Approximation Of Periodic Functions Research Articles

Best simultaneous approximation of periodic functions and their derivatives by trigonometric polynomials

Best simultaneous approximation of periodic functions and their derivatives by trigonometric polynomials

Exact estimates of approximation of periodic functions by linear polynomial methods of convolution type

Exact estimates of approximation of periodic functions by linear polynomial methods of convolution type

Simultaneous approximation of periodic functions and their derivatives by fourier sums

Simultaneous approximation of periodic functions and their derivatives by fourier sums

Approximation of periodic functions by splines of minimal defect

Approximation of periodic functions by splines of minimal defect

More about the approximation of periodic functions of the classes $$SH\begin{array}{*{20}c} r \\ * \\ p \\ \end{array}$$ by fourier sums

More about the approximation of periodic functions of the classes $$SH\begin{array}{*{20}c} r \\ * \\ p \\ \end{array}$$ by fourier sums

Approximation of periodic functions of two variables by Vall�e-Poussin sums

Approximation of periodic functions of two variables by Vall�e-Poussin sums

Best approximations of differentiable functions in the metric of the space L2

The relation between the best approximations of periodic functions by trigonometric polynomials and the moduli of continuity of their arbitrary derivatives in the metric of the space L2 is studied.

Approximation of periodic functions of the class $$SH\mathop *\limits_p^r $$ by Fourier sums

Approximation of periodic functions of the class $$SH\mathop *\limits_p^r $$ by Fourier sums

Direct and converse theorems of the theory of approximation in the metric of Lp for 0 < p < 1

A direct theorem of the Jackson type and several converse theorems are established for the approximation of periodic functions of period 2Π by trigonometrical polynomials in the metric of Lp, 0 < p < 1.

On extremal subspaces and approximation of periodic functions by splines of minimal defect

Н. П. Корнейчук СССР, ДНЕПРОПЕТРОВСК 320 625 ПР. ГАГАРИНА 72 ДНЕПРОПЕТРОВСКИЙ ГО СУДАРСТВЕННЫЙ УНИВЕ РСИТЕТ

Approximation of periodic functions by trigonometric polynomials in the mean

For arbitrary summation methods we obtain inequalities between upper bounds of deviations in the L metric and corresponding upper bounds in the C metric with respect to a certain class of functions. These inequalities constitute a generalization of known relationships due to S. M. Nikol'skii. We consider the cases wherein these inequalities become exact or asymptotic equalities.

An extension of a theorem of P. P. Korovkin to singular integrals with not necessarily positive kernels

Concerning the approximation of periodic functions by means of singular convolution integrals having positive trigonometric kernels, an equivalence theorem of P. P. Korovkin on convergence factors, i.e., on the Fourier coefficients of the kernel, is of great importance. A simple inversion formula between convergence factors and trigonometric moments of a general kernel immediately delivers an extension of this theorem to a class of singular integrals with not necessarily positive kernels as well as a simplified proof of the original result. The new theorem is applied to three representative examples of oscillating kernels.

The approximation of periodic functions by linear summation methods for Fourier series

The approximation of periodic functions by linear summation methods for Fourier series

Best approximation of periodic functions by trigonometric polynomials in L2

Estimates are gotten for the best approximations in L2 (0, 2π) of a periodic function by trigonometric polynomials in terms of its m-th continuity modulus or in terms of the continuity modulus of its r-th derivative. The inequality En−1(f)L2<(C 2m m ωm(2π/n;f)L2(1≠ const) is proved, where the constant (C 2m m )−1/2 is unimprovable for the whole space L2 (0, 2π). Two titles are cited in the bibliography.

Approximation of periodic functions in Orlicz spaces

Approximation of periodic functions in Orlicz spaces

Application of periodic functions approximation to antenna pattern synthesis and circuit theory

Recently, mathematicians gave results on the approximation of periodic functions f(x) by trigonometric sums P_{n}(x) . These results can be useful for antenna radiation and circuit theory problems. Rather than the least mean-square criterion which leads to Gibbs' phenomenon, it has been adopted that the maximum in the period of the error, | f - P_{n} | , is to be minimized. By linear transformation of the Fourier sum, a P_{n} sum can be obtained to give an error of the order 1/n^{p} . The Fourier sum would give \Log n/n^{p} . Limitations on the maximum of P_{n} derivatives are introduced allowing one to obtain the order of maximum error. Antenna power diagram synthesis is then looked at with these results. The power radiation v^{2} of an array of n isotropic independent sources equally spaced can always be written under the form of a P_{n} sum. Thus it is possible to give general limitations for the derivatives of v^{2} in the broadside case and the endfire case. These limitations depend upon the over-all antenna dimension vs wavelength a/\lambda and the maximum error. A practical problem of shaped beam antenna is examined. It is shown that, by using the mathematical theory, improvements can be made on the diagram from what is usually obtained. For circuit theory, physically evident limitations in time T and spectrum F allow one to write the most general function under the form of a P_{n} sum, and thus to apply the mathematical results to that field. Formal analogy allows comparison of antenna pattern and circuit theories.