On Approximating Two Distributions from a Single Complex-Valued Function

ABSTRACT

We consider the problem of approximating two, possibly unrelated probability distributions from a single complex-valued function and its Fourier transform. We show that this problem always has a solution within a specified degree of accuracy, provided the distributions satisfy the necessary regularity conditions. We describe the algorithm and construction of and provide examples of approximating several pairs of distributions using the algorithm.

We consider the problem of approximating two, possibly unrelated probability distributions from a single complex-valued function and its Fourier transform. We show that this problem always has a solution within a specified degree of accuracy, provided the distributions satisfy the necessary regularity conditions. We describe the algorithm and construction of and provide examples of approximating several pairs of distributions using the algorithm.

KEYWORDS

Probability Distribution, Sinc Approximation, Cardinal Function, Fourier Transform, Wavelet

Probability Distribution, Sinc Approximation, Cardinal Function, Fourier Transform, Wavelet

Cite this paper

nullW. Flanders and G. Japaridze, "On Approximating Two Distributions from a Single Complex-Valued Function,"*Applied Mathematics*, Vol. 1 No. 6, 2010, pp. 439-445. doi: 10.4236/am.2010.16058.

nullW. Flanders and G. Japaridze, "On Approximating Two Distributions from a Single Complex-Valued Function,"

References

[1] J. von Neumann, “Mathematical Foundations of Quantum Mechanics,” Princeton University Press, Princeton, 1955.

[2] W. Rudin, “Functional Analysis,” McGraw-Hill, New York, 1991.

[3] F. Stenger, “Summary of Sinc Numerical Methods,” Journal of Computational and Applied Mathematics, Vol. 121, No. 1-2, September 2000, pp. 379-420.

[4] P. L. Butzer, J. R. Higgins and R. L. Stens, “Classical and Approximate Sampling Theorems; Studies in the and the Uniform Norm,” Journal of Approximation Theory, Vol. 137, No. 2, December 2005, pp. 250-263.

[5] J. M. Whittaker, “Interpolation Function Theory,” Cambridge Tracts in Mathematics and Mathematical Physics, No. 33, Cambridge University Press, Cambridge, 1935.

[6] C. E. Shannon, “Communication in the Presence of Noise,” Proceedings of Institute of Radio Engineers, Vol. 37, No. 1, January 1949, pp. 10-21.

[7] I. Doubechis, “Ten Lectures on Wavelets,” CBMS-NSF Regional Conference Series for Applied Mathematics, 1992.

[1] J. von Neumann, “Mathematical Foundations of Quantum Mechanics,” Princeton University Press, Princeton, 1955.

[2] W. Rudin, “Functional Analysis,” McGraw-Hill, New York, 1991.

[3] F. Stenger, “Summary of Sinc Numerical Methods,” Journal of Computational and Applied Mathematics, Vol. 121, No. 1-2, September 2000, pp. 379-420.

[4] P. L. Butzer, J. R. Higgins and R. L. Stens, “Classical and Approximate Sampling Theorems; Studies in the and the Uniform Norm,” Journal of Approximation Theory, Vol. 137, No. 2, December 2005, pp. 250-263.

[5] J. M. Whittaker, “Interpolation Function Theory,” Cambridge Tracts in Mathematics and Mathematical Physics, No. 33, Cambridge University Press, Cambridge, 1935.

[6] C. E. Shannon, “Communication in the Presence of Noise,” Proceedings of Institute of Radio Engineers, Vol. 37, No. 1, January 1949, pp. 10-21.

[7] I. Doubechis, “Ten Lectures on Wavelets,” CBMS-NSF Regional Conference Series for Applied Mathematics, 1992.