**Improved Fourier coefficients for maps using phases from partial structures with errors**

Unrefined or partly refined models of macromolecules square measure usually incomplete and usually have giant coordinate errors. it’s shown that section likelihood equations applicable for an ideal partial structure cause inaccurate estimates of section possibilities in such cases. Therefore, it’s necessary to use equations that are derived granting errors within the partial structure. a technique is given to estimate the parameter [sigma]A in these section likelihood expressions from the ascertained and calculated structure issue amplitudes. From the variation of [sigma]A with resolution, one will estimate the mean coordinate error for the model. lepton density maps calculated exploitation partial structure phases square measure biased towards the partial structure. once there square measure coordinate errors, a brand new expression for the non-centric Fourier coefficients [(2m|FN| – D|FcP|) exp(i[alpha]cP)] is needed to suppress this model bias. Judged by correlation coefficients comparison lepton density maps with the right and also the partial structure maps, the Fourier coefficients derived here square measure superior to others presently in use. **[1]**

**Estimating frequency by interpolation using Fourier coefficients**

The periodogram of a time series that contains a sinusoidal element affords a crude estimate of its frequency parameter, the maximizer over the Fourier frequencies being inside O(T/sup -1/) of the frequency because the sample length T will increase. In the paper, a method for obtaining an estimator that has root imply square errors of order T/sup -three/2/ is offered, which includes simplest the Fourier components of the time series at three frequencies, The asymptotic variance of the estimator varies among, more or less, the asymptotic variance of the maximizer of the periodogram over all frequencies (the Cramer-Rao decrease sure) and 3 times this variance. The advantage of the brand new estimator is its computational simplicity. **[2]**

**Iterative frequency estimation by interpolation on Fourier coefficients**

The estimation of the frequency of a complex exponential is a trouble that is relevant to a big number of fields. In this paper, we advise and analyze new frequency estimators that interpolate on the Fourier coefficients of the received signal samples. The estimators are shown to acquire identical asymptotic performances. They are asymptotically impartial and usually allotted with a variance this is only 1.0147 times the asymptotic Crame/spl acute/r-Rao certain (ACRB) uniformly over the frequency estimation range.** [3]**

**The influence of statistical properties of Fourier coefficients on random Gaussian surfaces**

Many examples of natural systems can be defined by random Gaussian surfaces. Much may be found out by means of reading the Fourier enlargement of the surfaces, from which it’s far viable to decide the corresponding Hurst exponent and consequently establish the presence of scale invariance. We display that this symmetry isn’t tormented by the distribution of the modulus of the Fourier coefficients. Furthermore, we inspect the function of the Fourier levels of random surfaces. In specific, we display how the floor is tormented by a non-uniform distribution of phases. **[4]**

**Cusp Forms Whose Fourier Coefficients Involve Dirichlet Series**

We present a few cusp bureaucracy on the total modular institution ⌈1, the use of the residences of eigenfunctions, nonanalytic Poincare collection and Hecke operators Tn. Further, the Fourier coefficients of cusp forms Tnf on ⌈1 are given in terms of Dirichlet collection related to the Fourier coefficients of cusp form f of weight k.** [5]**

**Reference**

**[1]** Read, R.J., 1986. Improved Fourier coefficients for maps using phases from partial structures with errors. Acta Crystallographica Section A: Foundations of Crystallography, 42(3), (Web Link)

**[2]** Quinn, B.G., 1994. Estimating frequency by interpolation using Fourier coefficients. IEEE transactions on Signal Processing, 42(5), (Web Link)

**[3]** Aboutanios, E. and Mulgrew, B., 2005. Iterative frequency estimation by interpolation on Fourier coefficients. IEEE Transactions on Signal Processing, 53(4), (Web Link)

**[4]** The influence of statistical properties of Fourier coefficients on random Gaussian surfaces

C. P. de Castro, M. Luković, R. F. S. Andrade & H. J. Herrmann

Scientific Reports volume 7, Article number: 1961 (2017) (Web Link)

**[5]** Kırmacı, U. S. (2017) “Cusp Forms Whose Fourier Coefficients Involve Dirichlet Series”, Asian Research Journal of Mathematics, 2(2), (Web Link)