# Bushev D. M.

### Conditions of Convergence Almost Everywhere for the Convolution of a Function with Delta-Shaped Kernel to this Function

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1461-1476

We establish sufficient conditions for the convergence of the convolution of a function with delta-shaped kernel to this function. These conditions are used for the construction of the subspaces of solutions of differential equations and systems of these equations isometric to the spaces of real functions.

### Approximation of classes of periodic multivariable functions by linear positive operators

Bushev D. M., Kharkevych Yu. I.

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 12–19

In an *N*-dimensional space, we consider the approximation of classes of translation-invariant periodic functions by a linear operator whose kernel is the product of two kernels one of which is positive. We establish that the least upper bound of this approximation does not exceed the sum of properly chosen least upper bounds in *m*-and ((*N* ? *m*))-dimensional spaces. We also consider the cases where the inequality obtained turns into the equality.

### Approximation of classes of periodic functions with small smoothness

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 183-196

We prove that the approximations of classes of periodic functions with small smoothness in the metrics of the spaces *C* and *L* by different linear summation methods for Fourier series are asymptotically equal to the least upper bounds of the best approximations of these classes by trigonometric polynomials of degree not higher than (*n* - 1). We establish that the Fejér method is asymptotically the best among all positive linear approximation methods for these classes.

### Isometry of functional spaces with different number of variables

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1027–1045

We construct spaces of real functions of *n + k* variables that are isometric to spaces of real functions given on an *n*-dimensional Euclidean space. We present certain properties and examples of delta-like kernels used for the construction of isometric spaces of functions with different number of variables. We prove certain assertions that enable one to construct delta-like kernels with many variables by using delta-like kernels with smaller number of variables.

### Approximation of classes of convolutions by linear methods of summation of Fourier series

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 739–753

We consider a family of special linear methods of summation of Fourier series and establish exact equalities for the approximation of classes of convolutions with even and odd kernels by polynomials generated by these methods.

### Approximation of weakly differentiable periodic functions

Ukr. Mat. Zh. - 1990. - 42, № 3. - pp. 406-412

### Asymptotically best approximation of glasses of differentiable functions by linear positive operators

Ukr. Mat. Zh. - 1985. - 37, № 2. - pp. 154 – 162