Could You please clarify Your question? Sequence spaces deal with maps from the (positive) integers into the real or complex numbers. One can treat them as Lebesgue spaces by equipping them with the counting measure.

But there is no differentiability, so it would be complicated to define a Sobolev space for this setup. So did You mean function spaces? And what is Your idea of a usual basis?

One nice thing about wavelet bases and frames is that they have structure related to translation and scaling. And some wavelet bases consist of functions with compact support, giving locality or sparseness in function and operator representation. That they often come with a dual or biorthogonal basis for linear functionals is also sometimes useful. Of course, orthogonality of the basis as in the Daubechies wavelets is even better.

Are there papers where wavelet bases in H1 are constructed that are orthogonal in one of the H1 scalar products?

Regards, Lutz

There is a paper attached here with which gives an approximate wavelet basis for the given function space(lorentz space). My question is that ,isnt there any other basis for that space like standard basis /shauder basis etc.if so what is the need of finding such wavelet basis.

with regards,kiran kumar patra

No, there is no standard basis for function spaces. I'm not sure what other general function representations there are besides the waveler-related and Fourier transforms. Calderon-Zygmund-theory is basically the continuous wavelet transform, to continue the name dropping: Fefferman, Stein, Hardy-spaces.

Note that the finite element spaces (on regular grids) with refinements as in the multigrid methods may also be interpreted as variants of the discrete wavelet transform. In most cases they will lead to framelets, minimal generating systems that are not bases.

And the wavelets, if they form a basis, form it in the Schauder sense, that is, as topological basis, not as linear algebra Hamel basis.

thank you very much Mr Lutz for your excelent way of clearing my doubts.

What about Hamel bases?

Hamel bases, well, they exist. And are not really that useful. Except to demonstrate that continuity is, even in very weak forms, a useful condition.

How can we define the coefficients of the wavelet expansion of the given periodic function for different function spaces?