Abstract

The concept of fractal (self-similar) self-transform functions is examined. A general method to prove existence of these functions is introduced, and necessary conditions for this existence are derived. The results are general and apply to all transforms with product-type kernels.

© 2006 Optical Society of America

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References

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  1. E. Titchmarsh, Introduction to the Theory of Fourier Integrals (Clarendon, 1937).
  2. M. Caola, 'Self-Fourier functions,' J. Phys. A 24, L1143-L1144 (1991).
    [CrossRef]
  3. T. Horikis and M. McCallum, 'Self-Fourier functions and self-Fourier operators,' J. Opt. Soc. Am. A 23, 829-834 (2006).
    [CrossRef]
  4. A. Lakhtakia, 'Fractal self-Fourier functions,' Optik (Stuttgart) 94, 51-52 (1993).
  5. A. Lakhtakia and H. Caulfield, 'On some mathematical and optical integral transforms of self-similar (fractal) functions,' Optik (Stuttgart) 91, 131-133 (1992).

2006

1993

A. Lakhtakia, 'Fractal self-Fourier functions,' Optik (Stuttgart) 94, 51-52 (1993).

1992

A. Lakhtakia and H. Caulfield, 'On some mathematical and optical integral transforms of self-similar (fractal) functions,' Optik (Stuttgart) 91, 131-133 (1992).

1991

M. Caola, 'Self-Fourier functions,' J. Phys. A 24, L1143-L1144 (1991).
[CrossRef]

Caola, M.

M. Caola, 'Self-Fourier functions,' J. Phys. A 24, L1143-L1144 (1991).
[CrossRef]

Caulfield, H.

A. Lakhtakia and H. Caulfield, 'On some mathematical and optical integral transforms of self-similar (fractal) functions,' Optik (Stuttgart) 91, 131-133 (1992).

Horikis, T.

Lakhtakia, A.

A. Lakhtakia, 'Fractal self-Fourier functions,' Optik (Stuttgart) 94, 51-52 (1993).

A. Lakhtakia and H. Caulfield, 'On some mathematical and optical integral transforms of self-similar (fractal) functions,' Optik (Stuttgart) 91, 131-133 (1992).

McCallum, M.

Titchmarsh, E.

E. Titchmarsh, Introduction to the Theory of Fourier Integrals (Clarendon, 1937).

J. Opt. Soc. Am. A

J. Phys. A

M. Caola, 'Self-Fourier functions,' J. Phys. A 24, L1143-L1144 (1991).
[CrossRef]

Optik (Stuttgart)

A. Lakhtakia, 'Fractal self-Fourier functions,' Optik (Stuttgart) 94, 51-52 (1993).

A. Lakhtakia and H. Caulfield, 'On some mathematical and optical integral transforms of self-similar (fractal) functions,' Optik (Stuttgart) 91, 131-133 (1992).

Other

E. Titchmarsh, Introduction to the Theory of Fourier Integrals (Clarendon, 1937).

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Equations (17)

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F { f ( x ) } = f ̂ ( ω ) f ( x ) exp ( i ω x ) d x ,
F 1 { f ̂ ( ω ) } = f ( x ) 1 2 π f ̂ ( ω ) exp ( i ω x ) d ω
f ( a x ) = a d f ( x ) .
f ̂ ( x ) = λ f ( x ) ,
f ̂ ( x ) = + ω n f ( ω ) K ( ω x ) d ω ,
f ( a x ) ̂ = + ω n f ( a ω ) K ( ω x ) d ω
= 1 a n + 1 + u n f ( u ) K ( u a x ) d u
= 1 a n + 1 f ̂ ( x a ) = λ a n + 1 f ( x a ) ;
f ( a x ) ̂ = λ a n + d + 1 f ( x ) ,
f ( a x ) ̂ = a d f ̂ ( x ) = a d λ f ( x ) .
1 a n + d + 1 = a d 2 d + n + 1 = 0 d = n + 1 2 .
f ( a x ) = a n + 1 2 f ( x ) .
f ( a x ) ̂ = 1 a f ̂ ( x a )
f ( a x ) ̂ = 1 a f ̂ ( x a ) = λ a f ( x a ) .
f ( a x ) ̂ = λ a d + 1 f ( x ) .
f ( a x ) ̂ = λ f ( a x ) = λ a d f ( x ) .
1 a d + 1 = a d d = 1 2 ,

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