In this paper，we study the convergence of the generalized Lagrange interpolatingpolynomials at roots of unity in the complex plane，and obtain the order of approximationof the polynomials to f（z）∈A（｜z｜≤1） in the space L_p（｜z｜＝1）， 0＜p＜＋∞．

We give explicit systems of generators of the algebras of invariant polynomials in arbitrary many vector variables for the classical reflection groups (including the dihedral groups).

BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials

For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation.

We also prove a binomial formula for 6-parametric Koornwinder polynomials.

We also use known results about canonical bases forUq2 to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmannians), due to Lascoux-Schützenberger and Zelevinsky.

This paper attempts to answer the following two questions: (1) Is it possible to derive the law of distribution of hydrological frequency theoretically(2) What type of distribution curve should be adopted as the model of hydrological frequency curve and how to determine their parameters? The results obtained may be summarized as follows: 1. Hydrological phenomena are time series with concealed periodic fluctuations. The results from statistical analysis based upon the current assumption that hydrological phenomena...

This paper attempts to answer the following two questions: (1) Is it possible to derive the law of distribution of hydrological frequency theoretically(2) What type of distribution curve should be adopted as the model of hydrological frequency curve and how to determine their parameters? The results obtained may be summarized as follows: 1. Hydrological phenomena are time series with concealed periodic fluctuations. The results from statistical analysis based upon the current assumption that hydrological phenomena are independent stochastic variables should be accepted with due considerations. 2. In view of the regional nature of hydrological phenomena, the current parctice of analyzing samples taking from a single station only is, in effect, to narrow the sampling field arbitrarily from a larger area to a point, thus reducing the accuracy of the statistical results. Hence, the synthetic utilization of the data of all stations within the hydrologically homo- geneous region is an important measure to increase the accuracy of statistical analysis. 3. The belief that the flood frequency obeys the binomial theorem or Poisson's theorem is but to mix up the priori with the empirical probability problem. The binomial theorem, being a powerful weapon to deal with the problems of priori probability, has not been adquately and properly utilized in the hydrologieal frequency analysis. 4. Analyses have been made of the nature of distribution of shydrologieal series on the basis of Kaptyen's derivation of the skew distribution, which indicate: (1) That the theoretical interpretation of the log-probability law of the hydrologic phenomena by V. T. Chow is not sound; (2) that hydrologic phenomena being results of very complicated meteorological and hydrological processes, it is impossible to derive theoretically the law of distribution for the hydrological series. 5. The view that the flood frequency obeys the Gumbel's distribution is theoretically not sound and also not verified by actual data. 6. According to the nature of the mathematical treatments applied, the method of description of the empirical probability can be classified into three systems: (1) The methods of the generalization of the characteristic factors of the distributions, such as Pearson's curves, Goodrich's curves, etc.; (2) The methods of the modification of a fundamental distribution by series and polynomials, such as Gram-Charlier curves. curves, etc.; (3) The methods of transformed functions, such as the log-probability law, curves, etc. It should be remarked that not only Pearson's and Goodrich's curves are frequency curves of empirical nature, but even the theoretical laws, such as the normal law and the log-probability law, will be aceepted as curves of empirical nature, when used as models for empirical probability problem. 7. Hydrological frequency analysis should not be mystified and made absolute. Instead of free selections, the models of hydrological frequency curve should be uniquely selected and specified. Statistical parameters should be determined not solely by the short period data of single station, but also by the synthetic utilization of the data of possible more stations. 8. It is recommended that one of the two types of distribution, i.e. the log-normal frequency curve with both sides limited and the Pearson's type Ⅲ curve, may be selected as unified models. The author suggests that the K-value corresponding to recurrence intervals of say 10~4, 10~5, or 10~6 years may be selected as the upper and lower limits for the log-normal curve. For Pearson's type III curves, C_s should be treated not as independent but as dependent variables of C_v. 9. The proper way to select and determine the model frequency curve is to see whether it fits well with the actual data of grouped stations (stations to be grouped by regions for rainfall data and by C_v for runoff data) and the reasonableness of the extrapolating part. 10. Suggestions on the method of determination of x and C_v: For point rainfall, iso-x map may be utilized, and the mean C_v for each hydrologicregion may be adopted in order to minimize the errors from single stations and to avoid the discrepancies in results obtained from the same region. With regard to flood frequency analysis, flood mark reconnaissance must be utilized to determine the magnitude and the recurrence interval of the unusual flood. The x and C_v values of the floods and runoffs of hydrologically similiar river basins may be compared. Besides, the reasonableness of the results of frequency calculations as well as of the statistical parameters adopted therein may be checked by comparing runoffs and point-rainfall values of the same frequency.

In this paper the degree of approximation by n-dimensional Bernstein polynomials is discussed. By a k-dimensional Bemstein polynomial of a cominuous function f(x1,x2…xk) on the k-dimensional cube 0≤x1≤1,i=1,2,…k, it is defined by the following expression:whereThe following inequalities are established.Here the expressions ωf(δ1,δ2,…δk),ω2(δ) and ω4(δ) are moduli of continuity in different metrecs defined as followFurthermore, let us define the Bernstein polynomials on the k-dimensional simplex 0≤x1+x3+…xk≤1,xi≥0,i=1,2,…k,...

In this paper the degree of approximation by n-dimensional Bernstein polynomials is discussed. By a k-dimensional Bemstein polynomial of a cominuous function f(x1,x2…xk) on the k-dimensional cube 0≤x1≤1,i=1,2,…k, it is defined by the following expression:whereThe following inequalities are established.Here the expressions ωf(δ1,δ2,…δk),ω2(δ) and ω4(δ) are moduli of continuity in different metrecs defined as followFurthermore, let us define the Bernstein polynomials on the k-dimensional simplex 0≤x1+x3+…xk≤1,xi≥0,i=1,2,…k, by the following expression:Herethen the following inequality holds,

The assignment of aberration tolerances for optical systems suffering from wave-front aberrations greater than λ/4 is worth further studying. In this paper it is shown that "low" contrast resolving power of optical systems can be taken as an appropriate criterion for assessing image quality, where the optical transfer function can be evaluated approximately in terms of the sums of Legendre polynomials. Accordingly, the best program of aberration correction and tolerences in the case where certain aberration...

The assignment of aberration tolerances for optical systems suffering from wave-front aberrations greater than λ/4 is worth further studying. In this paper it is shown that "low" contrast resolving power of optical systems can be taken as an appropriate criterion for assessing image quality, where the optical transfer function can be evaluated approximately in terms of the sums of Legendre polynomials. Accordingly, the best program of aberration correction and tolerences in the case where certain aberration constants are inevitably present and others are adjustable in optical design can be obtained. The results so obtained as applied to aberrations of small magnitude comparable to λ/4 conform well to the Strehl Deffinitionshelligkeit method.