ,
( P θ g ˆ ) , ϕ = ( 2 π ) 1 / 2 θ g ˆ ( ξ ) ϕ ¯ ( ξ ) d ξ .
S n 1 θ | ξ | | g ˆ ( ξ ) | 2 d ξ d θ = C g L 2 C g D x r ,
S n 1 θ | P θ g ( x ) | ( 1 + | x | ) n d x d θ C g D x r ,
θ | P θ g ( x ) | ( 1 + | x | ) n d x <
S n 1 θ | P θ g ( x ) P θ g n ( x ) | ( 1 + | x | ) n d x d θ 0 .
θ | P θ g ( x ) P θ g n ( x ) | ( 1 + | x | ) n d x 0
( 2 π ) 1 / 2 θ g ˆ n ( ξ ) ϕ ¯ ( ξ ) d ξ = ( P θ g n ) ˆ , ϕ ( P θ g ) ˆ , ϕ .
S n 1 | ξ | 1 / 2 ( g ˆ g ˆ n ) L 2 ( θ ) d θ 0 .
| ξ | 1 / 2 ( g ˆ g ˆ n ) L 2 ( θ ) 0
ξ θ , | ξ | 1 | g ˆ ( ξ ) g ˆ n ( ξ ) | | ϕ ( ξ ) | d ξ ϕ L 2 | ξ | 1 / 2 ( g ˆ g ˆ n ) L 2 ( θ ) 0 .
ξ θ , | ξ | 1 | g ˆ ( ξ ) g ˆ n ( ξ ) | | ϕ ( ξ ) | d ξ ( | ξ | 1 | ξ | d ξ ) 1 / 2 | ξ | 1 / 2 ( g ˆ g ˆ n ) L 2 ( θ ) 0 .
S n 1 ξ θ , | ξ | 1 | g ˆ ( ξ ) | d ξ d θ = c | ξ | 1 | ξ | 1 | g ˆ ( ξ ) | d ξ .
S n 1 < P θ | g | , P θ e r > d θ = R 1 * | g | , e r 0 as r ,
H s = { g L 2 : | ξ | s g ˆ L 2 } , g H s 2 = g L 2 2 + | ξ | s g ˆ L 2 2 .
H s = { g L 2 : D j g L 2 for | j | s } .
(a) P θ e H loc 1 ( θ ) .
( D j P θ e ) ˆ = ( 2 π ) 1 / 2 i ξ j e ˆ on θ .
( Λ P θ e ) ˆ = ( 2 π ) 1 / 2 | ξ | e ˆ on θ .
e * f ( x ) = S n 1 k * P θ f ( E θ x ) d θ , with k = ( 1 / 2 ) C ( n , 1 ) Λ P θ e .
S n 1 θ | ξ | 2 | e ˆ ( ξ ) | 2 d ξ d θ = | S n 2 | | ξ | | e ˆ ( ξ ) | 2 d ξ <
Λ P θ e , P θ ϕ = ( 2 π ) 1 / 2 θ | ξ | e ˆ ( ξ ) ϕ ˆ ( ξ ) d ξ .
S n 1 Λ P θ e , P θ ϕ d θ = ( 2 π ) 1 / 2 | S n 2 | e , ϕ .
e * f ( x ) = ( 1 / 2 R ) A S n 1 L ˜ a f ( θ ) k ( E θ x E θ a ) d θ d a .
Λ ( e * f ) ( x ) = ( Λ e ) * f ( x ) = S n 1 k * P θ f ( E θ x ) d θ = ( 1 / 2 R ) A S n 1 L ˜ a f ( θ ) k ( E θ x E θ a ) d θ d a ,
k = ( 1 / 2 ) C ( n , 1 ) Δ P θ e .
e r ( x ) = r n e 1 ( x / r ) , k r ( x ) = r n k 1 ( x / r ) ,
R | x | R e ( x ) d x
e 1 ( x ) = { ( μ / 2 π ) ( 1 | x | 2 ) ( u 2 ) / 2 , | x | 1 , 0 , | x | > 1 ,
e 1 ( x ) = 4 0 k ( y ) ( | x | 2 y 2 ) 1 / 2 d y ,

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