Volume 42, Issue 3

In this paper, we extend the continuous wavelet transform to Schwartz distributions in D ′ ⁢ ( R n ) \mathcal{D}^{\prime}(\mathbb{R}^{n}) , n ≥ 1 n\geq 1 , and derive the corresponding wavelet inversion formula (valid modulo a constant distribution) interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is an element ψ ⁢ ( x ) \psi(x) of D ⁢ ( R n ) \mathcal{D}(\mathbb{R}^{n}) , n ≥ 1 n\geq 1 , which, when integrated along each of the real axes X 1 , X 2 , X 3 , … , X n X_{1},X_{2},X_{3},\ldots,X_{n} vanishes, but none of its moments ∫ R n ψ ⁢ ( x ) ⁢ x m ⁢ d x \int_{\mathbb{R}^{n}}\psi(x)x^{m}\,dx is zero; here x m = x 1 m 1 ⁢ x 2 m 2 ⁢ … ⁢ x n m n x^{m}=x_{1}^{{m_{1}}}\,x_{2}^{{m_{2}}}\ldots x_{n}^{{m_{n}}} , d ⁢ x = d ⁢ x 1 ⁢ d ⁢ x 2 ⁢ … ⁢ d ⁢ x n dx=dx_{1}\,dx_{2}\ldots dx_{n} and m = ( m 1 , m 2 , … , m n ) m=(m_{1},m_{2},\ldots,m_{n}) and each of m 1 , m 2 , … , m n m_{1},m_{2},\ldots,m_{n} is at least 1. The set of such kernel will be denoted by D m ⁢ ( R n ) \mathcal{D}_{m}(\mathbb{R}^{n}) . But the uniqueness theorem for our wavelet inversion formula is valid for the space D F ′ ⁢ ( R n ) \mathcal{D}_{F}^{\prime}(\mathbb{R}^{n}) obtained by filtering (deleting) (i) all non-zero constant distributions from the space D ′ ⁢ ( R n ) \mathcal{D}^{\prime}(\mathbb{R}^{n}) , (ii) all non-zero constants that appear with a distribution as a union as for example for x 1 2 + x 2 2 + ⋯ ⁢ x n 2 1 + x 1 2 + x 2 2 + ⋯ ⁢ x n 2 = 1 - 1 1 + x 1 2 + x 2 2 + ⋯ ⁢ x n 2 \frac{x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}}{1+x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}}=1-\frac{1}{1+x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}} , 1 is deleted and - 1 1 + x 1 2 + x 2 2 + ⋯ ⁢ x n 2 \frac{-1}{1+x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}} is retained.

This work discovers the Laplace transform using a generalized pathway fractional integral formula involving an extended Mittag-Leffler function in the kernel for various parameters. Our findings are fairly broad in scope. Some well-known and novel results can also be obtained here.

Discrete analogues of the index transforms, involving Bessel and the modified Bessel functions, are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.

In the present paper, some error bounds for weighted Ostrowski–Grüss type inequalities are found by using the weighted Čebyšev functional and Diaz–Metcalf inequality. These bounds give some better and new estimates. Applications are also obtained for numerical integration. Furthermore, this paper recaptures some previous established results as its special cases.

Various applications of degenerate polynomials in different areas call for the thoughtful study and research, and many extensions and variants can be found in the literature. In this paper, we introduce partially degenerate Bell–Bernoulli polynomials of the first kind and investigate their properties and identities. Furthermore, we introduce a generalized form of partially degenerate Bell–Bernoulli polynomials of the first kind and derive some interesting properties and identities. The results obtained are of general character and can be reduced to yield formulae and identities for relatively simple polynomials and numbers.

In this paper, we show that if the Nemytskii operator maps the ( p , α ) {(p,\alpha)} -bounded variation space into itself and satisfies some Lipschitz condition, then there are two functions g and h belonging to the ( p , α ) {(p,\alpha)} -bounded variation space such that f ⁢ ( t , y ) = g ⁢ ( t ) ⁢ y + h ⁢ ( t )   for all  ⁢ t ∈ [ a , b ] , y ∈ ℝ . f(t,y)=g(t)y+h(t)\quad\text{for all }t\in[a,b],\,y\in\mathbb{R}.

In this paper, the concept of left ϕ-approximate Connes amenability for a dual Banach algebra 𝒜 {\mathcal{A}} is studied, where ϕ is a weak * {{}^{*}} -continuous multiplicative linear functional on 𝒜 {\mathcal{A}} . Some characterizations for module extension dual Banach algebras and matrix algebras are given. Moreover, the notion of approximate left ϕ-Connes biprojectivity for dual Banach algebras is introduced and some concrete examples regarding this new notion are indicated.